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Practice › Mathematics › IMO › IMO 2023 ›

IMO 2023 Problem 6

IMO 2023 Problem 6 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 1m23s Problem Let $ABC$ be an equilateral triangle. Let $A_1,B_1,C_1$ be interior points of $ABC$ such that $BA_1=A_1C$, $CB_1=B_1A$, $AC_1=C_1B$, and $$ \angle BA_1C+\angle CB_1A+\angle AC_1B=480^\circ $$Let $BC_1$ and $CB_1$ meet at $A_2,$ let $CA_1$ and $AC_1$ meet at $B_2,$ and let $AB_1$ and $BA_1$ meet at $C_2.$ Prove that if triangle $A_1B_1C_1$ is scalene,...

Practice › Mathematics › IMO › IMO 2023 ›

IMO 2023 Problem 5

IMO 2023 Problem 5 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 43s Problem Let $n$ be a positive integer. A Japanese triangle consists of $1 + 2 + \dots + n$ circles arranged in an equilateral triangular shape such that for each $i = 1$, $2$, $\dots$, $n$, the $i^{th}$ row contains exactly $i$ circles, exactly one of which is coloured red. A ninja path in...

Practice › Mathematics › IMO › IMO 2023 ›

IMO 2023 Problem 4

IMO 2023 Problem 4 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 1m50s Problem Let $x_1, x_2, \cdots , x_{2023}$ be pairwise different positive real numbers such that $$ a_n = \sqrt{(x_1+x_2+ \text{···} +x_n)(\frac1{x_1} + \frac1{x_2} + \text{···} +\frac1{x_n})} $$ is an integer for every $n = 1,2,\cdots,2023$. Prove that $a_{2023} \ge 3034$. Exploration Write $$S_n=x_1+\cdots+x_n,\qquad T_n=\frac1{x_1}+\cdots+\frac1{x_n}.$$ Then $$a_n=\sqrt{S_nT_n}.$$ The first step is to understand how $a_n$...

Practice › Mathematics › IMO › IMO 2023 ›

IMO 2023 Problem 3

IMO 2023 Problem 3 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 1m49s Problem For each integer $k \geqslant 2$, determine all infinite sequences of positive integers $a_1, a_2, \ldots$ for which there exists a polynomial $P$ of the form $P(x)=x^k+c_{k-1} x^{k-1}+\cdots+c_1 x+c_0$, where $c_0, c_1, \ldots, c_{k-1}$ are non-negative integers, such that $$ P\left(a_n\right)=a_{n+1} a_{n+2} \cdots a_{n+k} $$ for every integer $n \geqslant 1$. Exploration The...

Practice › Mathematics › IMO › IMO 2023 ›

IMO 2023 Problem 2

IMO 2023 Problem 2 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 1m11s Problem Let $ABC$ be an acute-angled triangle with $AB < AC$. Let $\Omega$ be the circumcircle of $ABC$. Let $S$ be the midpoint of the arc $CB$ of $\Omega$ containing $A$. The perpendicular from $A$ to $BC$ meets $BS$ at $D$ and meets $\Omega$ again at $E \neq A$. The line through $D$ parallel...

Practice › Mathematics › IMO › IMO 2023 ›

IMO 2023 Problem 1

IMO 2023 Problem 1 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 1m14s Problem Determine all composite integers $n>1$ that satisfy the following property: if $d_1,d_2,\dots,d_k$ are all the positive divisors of $n$ with $1=d_1<d_2<\dots<d_k=n$, then $d_i$ divides $d_{i+1}+d_{i+2}$ for every $1\le i \le k-2$. Exploration The condition relates consecutive divisors in increasing order. A first check on prime powers is natural. If $n=p^r$, then the divisors...

Practice › Mathematics › IMO › IMO 2022 ›

IMO 2022 Problem 6

IMO 2022 Problem 6 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 1m27s Problem Let $n$ be a positive integer. A Nordic square is an $n \times n$ board containing all the integers from $1$ to $n^2$ so that each cell contains exactly one number. Two different cells are considered adjacent if they share an edge. Every cell that is adjacent only to cells containing larger numbers...

Practice › Mathematics › IMO › IMO 2022 ›

IMO 2022 Problem 5

IMO 2022 Problem 5 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 1m28s Problem Find all triples $(a,b,p)$ of positive integers with $p$ prime and $$ a^p = b! + p $$ Exploration The equation $a^p = b! + p$ forces a prime power to sit extremely close to a factorial. Small values of $b$ suggest very few possibilities: factorials $2!,3!,4!,5!$ can be shifted by a prime...

Practice › Mathematics › IMO › IMO 2022 ›

IMO 2022 Problem 4

IMO 2022 Problem 4 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 50s Problem Let $ABCDE$ be a convex pentagon such that $BC = DE$. Assume that there is a point $T$ inside $ABCDE$ with $TB = TD$, $TC = TE$ and $\angle ABT = \angle TEA$. Let line $AB$ intersect lines $CD$ and $CT$ at points $P$ and $Q$, respectively. Assume that the points $P, B,...

Practice › Mathematics › IMO › IMO 2022 ›

IMO 2022 Problem 3

IMO 2022 Problem 3 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 42s Problem Let $k$ be a positive integer and let $S$ be a finite set of odd prime numbers. Prove that there is at most one way (up to rotation and reflection) to place the elements of $S$ around a circle such that the product of any two neighbours is of the form $x^2 +...

Practice › Mathematics › IMO › IMO 2022 ›

IMO 2022 Problem 2

IMO 2022 Problem 2 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 43s Problem Let $\mathbb{R}^+$ denote the set of positive real numbers. Find all functions $f : \mathbb{R}^+ \to \mathbb{R}^+$ such that for each $x \in \mathbb{R}^+$, there is exactly one $y \in \mathbb{R}^+$ satisfying $$ xf (y) + yf (x) \le 2 $$. Working

Practice › Mathematics › IMO › IMO 2022 ›

IMO 2022 Problem 1

IMO 2022 Problem 1 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 1m06s Problem The Bank of Oslo issues two types of coin: aluminium (denoted A) and bronze (denoted B). Marianne has $n$ aluminium coins and $n$ bronze coins, arranged in a row in some arbitrary initial order. A chain is any subsequence of consecutive coins of the same type. Given a fixed positive integer $k\le 2n$,...

Practice › Mathematics › IMO › IMO 2021 ›

IMO 2021 Problem 6

IMO 2021 Problem 6 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 1m01s Problem Let $m \ge 2$ be an integer, $A$ be a finite set of (not necessarily positive) integers, and $B_1, B_2, B_3 , \ldots, B_m$ be subsets of $A$. Assume that for each $k = 1, 2,...,m$ the sum of the elements of $B_k$ is $m^k$. Prove that $A$ contains at least $m/2$ elements....

Practice › Mathematics › IMO › IMO 2021 ›

IMO 2021 Problem 5

IMO 2021 Problem 5 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 1m01s Problem Two squirrels, Bushy and Jumpy, have collected 2021 walnuts for the winter. Jumpy numbers the walnuts from 1 through 2021, and digs 2021 little holes in a circular pattern in the ground around their favourite tree. The next morning Jumpy notices that Bushy had placed one walnut into each hole, but had paid...

Practice › Mathematics › IMO › IMO 2021 ›

IMO 2021 Problem 4

IMO 2021 Problem 4 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 1m20s Problem Let $\Gamma$ be a circle with centre $I$, and $ABCD$ a convex quadrilateral such that each of the segments $AB, BC, CD$ and $DA$ is tangent to $\Gamma$. Let $\Omega$ be the circumcircle of the triangle $AIC$. The extension of $BA$ beyond $A$ meets $\Omega$ at $X$, and the extension of $BC$ beyond...

Practice › Mathematics › IMO › IMO 2021 ›

IMO 2021 Problem 3

IMO 2021 Problem 3 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 49s Problem Let $D$ be an interior point of the acute triangle $ABC$ with $AB > AC$ so that $\angle DAB= \angle CAD$. The point $E$ on the segment $AC$ satisfies $\angle ADE= \angle BCD$, the point $F$ on the segment $AB$ satisfies $\angle FDA= \angle DBC$, and the point $X$ on the line $AC$...

Practice › Mathematics › IMO › IMO 2021 ›

IMO 2021 Problem 2

IMO 2021 Problem 2 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 58s Problem Show that the inequality $$ \sum_{i=1}^n \sum_{j=1}^n \sqrt{|x_i-x_j|} \le \sum_{i=1}^n \sum_{j=1}^n \sqrt{|x_i+x_j|} $$ holds for all real numbers $x_1,x_2,\dots,x_n$. Exploration For small values of $n$, the inequality reduces to comparing pairwise contributions. When $n=1$, both sides are $0$. When $n=2$, the inequality becomes $$\sqrt{|x_1-x_1|}+\sqrt{|x_1-x_2|}+\sqrt{|x_2-x_1|}+\sqrt{|x_2-x_2|} \le \sqrt{|x_1+x_1|}+\sqrt{|x_1+x_2|}+\sqrt{|x_2+x_1|}+\sqrt{|x_2+x_2|},$$ which simplifies to $$2\sqrt{|x_1-x_2|} \le 2\sqrt{|x_1+x_2|},$$ so...

Practice › Mathematics › IMO › IMO 2021 ›

IMO 2021 Problem 1

IMO 2021 Problem 1 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 1m06s Problem Let $n \geq 100$ be an integer. Ivan writes the numbers $n, n+1, \ldots, 2 n$ each on different cards. He then shuffles these $n+1$ cards, and divides them into two piles. Prove that at least one of the piles contains two cards such that the sum of their numbers is a perfect...

Practice › Mathematics › IMO › IMO 2020 ›

IMO 2020 Problem 6

IMO 2020 Problem 6 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 1m17s Problem Prove that there exists a positive constant $c$ such that the following statement is true: Consider an integer $n > 1$, and a set $S$ of n points in the plane such that the distance between any two different points in $S$ is at least $1$. It follows that there is a line...

Practice › Mathematics › IMO › IMO 2020 ›

IMO 2020 Problem 5

IMO 2020 Problem 5 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 1m09s Problem A deck of $n > 1$ cards is given. A positive integer is written on each card. The deck has the property that the arithmetic mean of the numbers on each pair of cards is also the geometric mean of the numbers on some collection of one or more cards. For which $n$...

Practice › Mathematics › IMO › IMO 2020 ›

IMO 2020 Problem 4

IMO 2020 Problem 4 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 1m01s Problem There is an integer $n > 1$. There are $n^2$ stations on a slope of a mountain, all at different altitudes. Each of two cable car companies, $A$ and $B$, operates $k$ cable cars; each cable car provides a transfer from one of the stations to a higher one (with no intermediate stops)....

Practice › Mathematics › IMO › IMO 2020 ›

IMO 2020 Problem 3

IMO 2020 Problem 3 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 43s Problem There are $4n$ pebbles of weights $1, 2, 3, . . . , 4n$. Each pebble is colored in one of $n$ colors and there are four pebbles of each color. Show that we can arrange the pebbles into two piles so that the following two conditions are both satisfied: The total weights...

Practice › Mathematics › IMO › IMO 2020 ›

IMO 2020 Problem 2

IMO 2020 Problem 2 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 1m15s Problem The real numbers $a$, $b$, $c$, $d$ are such that $a \geq b \geq c \geq d > 0$ and $a + b + c + d = 1$. Prove that$$ (a + 2b + 3c + 4d) a^a b^b c^c d^d < 1. $$ Exploration The expression splits into two components: a...

Practice › Mathematics › IMO › IMO 2020 ›

IMO 2020 Problem 1

IMO 2020 Problem 1 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 1m09s Problem Consider the convex quadrilateral $ABCD$. The point $P$ is in the interior of $ABCD$. The following ratio equalities hold: $$ \angle PAD : \angle PBA : \angle DPA = 1 : 2 : 3 = \angle CBP : \angle BAP : \angle BPC. $$ Prove that the following three lines meet in a...

Practice › Mathematics › IMO › IMO 2019 ›

IMO 2019 Problem 6

IMO 2019 Problem 6 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 1m13s Problem Let $I$ be the incenter of acute triangle $ABC$ with $AB \neq AC$. The incircle $\omega$ of $ABC$ is tangent to sides $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. The line through $D$ perpendicular to $EF$ meets $\omega$ again at $R$. Line $AR$ meets $\omega$ again at $P$. The circumcircles...

Practice › Mathematics › IMO › IMO 2019 ›

IMO 2019 Problem 5

IMO 2019 Problem 5 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 1m00s Problem The Bank of Bath issues coins with an $H$ on one side and a $T$ on the other. Harry has $n$ of these coins arranged in a line from left to right. He repeatedly performs the following operation: If there are exactly $k > 0$ coins showing $H$, then he turns over the...

Practice › Mathematics › IMO › IMO 2019 ›

IMO 2019 Problem 4

IMO 2019 Problem 4 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 58s Problem Find all pairs $(k,n)$ of positive integers such that $$ k!=(2^n-1)(2^n-2)(2^n-4)\dots(2^n-2^{n-1}). $$ Exploration The expression on the right-hand side is $$(2^n-1)(2^n-2)(2^n-4)\cdots(2^n-2^{n-1}),$$ which is a product of $n$ consecutive terms of the form $2^n-2^i$ for $i=0,1,\dots,n-1$. Each factor is divisible by $2^i$ except the first one, and the total $2$-adic valuation is strongly constrained...

Practice › Mathematics › IMO › IMO 2019 ›

IMO 2019 Problem 3

IMO 2019 Problem 3 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 59s Problem A social network has $2019$ users, some pairs of whom are friends. Whenever user $A$ is friends with user $B$, user $B$ is also friends with user $A$. Events of the following kind may happen repeatedly, one at a time: Three users $A$, $B$, and $C$ such that $A$ is friends with both...

Practice › Mathematics › IMO › IMO 2019 ›

IMO 2019 Problem 2

IMO 2019 Problem 2 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 1m02s Problem In triangle $ABC$, point $A_1$ lies on side $BC$ and point $B_1$ lies on side $AC$. Let $P$ and $Q$ be points on segments $AA_1$ and $BB_1$, respectively, such that $PQ$ is parallel to $AB$. Let $P_1$ be a point on line $PB_1$, such that $B_1$ lies strictly between $P$ and $P_1$, and...

Practice › Mathematics › IMO › IMO 2019 ›

IMO 2019 Problem 1

IMO 2019 Problem 1 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 1m01s Problem Let $\mathbb{Z}$ be the set of integers. Determine all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that, for all integers $a$ and $b$, $$ f(2a) + 2f(b) = f(f(a + b)). $$ Exploration The equation couples values of $f$ at linear arguments with values of $f$ applied twice: $$f(2a) + 2f(b) = f(f(a+b)).$$...

Practice › Mathematics › IMO › IMO 2018 ›

IMO 2018 Problem 6

IMO 2018 Problem 6 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 59s Problem A convex quadrilateral $ABCD$ satisfies $AB\cdot CD=BC \cdot DA.$ Point $X$ lies inside $ABCD$ so that $\angle XAB = \angle XCD$ and $\angle XBC = \angle XDA.$ Prove that $\angle BXA + \angle DXC = 180^{\circ}$. 2018 IMO ( Problems ) • Resources Preceded by 2017 IMO Problems 1 • 2 • 3...

Practice › Mathematics › IMO › IMO 2018 ›

IMO 2018 Problem 5

IMO 2018 Problem 5 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 1m00s Problem Let $a_1, a_2, \dots$ be an infinite sequence of positive integers. Suppose that there is an integer$N > 1$ such that, for each $n \geq N$, the number $\frac{a_1}{a_2}+\frac{a_2}{a_3}+\dots +\frac{a_{n-1}}{a_n}+\frac{a_n}{a_1}$ is an integer. Prove that there is a positive integer $M$ such that $a_m = a_{m+1}$ for all $m \geq M.$ Exploration Let...

Practice › Mathematics › IMO › IMO 2018 ›

IMO 2018 Problem 4

IMO 2018 Problem 4 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 9m14s Problem A site is any point $(x, y)$ in the plane such that $x$ and $y$ are both positive integers less than or equal to 20. Initially, each of the 400 sites is unoccupied. Amy and Ben take turns placing stones with Amy going first. On her turn, Amy places a new red stone...

Practice › Mathematics › IMO › IMO 2017 ›

IMO 2017 Problem 6

IMO 2017 Problem 6 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 16m42s Problem An ordered pair $(x, y)$ of integers is a primitive point if the greatest common divisor of $x$ and $y$ is $1$. Given a finite set $S$ of primitive points, prove that there exist a positive integer $n$ and integers $a_0, a_1, \ldots , a_n$ such that, for each $(x, y)$ in $S$,...

Practice › Mathematics › IMO › IMO 2017 ›

IMO 2017 Problem 5

IMO 2017 Problem 5 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 15m44s Problem An integer $N \ge 2$ is given. A collection of $N(N + 1)$ soccer players, no two of whom are of the same height, stand in a row. Sir Alex wants to remove $N(N - 1)$ players from this row leaving a new row of $2N$ players in which the following $N$ conditions...

Practice › Mathematics › IMO › IMO 2017 ›

IMO 2017 Problem 4

IMO 2017 Problem 4 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 12m23s Problem Let $R$ and $S$ be different points on a circle $\Omega$ such that $RS$ is not a diameter. Let $\ell$ be the tangent line to $\Omega$ at $R$. Point $T$ is such that $S$ is the midpoint of the line segment $RT$. Point $J$ is chosen on the shorter arc $RS$ of $\Omega$...

Practice › Mathematics › IMO › IMO 2017 ›

IMO 2017 Problem 3

IMO 2017 Problem 3 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 8m57s Problem A hunter and an invisible rabbit play a game in the Euclidean plane. The rabbit's starting point, $A_0$, and the hunter's starting point, $B_0$, are the same. After $n-1$ rounds of the game, the rabbit is at point $A_{n-1}$ and the hunter is at point $B_{n-1}$. In the nth round of the game,...

Practice › Mathematics › IMO › IMO 2017 ›

IMO 2017 Problem 2

IMO 2017 Problem 2 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 3m33s Problem Let $\mathbb{R}$ be the set of real numbers , determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for any real numbers $x$ and $y$ $$ {f(f(x)f(y)) + f(x+y)} =f(xy) $$ Exploration The equation couples three expressions: $f(xy)$, $f(x+y)$, and $f(f(x)f(y))$. The presence of $f$ applied to a product of values suggests that the image of...

Practice › Mathematics › IMO › IMO 2017 ›

IMO 2017 Problem 1

IMO 2017 Problem 1 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 8m02s Problem For each integer $a_0 > 1$, define the sequence $a_0, a_1, a_2, \ldots$ for $n \geq 0$ as $$ a_{n+1} = \begin{cases} \sqrt{a_n} & \text{if } \sqrt{a_n} \text{ is an integer,} \ a_n + 3 & \text{otherwise.} \end{cases} $$Determine all values of $a_0$ such that there exists a number $A$ such that $a_n...

Practice › Mathematics › IMO › IMO 2016 ›

IMO 2016 Problem 6

IMO 2016 Problem 6 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 13m15s Problem There are $n\ge 2$ line segments in the plane such that every two segments cross and no three segments meet at a point. Geoff has to choose an endpoint of each segment and place a frog on it facing the other endpoint. Then he will clap his hands $n-1$ times. Every time he...

Practice › Mathematics › IMO › IMO 2016 ›

IMO 2016 Problem 5

IMO 2016 Problem 5 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 9m15s Problem The equation $(x-1)(x-2)\cdots(x-2016)=(x-1)(x-2)\cdots (x-2016)$ is written on the board, with $2016$ linear factors on each side. What is the least possible value of $k$ for which it is possible to erase exactly $k$ of these $4032$ linear factors so that at least one factor remains on each side and the resulting equation has...

Practice › Mathematics › IMO › IMO 2016 ›

IMO 2016 Problem 4

IMO 2016 Problem 4 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 10m36s Problem A set of positive integers is called fragrant if it contains at least two elements and each of its elements has a prime factor in common with at least one of the other elements. Let $P(n)=n^2+n+1$. What is the least possible positive integer value of $b$ such that there exists a non-negative integer...

Practice › Mathematics › IMO › IMO 2016 ›

IMO 2016 Problem 3

IMO 2016 Problem 3 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 5m12s Problem Let $P = A_1A_2 \cdots A_k$ be a convex polygon in the plane. The vertices $A_1,A_2,\dots, A_k$ have integer coordinates and lie on a circle. Let $S$ be the area of $P$. An odd positive integer $n$ is given such that the squares of the side lengths of $P$ are integers divisible by...

Practice › Mathematics › IMO › IMO 2016 ›

IMO 2016 Problem 2

IMO 2016 Problem 2 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 4m59s Problem Find all integers $n$ for which each cell of $n \times n$ table can be filled with one of the letters $I,M$ and $O$ in such a way that: - in each row and each column, one third of the entries are $I$, one third are $M$ and one third are $O$; and...

Practice › Mathematics › IMO › IMO 2016 ›

IMO 2016 Problem 1

IMO 2016 Problem 1 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 10m29s Problem Triangle $BCF$ has a right angle at $B$. Let $A$ be the point on line $CF$ such that $FA=FB$ and $F$ lies between $A$ and $C$. Point $D$ is chosen so that $DA=DC$ and $AC$ is the bisector of $\angle{DAB}$. Point $E$ is chosen so that $EA=ED$ and $AD$ is the bisector of...

Practice › Mathematics › IMO › IMO 2015 ›

IMO 2015 Problem 6

IMO 2015 Problem 6 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 13m40s Problem The sequence $a_1,a_2,\dots$ of integers satisfies the conditions: (i) $1\le a_j\le2015$ for all $j\ge1$, (ii) $k+a_k\neq \ell+a_\ell$ for all $1\le k<\ell$. Prove that there exist two positive integers $b$ and $N$ for which$$ \left\vert\sum_{j=m+1}^n(a_j-b)\right\vert\le1007^2 $$for all integers $m$ and $n$ such that $n>m\ge N$. Exploration Condition (ii) states that the integers $k+a_k$ are...

Practice › Mathematics › IMO › IMO 2015 ›

IMO 2015 Problem 5

IMO 2015 Problem 5 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 5m39s Problem Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfying the equation $f(x+f(x+y))+f(xy) = x+f(x+y)+yf(x)$ for all real numbers $x$ and $y$. Exploration Substituting $y=0$ produces a relation linking $f(x+f(x))$, $f(0)$, and $f(x)$. This suggests that the value $f(0)$ plays a structural role and may be eliminated early. Testing linear...

Practice › Mathematics › IMO › IMO 2015 ›

IMO 2015 Problem 4

IMO 2015 Problem 4 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 3m53s Problem Triangle $ABC$ has circumcircle $\Omega$ and circumcenter $O$. A circle $\Gamma$ with center $A$ intersects the segment $BC$ at points $D$ and $E$, such that $B$, $D$, $E$, and $C$ are all different and lie on line $BC$ in this order. Let $F$ and $G$ be the points of intersection of $\Gamma$ and...

Practice › Mathematics › IMO › IMO 2015 ›

IMO 2015 Problem 1

IMO 2015 Problem 1 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 3m58s Problem We say that a finite set $\mathcal{S}$ in the plane is balanced if, for any two different points $A$, $B$ in $\mathcal{S}$, there is a point $C$ in $\mathcal{S}$ such that $AC=BC$. We say that $\mathcal{S}$ is centre-free if for any three points $A$, $B$, $C$ in $\mathcal{S}$, there is no point $P$...

Practice › Mathematics › IMO › IMO 2014 ›

IMO 2014 Problem 6

IMO 2014 Problem 6 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 5m10s Problem A set of lines in the plane is in $\textit{general position}$ if no two are parallel and no three pass through the same point. A set of lines in general position cuts the plane into regions, some of which have finite are; we call these its $\textit{finite regions}$. Prove that for all sufficiently...

Practice › Mathematics › IMO › IMO 2014 ›

IMO 2014 Problem 5

IMO 2014 Problem 5 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 4m48s Problem For each positive integer $n$, the Bank of Cape Town issues coins of denomination $\tfrac{1}{n}$. Given a finite collection of such coins (of not necessarily different denominations) with total value at most $99+\tfrac{1}{2}$, prove that it is possible to split this collection into $100$ or fewer groups, such that each group has total...

Practice › Mathematics › IMO › IMO 2014 ›

IMO 2014 Problem 4

IMO 2014 Problem 4 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 4m59s Problem Points $P$ and $Q$ lie on side $BC$ of acute-angled $\triangle{ABC}$ so that $\angle{PAB}=\angle{BCA}$ and $\angle{CAQ}=\angle{ABC}$. Points $M$ and $N$ lie on lines $AP$ and $AQ$, respectively, such that $P$ is the midpoint of $AM$, and $Q$ is the midpoint of $AN$. Prove that lines $BM$ and $CN$ intersect on the circumcircle of...

Practice › Mathematics › IMO › IMO 2014 ›

IMO 2014 Problem 3

IMO 2014 Problem 3 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 8m55s Problem Convex quadrilateral $ABCD$ has $\angle{ABC}=\angle{CDA}=90^{\circ}$. Point $H$ is the foot of the perpendicular from $A$ to $BD$. Points $S$ and $T$ lie on sides $AB$ and $AD$, respectively, such that $H$ lies inside $\triangle{SCT}$ and $$ \angle{CHS}-\angle{CSB}=90^{\circ},\quad \angle{THC}-\angle{DTC} = 90^{\circ}. $$ Prove that line $BD$ is tangent to the circumcircle of $\triangle{TSH}.$ Exploration...

Practice › Mathematics › IMO › IMO 2014 ›

IMO 2014 Problem 2

IMO 2014 Problem 2 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 9m03s Problem Let $n\ge2$ be an integer. Consider an $n\times n$ chessboard consisting of $n^2$ unit squares. A configuration of $n$ rooks on this board is $peaceful$ if every row and every column contains exactly one rook. Find the greatest positive integer $k$ such that, for each peaceful configuration of $n$ rooks, there is a...

Practice › Mathematics › IMO › IMO 2014 ›

IMO 2014 Problem 1

IMO 2014 Problem 1 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 5m10s Problem Let $a_0<a_1<a_2<\cdots \quad$ be an infinite sequence of positive integers, Prove that there exists a unique integer $n\ge1$ such that $$ a_n<\frac{a_0+a_1+\cdots + a_n}{n}\le a_{n+1}. $$ Exploration The condition compares a partial average of an increasing integer sequence with two consecutive terms. Writing $$A_n=\frac{a_0+a_1+\cdots+a_n}{n},$$ the problem asserts the existence and uniqueness of an...

Practice › Mathematics › IMO › IMO 2013 ›

IMO 2013 Problem 6

IMO 2013 Problem 6 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 8m44s Problem Let $n \ge 3$ be an integer, and consider a circle with $n + 1$ equally spaced points marked on it. Consider all labellings of these points with the numbers $0, 1, ... , n$ such that each label is used exactly once; two such labellings are considered to be the same if...

Practice › Mathematics › IMO › IMO 2013 ›

IMO 2013 Problem 5

IMO 2013 Problem 5 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 5m21s Problem Let $\mathbb Q_{>0}$ be the set of all positive rational numbers. Let $f:\mathbb Q_{>0}\to\mathbb R$ be a function satisfying the following three conditions: (i) for all $x,y\in\mathbb Q_{>0}$, we have $f(x)f(y)\geq f(xy)$; (ii) for all $x,y\in\mathbb Q_{>0}$, we have $f(x+y)\geq f(x)+f(y)$; (iii) there exists a rational number $a>1$ such that $f(a)=a$. Prove that...

Practice › Mathematics › IMO › IMO 2013 ›

IMO 2013 Problem 4

IMO 2013 Problem 4 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 4m45s Problem Let $ABC$ be an acute triangle with orthocenter $H$, and let $W$ be a point on the side $BC$, lying strictly between $B$ and $C$. The points $M$ and $N$ are the feet of the altitudes from $B$ and $C$, respectively. Denote by $\omega_1$ is [sic] the circumcircle of $BWN$, and let $X$...

Practice › Mathematics › IMO › IMO 2013 ›

IMO 2013 Problem 3

IMO 2013 Problem 3 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 10m31s Problem Let the excircle of triangle $ABC$ opposite the vertex $A$ be tangent to the side $BC$ at the point $A_1$. Define the points $B_1$ on $CA$ and $C_1$ on $AB$ analogously, using the excircles opposite $B$ and $C$, respectively. Suppose that the circumcentre of triangle $A_1B_1C_1$ lies on the circumcircle of triangle $ABC$....

Practice › Mathematics › IMO › IMO 2013 ›

IMO 2013 Problem 2

IMO 2013 Problem 2 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 7m20s Problem A configuration of $4027$ points in the plane is called Colombian if it consists of $2013$ red points and $2014$ blue points, and no three of the points of the configuration are collinear. By drawing some lines, the plane is divided into several regions. An arrangement of lines is good for a Colombian...

Practice › Mathematics › IMO › IMO 2013 ›

IMO 2013 Problem 1

IMO 2013 Problem 1 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 11m44s Problem Prove that for any pair of positive integers $k$ and $n$, there exist $k$ positive integers $m_1,m_2,...,m_k$ (not necessarily different) such that $1+\frac{2^k-1}{n}=\left(1+\frac{1}{m_1}\right)\left(1+\frac{1}{m_2}\right)\cdots\left(1+\frac{1}{m_k}\right).$ Exploration The reviewer correctly identifies that any attempt based on naive telescoping of expressions like $a_{i+1}/a_i$ or products of $(a_i+1)/a_i$ fails because the denominators and numerators do not align in...

Practice › Mathematics › IMO › IMO 2012 ›

IMO 2012 Problem 6

IMO 2012 Problem 6 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 5m12s Problem Find all positive integers n for which there exist non-negative integers $a_1$, $a_2$, $\ldots$ , $a_n$ such that $\frac{1}{{{2}^{{{a} {1}}}}}+\frac{1}{{{2}^{{{a} {2}}}}}+\cdots +\frac{1}{{{2}^{{{a} {n}}}}}=\frac{1}{{{3}^{{{a} {1}}}}}+\frac{2}{{{3}^{{{a} {2}}}}}+\cdots +\frac{n}{{{3}^{{{a} {n}}}}}=1$ Author: Dušan Djukić, Serbia Exploration For small values of $n$, direct construction can be tested. When $n=1$, taking $a_1=0$ satisfies both equalities. When $n=2$, the condition...

Practice › Mathematics › IMO › IMO 2012 ›

IMO 2012 Problem 5

IMO 2012 Problem 5 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 3m28s Problem Let $ABC$ be a triangle with $\angle BCA=90{}^\circ$, and let $D$ be the foot of the altitude from $C$. Let $X$ be a point in the interior of the segment $CD$. Let K be the point on the segment $AX$ such that $BK = BC$. Similarly, let $L$ be the point on the...

Practice › Mathematics › IMO › IMO 2012 ›

IMO 2012 Problem 4

IMO 2012 Problem 4 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 5m22s Problem Find all functions $f:\mathbb{Z}\to \mathbb{Z}$ such that, for all integers $a$, $b$, $c$ that satisfy $a+b+c = 0$, the following equality holds: $$ f(a)^2 + f(b)^2 + f(c)^2 = 2f(a)f(b) + 2f(b)f(c) + 2f(c)f(a). $$ (Here $\mathbb{Z}$ denotes the set of integers.) Author: Liam Baker, South Africa Exploration The equation is symmetric and...

Practice › Mathematics › IMO › IMO 2012 ›

IMO 2012 Problem 3

IMO 2012 Problem 3 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 9m01s Problem The liar’s guessing game is a game played between two players $A$ and $B$. The rules of the game depend on two positive integers $k$ and $n$ which are known to both players. At the start of the game A chooses integers $x$ and $N$ with $1\le x\le N$. Player $A$ keeps $x$...

Practice › Mathematics › IMO › IMO 2012 ›

IMO 2012 Problem 2

IMO 2012 Problem 2 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 7m50s Problem Let ${{a} {2}}, {{a} {3}}, \cdots, {{a} {n}}$ be positive real numbers that satisfy ${{a} {2}}\cdot {{a} {3}}\cdots {{a} {n}}=1$ . Prove that $$ \left(a_2+1\right)^2\cdot \left(a_3+1\right)^3\cdots \left(a_n+1\right)^n> n^n $$ Author: Angelo di Pasquale, Australia Exploration Small cases are tested first. For $n=2$, the expression is $(a_2+1)^2$ with $a_2=1$, giving $4$, and the right-hand...

Practice › Mathematics › IMO › IMO 2012 ›

IMO 2012 Problem 1

IMO 2012 Problem 1 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 5m12s Problem Given $\triangle ABC$ the point $J$ is the centre of the excircle opposite the vertex $A$. This excircle is tangent to the side $BC$ at $M$, and to the lines $AB$ and $AC$ at $K$ and $L$, respectively. The lines $LM$ and $BJ$ meet at $F$, and the lines $KM$ and $CJ$ meet...

Practice › Mathematics › IMO › IMO 2011 ›

IMO 2011 Problem 6

IMO 2011 Problem 6 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 4m51s Problem Let $ABC$ be an acute triangle with circumcircle $\Gamma$. Let $l$ be a tangent line to $\Gamma$, and let $l_a$, $l_b$ and $l_c$ be the lines obtained by reflecting $l$ in the lines $BC$, $CA$ and $AB$, respectively. Show that the circumcircle of the triangle determined by the lines $l_a$, $l_b$ and $l_c$...

Practice › Mathematics › IMO › IMO 2011 ›

IMO 2011 Problem 5

IMO 2011 Problem 5 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 5m14s Problem Let $f$ be a function from the set of integers to the set of positive integers. Suppose that, for any two integers $m$ and $n$, the difference $f(m) - f(n)$ is divisible by $f(m - n)$. Prove that, for all integers $m$ and $n$ with $f(m) \le f(n)$, the number $f(n)$ is divisible...

Practice › Mathematics › IMO › IMO 2011 ›

IMO 2011 Problem 4

IMO 2011 Problem 4 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 5m05s Problem Let $n > 0$ be an integer. We are given a balance and $n$ weights of weight $2^0, 2^1,\ldots, 2^{n-1}$ . We are to place each of the $n$ weights on the balance, one after another, in such a way that the right pan is never heavier than the left pan. At each...

Practice › Mathematics › IMO › IMO 2011 ›

IMO 2011 Problem 3

IMO 2011 Problem 3 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 5m02s Problem Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a real-valued function defined on the set of real numbers that satisfies $f(x + y) \le yf(x) + f(f(x))$ for all real numbers $x$ and $y$. Prove that $f(x)=0$ for all $x \le 0$. Author: Igor Voronovich, Belarus Exploration The inequality couples three expressions: $f(x+y)$, $yf(x)$,...

Practice › Mathematics › IMO › IMO 2011 ›

IMO 2011 Problem 2

IMO 2011 Problem 2 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 7m48s Problem Let $S$ be a finite set of at least two points in the plane. Assume that no three points of $S$ are collinear. A windmill is a process that starts with a line $l$ going through a single point $P \in S$. The line rotates clockwise about the pivot $P$ until the first...

Practice › Mathematics › IMO › IMO 2011 ›

IMO 2011 Problem 1

IMO 2011 Problem 1 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 3m33s Problem Given any set $A = {a_1, a_2, a_3, a_4}$ of four distinct positive integers, let $s(A)=a_1+a_2+a_3+a_4$ be the sum of the positive integers in $A$. Let $n(A)$ denote the number of pairs $(i,j)$ with $1 \le i < j \le 4$ for which $a_i+a_j$ divides $s(A)$. Find all sets $A$ of four distinct...

Practice › Mathematics › IMO › IMO 2010 ›

IMO 2010 Problem 6

IMO 2010 Problem 6 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 4m59s Problem Let $a_1, a_2, a_3, \ldots$ be a sequence of positive real numbers, and $s$ be a positive integer, such that $$ a_n = \max { a_k + a_{n-k} \mid 1 \leq k \leq n-1 } \ \textrm{ for all } \ n > s. $$ Prove there exist positive integers $\ell \leq s$...

Practice › Mathematics › IMO › IMO 2010 ›

IMO 2010 Problem 5

IMO 2010 Problem 5 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 8m57s Problem Each of the six boxes $B_1$, $B_2$, $B_3$, $B_4$, $B_5$, $B_6$ initially contains one coin. The following operations are allowed Type 1) Choose a non-empty box $B_j$, $1\leq j \leq 5$, remove one coin from $B_j$ and add two coins to $B_{j+1}$; Type 2) Choose a non-empty box $B_k$, $1\leq k \leq 4$,...

Practice › Mathematics › IMO › IMO 2010 ›

IMO 2010 Problem 4

IMO 2010 Problem 4 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 4m58s Problem Let $P$ be a point interior to triangle $ABC$ (with $CA \neq CB$). The lines $AP$, $BP$ and $CP$ meet again its circumcircle $\Gamma$ at $K$, $L$, respectively $M$. The tangent line at $C$ to $\Gamma$ meets the line $AB$ at $S$. Show that from $SC = SP$ follows $MK = ML$. Author:...

Practice › Mathematics › IMO › IMO 2010 ›

IMO 2010 Problem 3

IMO 2010 Problem 3 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 5m01s Problem Find all functions $g:\mathbb{N}\rightarrow\mathbb{N}$ such that $\left(g(m)+n\right)\left(g(n)+m\right)$ is a perfect square for all $m,n\in\mathbb{N}.$ Author: Gabriel Carroll, USA Exploration The condition states that for all positive integers $m,n$, the product $\left(g(m)+n\right)\left(g(n)+m\right)$ is a perfect square. The expression is symmetric in $m$ and $n$, which suggests that the function is strongly constrained by additive...

Practice › Mathematics › IMO › IMO 2010 ›

IMO 2010 Problem 2

IMO 2010 Problem 2 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 5m17s Problem Given a triangle $ABC$, with $I$ as its incenter and $\Gamma$ as its circumcircle, $AI$ intersects $\Gamma$ again at $D$. Let $E$ be a point on arc $BDC$, and $F$ a point on the segment $BC$, such that $\angle BAF=\angle CAE< \frac12\angle BAC$. If $G$ is the midpoint of $IF$, prove that the...

Practice › Mathematics › IMO › IMO 2010 ›

IMO 2010 Problem 1

IMO 2010 Problem 1 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 10m09s Problem Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x,y\in\mathbb{R}$ the following equality holds $$ f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor $$ where $\left\lfloor a\right\rfloor$ is greatest integer not greater than $a.$ Author: Pierre Bornsztein, France Exploration Assume a nonzero function exists. If there is a point $y_0$ with $k=\lfloor f(y_0)\rfloor\neq 0$, then the equation forces...

Practice › Mathematics › IMO › IMO 2009 ›

IMO 2009 Problem 6

IMO 2009 Problem 6 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 13m20s Problem Let $a_1,a_2,\ldots,a_n$ be distinct positive integers and let $M$ be a set of $n-1$ positive integers not containing $s=a_1+a_2+\ldots+a_n$. A grasshopper is to jump along the real axis, starting at the point $0$ and making $n$ jumps to the right with lengths $a_1,a_2,\ldots,a_n$ in some order. Prove that the order can be chosen...

Practice › Mathematics › IMO › IMO 2009 ›

IMO 2009 Problem 5

IMO 2009 Problem 5 Proposed by: - Verified: no Verdicts: FAIL + UNKNOWN Solve time: 5m57s Problem Determine all functions $f$ from the set of positive integers to the set of positive integers such that, for all positive integers $a$ and $b$, there exists a non-degenerate triangle with sides of lengths $a,f(b)$ and $f(b+f(a)-1)$. (A triangle is non-degenerate if its vertices are not collinear.) Author: Bruno Le Floch, France Exploration...

Practice › Mathematics › IMO › IMO 2009 ›

IMO 2009 Problem 4

IMO 2009 Problem 4 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 5m10s Problem Let $ABC$ be a triangle with $AB=AC$. The angle bisectors of $\angle CAB$ and $\angle ABC$ meet the sides $BC$ and $CA$ at $D$ and $E$, respectively. Let $K$ be the incentre of triangle $ADC$. Suppose that $\angle BEK=45^\circ$. Find all possible values of $\angle CAB$. Authors: Jan Vonk and Peter Vandendriessche, Belgium,...

Practice › Mathematics › IMO › IMO 2009 ›

IMO 2009 Problem 3

IMO 2009 Problem 3 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 4m57s Problem Suppose that $s_1,s_2,s_3,\ldots$ is a strictly increasing sequence of positive integers such that the subsequences $s_{s_1},s_{s_2},s_{s_3},\ldots$ and $s_{s_1+1},s_{s_2+1},s_{s_3+1},\ldots$ are both arithmetic progressions. Prove that the sequence $s_1,s_2,s_3,\ldots$ is itself an arithmetic progression. Author: Gabriel Carroll, USA Exploration The condition states that a strictly increasing integer sequence $s_1,s_2,s_3,\ldots$ has the property that when it...

Practice › Mathematics › IMO › IMO 2009 ›

IMO 2009 Problem 2

IMO 2009 Problem 2 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 5m00s Problem Let $ABC$ be a triangle with circumcentre $O$. The points $P$ and $Q$ are interior points of the sides $CA$ and $AB$ respectively. Let $K,L$ and $M$ be the midpoints of the segments $BP,CQ$ and $PQ$, respectively, and let $\Gamma$ be the circle passing through $K,L$ and $M$. Suppose that the line $PQ$...

Practice › Mathematics › IMO › IMO 2009 ›

IMO 2009 Problem 1

IMO 2009 Problem 1 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 9m57s Problem Let $n$ be a positive integer and let $a_1,\ldots,a_k (k\ge2)$ be distinct integers in the set ${1,\ldots,n}$ such that $n$ divides $a_i(a_{i+1}-1)$ for $i=1,\ldots,k-1$. Prove that $n$ doesn't divide $a_k(a_1-1)$. Author: Ross Atkins, Australia Exploration Testing small cases clarifies the structure. For $n=2$, the condition forces each step to satisfy $2 \mid a_i(a_{i+1}-1)$,...

Practice › Mathematics › IMO › IMO 2008 ›

IMO 2008 Problem 6

IMO 2008 Problem 6 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 3m37s Problem Let $ABCD$ be a convex quadrilateral with $BA$ different from $BC$. Denote the incircles of triangles $ABC$ and $ADC$ by $k_{1}$ and $k_{2}$ respectively. Suppose that there exists a circle $k$ tangent to ray $BA$ beyond $A$ and to the ray $BC$ beyond $C$, which is also tangent to the lines $AD$ and...

Practice › Mathematics › IMO › IMO 2008 ›

IMO 2008 Problem 5

IMO 2008 Problem 5 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 5m23s Problem Let $n$ and $k$ be positive integers with $k \geq n$ and $k - n$ an even number. Let $2n$ lamps labelled $1, 2, \dots, 2n$ be given, each of which can be either on or off . Initially all the lamps are off. We consider sequences of steps: at each step one...

Practice › Mathematics › IMO › IMO 2008 ›

IMO 2008 Problem 4

IMO 2008 Problem 4 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 8m48s Problem Find all functions $f: (0, \infty) \mapsto (0, \infty)$ (so $f$ is a function from the positive real numbers) such that $\frac {\left( f(w) \right)^2 + \left( f(x) \right)^2}{f(y^2) + f(z^2) } = \frac {w^2 + x^2}{y^2 + z^2}$ for all positive real numbes $w,x,y,z,$ satisfying $wx = yz.$ Exploration A direct attempt...

Practice › Mathematics › IMO › IMO 2008 ›

IMO 2008 Problem 3

IMO 2008 Problem 3 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 7m55s Problem Prove that there are infinitely many positive integers $n$ such that $n^{2} + 1$ has a prime divisor greater than $2n + \sqrt {2n}$. Exploration Testing small values shows the statement is plausible but not uniform. For $n=1$, $n^2+1=2$ and $2>2+\sqrt2$ fails. For $n=2$, $5>4+\sqrt4=6$ fails. For $n=3$, $10$ has largest prime $5<6+\sqrt6$...

Practice › Mathematics › IMO › IMO 2008 ›

IMO 2008 Problem 2

IMO 2008 Problem 2 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 5m03s Problem Let $x, y, z\neq 1$ be three real numbers, such that $xyz = 1$ (i) Prove that; $\frac {x^{2}}{\left(x - 1\right)^{2}} + \frac {y^{2}}{\left(y - 1\right)^{2}} + \frac {z^{2}}{\left(z - 1\right)^{2}} \geq 1$. (ii) Prove that $\frac {x^{2}}{\left(x - 1\right)^{2}} + \frac {y^{2}}{\left(y - 1\right)^{2}} + \frac {z^{2}}{\left(z - 1\right)^{2}} = 1$ for...

Practice › Mathematics › IMO › IMO 2008 ›

IMO 2008 Problem 1

IMO 2008 Problem 1 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 5m09s Problem Let $H$ be the orthocenter of an acute-angled triangle $ABC$. The circle $\Gamma_{A}$ centered at the midpoint of $BC$ and passing through $H$ intersects line $BC$ at points $A_{1}$ and $A_{2}$. Similarly, define the points $B_{1}$, $B_{2}$, $C_{1}$ and $C_{2}$. Prove that six points $A_{1}$ , $A_{2}$, $B_{1}$, $B_{2}$, $C_{1}$ and $C_{2}$ are...

Practice › Mathematics › IMO › IMO 2007 ›

IMO 2007 Problem 6

IMO 2007 Problem 6 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 3m48s Problem Let $n$ be a positive integer. Consider $$ S={(x,y,z)~:~x,y,z\in {0,1,\ldots,n },~x+y+z>0} $$ as a set of $(n+1)^3-1$ points in three-dimensional space. Determine the smallest possible number of planes, the union of which contain $S$ but does not include $(0,0,0)$. 2007 IMO ( Problems ) • Resources Preceded by 2006 IMO Problems 1 •...

Practice › Mathematics › IMO › IMO 2007 ›

IMO 2007 Problem 5

IMO 2007 Problem 5 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 7m16s Problem ( Kevin Buzzard and Edward Crane, United Kingdom ) Let $a$ and $b$ be positive integers. Show that if $4ab-1$ divides $(4a^2-1)^2$, then $a=b$. Exploration Small cases are checked directly. For $a=1$, the condition becomes $4b-1 \mid 9$, giving $4b-1 \in {1,3,9}$, hence $b \in {1,1,2.5}$ and only $b=1$ is valid. For $a=2$,...

Practice › Mathematics › IMO › IMO 2007 ›

IMO 2007 Problem 4

IMO 2007 Problem 4 Proposed by: - Verified: no Verdicts: FAIL + PASS Solve time: 24m42s Problem In $\triangle ABC$ the bisector of $\angle{BCA}$ intersects the circumcircle again at $R$, the perpendicular bisector of $BC$ at $P$, and the perpendicular bisector of $AC$ at $Q$. The midpoint of $BC$ is $K$ and the midpoint of $AC$ is $L$. Prove that the triangles $RPK$ and $RQL$ have the same area. Exploration...

Practice › Mathematics › IMO › IMO 2007 ›

IMO 2007 Problem 3

IMO 2007 Problem 3 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 5m06s Problem In a mathematical competition some competitors are friends. Friendship is always mutual. Call a group of competitors a clique if each two of them are friends. (In particular, any group of fewer than two competitors is a clique.) The number of members of a clique is called its size. Given that, in this...

Practice › Mathematics › IMO › IMO 2007 ›

IMO 2007 Problem 2

IMO 2007 Problem 2 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 3m29s Problem Consider five points $A,B,C,D$, and $E$ such that $ABCD$ is a parallelogram and $BCED$ is a cyclic quadrilateral. Let $\ell$ be a line passing through $A$. Suppose that $\ell$ intersects the interior of the segment $DC$ at $F$ and intersects line $BC$ at $G$. Suppose also that $EF=EG=EC$. Prove that $\ell$ is the...

Practice › Mathematics › IMO › IMO 2007 ›

IMO 2007 Problem 1

IMO 2007 Problem 1 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 13m25s Problem Real numbers $a_1, a_2, \dots , a_n$ are given. For each $i$ ($1\le i\le n$) define $$ d_i=\max{a_j:1\le j\le i}-\min{a_j:i\le j\le n} $$ and let $$ d=\max{d_i:1\le i\le n}. $$ (a) Prove that, for any real numbers $x_1\le x_2\le \cdots\le x_n$, $$ \max{|x_i-a_i|:1\le i\le n}\ge \dfrac{d}{2} \qquad (*) $$ (b) Show that there...

Practice › Mathematics › IMO › IMO 2006 ›

IMO 2006 Problem 6

IMO 2006 Problem 6 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 8m40s Problem Let $P$ be a convex $n$-sided polygon with vertices $V_1, V_2, \dots, V_n,$ and sides $S_1, S_2, \dots, S_n.$ For a given side $S_i,$ let $A_i$ be the maximum possible area of a triangle with vertices among $V_1, V_2, \dots, V_n$ and with $S_i$ as a side. Show that the sum of the...

Practice › Mathematics › IMO › IMO 2006 ›

IMO 2006 Problem 5

IMO 2006 Problem 5 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 11m29s Problem Let $P(x)$ be a polynomial of degree $n>1$ with integer coefficients, and let $k$ be a positive integer. Consider the polynomial $Q(x) = P( P ( \ldots P(P(x)) \ldots ))$, where $P$ occurs $k$ times. Prove that there are at most $n$ integers $t$ such that $Q(t)=t$. Exploration The previous argument fails precisely...

Practice › Mathematics › IMO › IMO 2006 ›

IMO 2006 Problem 4

IMO 2006 Problem 4 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 13m44s Problem Determine all pairs $(x, y)$ of integers such that $$ 1+2^{x}+2^{2x+1}= y^{2}. $$ Exploration The equation is rewritten as $y^2=1+2^x+2^{2x+1}$. Direct substitution $t=2^x$ gives $y^2=2t^2+t+1$ with $t=2^x\ge 1$. Testing small values confirms $t=1$ gives $y^2=4$ and $t=16$ gives $y^2=529$. These correspond to $(x,y)=(0,\pm2)$ and $(4,\pm23)$. Substituting nearby powers of two, $t=2,4,8,32$, shows the...

Practice › Mathematics › IMO › IMO 2006 ›

IMO 2006 Problem 3

IMO 2006 Problem 3 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 6m39s Problem Determine the least real number $M$ such that the inequality $$ \left| ab\left(a^{2}-b^{2}\right)+bc\left(b^{2}-c^{2}\right)+ca\left(c^{2}-a^{2}\right)\right|\leq M\left(a^{2}+b^{2}+c^{2}\right)^{2} $$ holds for all real numbers $a,b$ and $c$ Exploration Homogeneity allows normalization $a^2+b^2+c^2=1$, so the task becomes maximizing $\left|ab(a^2-b^2)+bc(b^2-c^2)+ca(c^2-a^2)\right|.$ The factorization $ab(a^2-b^2)+bc(b^2-c^2)+ca(c^2-a^2)=-(a-b)(b-c)(c-a)(a+b+c)$ is algebraically correct and reduces the problem to optimizing a product of a Vandermonde term and...

Practice › Mathematics › IMO › IMO 2006 ›

IMO 2006 Problem 2

IMO 2006 Problem 2 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 4m03s Problem Let $P$ be a regular 2006 sided polygon. A diagonal of $P$ is called good if its endpoints divide the boundary of $P$ into two parts, each composed of an odd number of sides of $P$. The sides of $P$ are also called good. Suppose $P$ has been dissected into triangles by 2003...

Practice › Mathematics › IMO › IMO 2006 ›

IMO 2006 Problem 1

IMO 2006 Problem 1 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 5m23s Problem Let $ABC$ be a triangle with incentre $I.$ A point $P$ in the interior of the triangle satisfies $\angle PBA + \angle PCA = \angle PBC + \angle PCB$. Show that $AP \ge AI,$ and that equality holds if and only if $P = I.$ Exploration The condition $$\angle PBA + \angle PCA...

Practice › Mathematics › IMO › IMO 2005 ›

IMO 2005 Problem 6

IMO 2005 Problem 6 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 21m13s Problem In a mathematical competition, in which 6 problems were posed to the participants, every two of these problems were solved by more than 2/5 of the contestants. Moreover, no contestant solved all the 6 problems. Show that there are at least 2 contestants who solved exactly 5 problems each. Exploration The earlier approach...

Practice › Mathematics › IMO › IMO 2005 ›

IMO 2005 Problem 1

IMO 2005 Problem 1 Proposed by: - Verified: no Verdicts: UNKNOWN + FAIL Solve time: 14m57s Problem Six points are chosen on the sides of an equilateral triangle $ABC$: $A_1, A_2$ on $BC$, $B_1$, $B_2$ on $CA$ and $C_1$, $C_2$ on $AB$, such that they are the vertices of a convex hexagon $A_1A_2B_1B_2C_1C_2$ with equal side lengths. Prove that the lines $A_1B_2, B_1C_2$ and $C_1A_2$ are concurrent. Exploration The configuration...

Practice › Mathematics › IMO › IMO 2004 ›

IMO 2004 Problem 6

IMO 2004 Problem 6 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 13m30s Problem We call a positive integer alternating if every two consecutive digits in its decimal representation have a different parity. Find all positive integers $n$ such that $n$ has a multiple which is alternating. Exploration An integer is alternating when its decimal digits switch parity at every step. The condition depends only on parity...

Practice › Mathematics › IMO › IMO 2003 ›

IMO 2003 Problem 3

IMO 2003 Problem 3 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 14m15s Problem Each pair of opposite sides of convex hexagon has the property that the distance between their midpoints is $\frac{\sqrt{3}}{2}$ times the sum of their lengths. Prove that the hexagon is equiangular. Exploration A convex hexagon has three pairs of opposite sides. The condition relates each pair through the distance between midpoints of opposite...

Practice › Mathematics › IMO › IMO 2003 ›

IMO 2003 Problem 2

IMO 2003 Problem 2 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 13m49s Problem Determine all pairs of positive integers $(a,b)$ such that $$ \frac{a^2}{2ab^2-b^3+1} $$ is a positive integer. Exploration The expression $$\frac{a^2}{2ab^2-b^3+1}$$ is an integer, so the denominator must divide $a^2$ and must be positive. A first structural observation concerns the denominator $$D = 2ab^2 - b^3 + 1 = b^2(2a - b) + 1.$$...

Practice › Mathematics › IMO › IMO 2003 ›

IMO 2003 Problem 1

IMO 2003 Problem 1 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 13m40s Problem $S$ is the set ${1, 2, 3, \dots ,1000000}$. Show that for any subset $A$ of $S$ with $101$ elements we can find $100$ distinct elements $x_i$ of $S$, such that the sets ${a + x_i \mid a \in A}$ are all pairwise disjoint. Exploration Let $S={1,2,\dots,10^6}$ and let $A\subset S$ with $|A|=101$....

Practice › Mathematics › IMO › IMO 2002 ›

IMO 2002 Problem 6

IMO 2002 Problem 6 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 14m29s Problem Let $n \ge 3$ be a positive integer. Let $C_1,C_2,...,C_n$ be unit circles in the plane, with centers $O_1,O_2,...,O_n$ respectively. If no line meets more than two of the circles, prove that $$ \sum_{1\le i< j \le n}^{}\frac{1}{O_iO_j}\le\frac{(n-1)\pi}{4} $$ Exploration The configuration consists of $n$ unit disks whose centers are pairwise unconstrained except...

Practice › Mathematics › IMO › IMO 2002 ›

IMO 2002 Problem 5

IMO 2002 Problem 5 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 14m11s Problem Find all functions $f:\Bbb{R}\to \Bbb{R}$ such that $$ (f(x)+f(z))(f(y)+f(t))=f(xy-zt)+f(xt+yz) $$ for all real numbers $x,y,z,t$. Exploration Substituting zero values isolates the role of $f(0)$ and reduces the functional equation to a simpler constraint linking constant terms and general values. This typically separates constant solutions from genuinely nonlinear ones. The expression $$f(xy-zt)+f(xt+yz)$$ suggests a...

Practice › Mathematics › IMO › IMO 2002 ›

IMO 2002 Problem 4

IMO 2002 Problem 4 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 14m10s Problem Let $n>1$ be an integer and let $1=d_{1}<d_{2}<d_{3} \cdots <d_{r}=n$ be all of its positive divisors in increasing order. Show that $$ d=d_1d_2+d_2d_3+ \cdots +d_{r-1}d_r <n^2. $$ Exploration Compute small examples to understand structure. For a prime $n=p$, the divisors are $1,p$, so $d=1\cdot p=p$, which is far below $p^2$. For $n=4$, divisors...

Practice › Mathematics › IMO › IMO 2002 ›

IMO 2002 Problem 3

IMO 2002 Problem 3 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 15m09s Problem Find all pairs of positive integers $m,n \ge 3$ for which here exist infinitely many positive integers $a$ such that $$ \frac{a^m+a-1}{a^n+a^2-1} $$ is itself an integer. Exploration Let $$\frac{a^m+a-1}{a^n+a^2-1}$$ be an integer for infinitely many positive integers $a$, with $m,n \ge 3$. For large $a$, the expression behaves like $a^{m-n}$, so integrality...

Practice › Mathematics › IMO › IMO 2002 ›

IMO 2002 Problem 2

IMO 2002 Problem 2 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 14m01s Problem $BC$ is a diameter of a circle center $O$. $A$ is any point on the circle with $\angle AOC \not\le 60^\circ$. $EF$ is the chord which is the perpendicular bisector of $AO$. $D$ is the midpoint of the minor arc $AB$. The line through $O$ parallel to $AD$ meets $AC$ at $J$. Show...

Practice › Mathematics › IMO › IMO 2002 ›

IMO 2002 Problem 1

IMO 2002 Problem 1 Proposed by: - Verified: no Verdicts: FAIL + PASS Solve time: 8m44s Problem $S$ is the set of all $(h,k)$ with $h,k$ non-negative integers such that $h + k < n$. Each element of $S$ is colored red or blue, so that if $(h,k)$ is red and $h' \le h,k' \le k$, then $(h',k')$ is also red. A type $1$ subset of $S$ has $n$ blue...

Practice › Mathematics › IMO › IMO 2001 ›

IMO 2001 Problem 6

IMO 2001 Problem 6 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 7m15s Problem $k > l > m > n$ are positive integers such that $km + ln = (k+l-m+n)(-k+l+m+n)$. Prove that $kl+mn$ is not prime. Exploration The key relation is rewritten as $km+ln=(l+n)^2-(k-m)^2.$ Testing small admissible tuples such as $(k,l,m,n)=(5,4,3,2)$ and $(6,4,3,1)$ confirms that $kl+mn$ takes values $26$ and $27$ respectively, both composite or non-prime,...

Practice › Mathematics › IMO › IMO 2001 ›

IMO 2001 Problem 5

IMO 2001 Problem 5 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 7m02s Problem $ABC$ is a triangle . $X$ lies on $BC$ and $AX$ bisects angle $A$. $Y$ lies on $CA$ and $BY$ bisects angle $B$. Angle $A$ is $60^{\circ}$. $AB + BX = AY + YB$. Find all possible values for angle $B$. Exploration The earlier approach failed because it replaced a non-collinear segment $YB$...

Practice › Mathematics › IMO › IMO 2001 ›

IMO 2001 Problem 4

IMO 2001 Problem 4 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 12m18s Problem Let $n_1, n_2, \dots , n_m$ be integers where $m>1$ is odd. Let $x = (x_1, \dots , x_m)$ denote a permutation of the integers $1, 2, \cdots , m$. Let $f(x) = x_1n_1 + x_2n_2 + ... + x_mn_m$. Show that for some distinct permutations $a$, $b$ the difference $f(a) - f(b)$...

Practice › Mathematics › IMO › IMO 2001 ›

IMO 2001 Problem 3

IMO 2001 Problem 3 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 13m45s Problem Twenty-one girls and twenty-one boys took part in a mathematical competition. It turned out that each contestant solved at most six problems, and for each pair of a girl and a boy, there was at least one problem that was solved by both the girl and the boy. Show that there is a...

Practice › Mathematics › IMO › IMO 2001 ›

IMO 2001 Problem 2

IMO 2001 Problem 2 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 16m07s Problem Let $a,b,c$ be positive real numbers. Prove that $\frac{a}{\sqrt{a^{2}+8bc}}+\frac{b}{\sqrt{b^{2}+8ca}}+\frac{c}{\sqrt{c^{2}+8ab}}\ge 1$. Exploration Direct substitutions such as $a=x^2$ and linearization of the denominator repeatedly reduce the expression to cyclic rational forms that become weaker than the target bound. Testing asymmetric configurations like $a=1$, $b=c=\varepsilon$ shows that any reduction to expressions of type $\sum \frac{a}{a+2b+2c}$ or...

Practice › Mathematics › IMO › IMO 2001 ›

IMO 2001 Problem 1

IMO 2001 Problem 1 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 14m04s Problem Consider an acute triangle $\triangle ABC$. Let $P$ be the foot of the altitude of triangle $\triangle ABC$ issuing from the vertex $A$, and let $O$ be the circumcenter of triangle $\triangle ABC$. Assume that $\angle C \geq \angle B+30^{\circ}$. Prove that $\angle A+\angle COP < 90^{\circ}$. Exploration Let $A,B,C$ be the angles...

Practice › Mathematics › IMO › IMO 2000 ›

IMO 2000 Problem 6

IMO 2000 Problem 6 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 14m04s Problem Let $\overline{AH_1}$, $\overline{BH_2}$, and $\overline{CH_3}$ be the altitudes of an acute triangle $ABC$. The incircle $\omega$ of triangle $ABC$ touches the sides $BC$, $CA$, and $AB$ at $T_1$, $T_2$, and $T_3$, respectively. Consider the reflections of the lines $H_1H_2$, $H_2H_3$, and $H_3H_1$ with respect to the lines $T_1T_2$, $T_2T_3$, and $T_3T_1$. Prove that...

Practice › Mathematics › IMO › IMO 2000 ›

IMO 2000 Problem 5

IMO 2000 Problem 5 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 14m28s Problem Does there exist a positive integer $n$ such that $n$ has exactly 2000 prime divisors and $n$ divides $2^n+1$? Exploration Let $n$ be a positive integer with exactly $2000$ prime divisors, counted with multiplicity. Thus $n$ has the form $$n = \prod_{i=1}^{k} p_i^{a_i}, \quad \sum_{i=1}^{k} a_i = 2000,$$ where the $p_i$ are distinct...

Practice › Mathematics › IMO › IMO 2000 ›

IMO 2000 Problem 4

IMO 2000 Problem 4 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 13m49s Problem A magician has one hundred cards numbered $1$ to $100$. He puts them into three boxes, a red one, a white one and a blue one, so that each box contains at least one card. A member of the audience selects two of the three boxes, chooses one card from each and announces...

Practice › Mathematics › IMO › IMO 2000 ›

IMO 2000 Problem 3

IMO 2000 Problem 3 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 14m29s Problem Let $n \ge 2$ be a positive integer and $\lambda$ a positive real number. Initially there are $n$ fleas on a horizontal line, not all at the same point. We define a move as choosing two fleas at some points $A$ and $B$ to the left of $B$, and letting the flea from...

Practice › Mathematics › IMO › IMO 2000 ›

IMO 2000 Problem 2

IMO 2000 Problem 2 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 13m46s Problem Let $a, b, c$ be positive real numbers with $abc=1$. Show that $$ \left( a-1+\frac{1}{b} \right)\left( b-1+\frac{1}{c} \right)\left( c-1+\frac{1}{a} \right) \le 1 $$ Exploration The expression is cyclic in $a,b,c$ under the constraint $abc=1$, suggesting a symmetric inequality that may become simpler after a substitution such as $a=\frac{x}{y}$, $b=\frac{y}{z}$, $c=\frac{z}{x}$ or by introducing...

Practice › Mathematics › IMO › IMO 2000 ›

IMO 2000 Problem 1

IMO 2000 Problem 1 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 14m23s Problem Two circles $G_1$ and $G_2$ intersect at two points $M$ and $N$. Let $AB$ be the line tangent to these circles at $A$ and $B$, respectively, so that $M$ lies closer to $AB$ than $N$. Let $CD$ be the line parallel to $AB$ and passing through the point $M$, with $C$ on $G_1$...

Practice › Mathematics › IMO › IMO 1999 ›

IMO 1999 Problem 6

IMO 1999 Problem 6 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 13m58s Problem Determine all functions $f:\Bbb{R}\to \Bbb{R}$ such that $$ f(x-f(y))=f(f(y))+xf(y)+f(x)-1 $$ for all real numbers $x,y$. Exploration Substituting special values is the first constraint. Setting $x=0$ converts the equation into a relation involving only $f(-f(y))$, $f(f(y))$, and constants. Setting $y=0$ gives another functional constraint linking $f(x-f(0))$ and $f(x)$, suggesting that the value $f(0)$ plays...

Practice › Mathematics › IMO › IMO 1999 ›

IMO 1999 Problem 5

IMO 1999 Problem 5 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 13m54s Problem Two circles $G_{1}$ and $G_{2}$ are contained inside the circle $G$, and are tangent to $G$ at the distinct points $M$ and $N$, respectively. $G_{1}$ passes through the center of $G_{2}$. The line passing through the two points of intersection of $G_{1}$ and $G_{2}$ meets $G$ at $A$ and $B$. The lines $MA$...

Practice › Mathematics › IMO › IMO 1999 ›

IMO 1999 Problem 4

IMO 1999 Problem 4 Proposed by: - Verified: no Verdicts: FAIL + UNKNOWN Solve time: 13m43s Problem Determine all pairs $(n,p)$ of positive integers such that $p$ is a prime, $n$ not exceeded $2p$, and $(p-1)^{n}+1$ is divisible by $n^{p-1}$. Exploration The condition states that $n^{p-1}$ divides $(p-1)^n+1$, with $p$ prime and $n \le 2p$. The expression depends on two competing structures: a fixed base $p-1$ raised to a variable...

Practice › Mathematics › IMO › IMO 1999 ›

IMO 1999 Problem 3

IMO 1999 Problem 3 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 14m45s Problem Consider an $n \times n$ square board, where $n$ is a fixed even positive integer. The board is divided into $n^{2}$ units squares. We say that two different squares on the board are adjacent if they have a common side. $N$ unit squares on the board are marked in such a way that...

Practice › Mathematics › IMO › IMO 1999 ›

IMO 1999 Problem 2

IMO 1999 Problem 2 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 13m57s Problem Let $n \geq 2$ be a fixed integer. (a) Find the least constant $C$ such that for all nonnegative real numbers $x_1, \dots, x_n$, $$ \sum_{1\leq i<j \leq n} x_ix_j (x_i^2 + x_j^2) \leq C \left( \sum_{i=1}^n x_i \right)^4. $$ (b) Determine when equality occurs for this value of $C$. Exploration Let $S=\sum_{1\le...

Practice › Mathematics › IMO › IMO 1999 ›

IMO 1999 Problem 1

IMO 1999 Problem 1 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 14m15s Problem Determine all finite sets $S$ of at least three points in the plane which satisfy the following condition: For any two distinct points $A$ and $B$ in $S$, the perpendicular bisector of the line segment $AB$ is an axis of symmetry of $S$. Exploration The condition says that for every pair of distinct...

Practice › Mathematics › IMO › IMO 1998 ›

IMO 1998 Problem 6

IMO 1998 Problem 6 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 4m04s Problem Determine the least possible value of $f(1998),$ where $f:\Bbb{N}\to \Bbb{N}$ is a function such that for all $m,n\in {\Bbb N}$, $$ f\left( n^{2}f(m)\right) =m\left( f(n)\right) ^{2}. $$ Exploration Substituting small values reveals structural rigidity. Taking $n=1$ gives $$f(f(m)) = m \bigl(f(1)\bigr)^2,$$ which shows that repeated application of $f$ acts like multiplication by a...

Practice › Mathematics › IMO › IMO 1998 ›

IMO 1998 Problem 5

IMO 1998 Problem 5 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 4m04s Problem Let $I$ be the incenter of triangle $ABC$. Let the incircle of $ABC$ touch the sides $BC$, $CA$, and $AB$ at $K$, $L$, and $M$, respectively. The line through $B$ parallel to $MK$ meets the lines $LM$ and $LK$ at $R$ and $S$, respectively. Prove that angle $RIS$ is acute. Exploration The configuration...

Practice › Mathematics › IMO › IMO 1998 ›

IMO 1998 Problem 4

IMO 1998 Problem 4 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 14m26s Problem Determine all pairs $(a, b)$ of positive integers such that $ab^{2} + b + 7$ divides $a^{2}b + a + b$. Exploration The divisibility condition is $ab^{2}+b+7 \mid a^{2}b+a+b$. Writing $D=ab^{2}+b+7$ and $N=a^{2}b+a+b$, the combination $bN-aD$ eliminates the highest degree terms and gives $b^{2}-7a$. This shows $D \mid (b^{2}-7a)$. Substituting small values suggests...

Practice › Mathematics › IMO › IMO 1998 ›

IMO 1998 Problem 3

IMO 1998 Problem 3 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 15m21s Problem For any positive integer $n$, let $d(n)$ denote the number of positive divisors of $n$ (including 1 and $n$ itself). Determine all positive integers $k$ such that $\frac{d(n^2)}{d(n)} = k$ for some $n$. Exploration For each prime exponent $a$ in $n=\prod p_i^{a_i}$ the ratio contributes a factor $$\frac{2a+1}{a+1}.$$ Small values give $$a=1:\frac{3}{2},\quad a=2:\frac{5}{3},\quad...

Practice › Mathematics › IMO › IMO 1998 ›

IMO 1998 Problem 2

IMO 1998 Problem 2 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 4m11s Problem In a competition, there are $a$ contestants and $b$ judges, where $b\ge3$ is an odd integer. Each judge rates each contestant as either “pass” or “fail”. Suppose $k$ is a number such that, for any two judges, their ratings coincide for at most $k$ contestants. Prove that $\frac{k}{a}\ge\frac{b-1}{2b}$. Exploration For each contestant, the...

Practice › Mathematics › IMO › IMO 1998 ›

IMO 1998 Problem 1

IMO 1998 Problem 1 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 21m23s Problem In the convex quadrilateral $ABCD$, the diagonals $AC$ and $BD$ are perpendicular and the opposite sides $AB$ and $DC$ are not parallel. Suppose that the point $P$, where the perpendicular bisectors of $AB$ and $DC$ meet, is inside $ABCD$. Prove that $ABCD$ is a cyclic quadrilateral if and only if the triangles $ABP$...

Practice › Mathematics › IMO › IMO 1997 ›

IMO 1997 Problem 6

IMO 1997 Problem 6 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 6m59s Problem For each positive integer $n$, let $f(n)$ denote the number of ways of representing $n$ as a sum of powers of $2$ with nonnegative integer exponents. Representations which differ only in the ordering of their summands are considered to be the same. For instance, $f(4)=4$, because the number 4 can be represented in...

Practice › Mathematics › IMO › IMO 1997 ›

IMO 1997 Problem 5

IMO 1997 Problem 5 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 14m21s Problem Find all pairs $(a,b)$ of integers $a,b \ge 1$ that satisfy the equation $a^{b^{2}}=b^{a}$ Exploration The equation is $a^{b^2}=b^a,$ with $a,b\in \mathbb{Z}_{\ge 1}$. Testing small values directly is consistent and does not introduce hidden assumptions. For $b=1$, the equation becomes $a=1$, giving $(1,1)$. For $b=2$, the equation is $a^4=2^a$. Checking small integers gives...

Practice › Mathematics › IMO › IMO 1997 ›

IMO 1997 Problem 4

IMO 1997 Problem 4 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 13m50s Problem An $n \times n$ matrix whose entries come from the set $S={1,2,...,2n-1}$ is called a $\textit{silver}$ matrix if, for each $i=1,2,...,n$, the $i$th row and the $i$th column together contain all elements of $S$. Show that (a) there is no $\textit{silver}$ matrix for $n=1997$; (b) $\textit{silver}$ matrices exist for infinitely many values of...

Practice › Mathematics › IMO › IMO 1997 ›

IMO 1997 Problem 3

IMO 1997 Problem 3 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 13m29s Problem Let $x_{1}$, $x_{2}$,...,$x_{n}$ be real numbers satisfying the conditions $|x_{1}+x_{2}+...+x_{n}|=1$ and $|x_{i}| \le \frac{n+1}{2}$, for $i=1,2,...,n$ Show that there exists a permutation $y_{1}$, $y_{2}$,...,$y_{n}$ of $x_{1}$, $x_{2}$,...,$x_{n}$ such that $|y_{1}+2y_{2}+...+ny_{n}|\le \frac{n+1}{2}$ Exploration The previous argument fails because it attempts to control prefix sums via a greedy choice that is not guaranteed to be...

Practice › Mathematics › IMO › IMO 1997 ›

IMO 1997 Problem 2

IMO 1997 Problem 2 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 14m01s Problem The angle at $A$ is the smallest angle of triangle $ABC$. The points $B$ and $C$ divide the circumcircle of the triangle into two arcs. Let $U$ be an interior point of the arc between $B$ and $C$ which does not contain $A$. The perpendicular bisectors of $AB$ and $AC$ meet the line...

Practice › Mathematics › IMO › IMO 1997 ›

IMO 1997 Problem 1

IMO 1997 Problem 1 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 4m23s Problem In the plane the points with integer coordinates are the vertices of unit squares. The squares are colored alternatively black and white (as on a chessboard). For any pair of positive integers $m$ and $n$, consider a right-angled triangle whose vertices have integer coordinates and whose legs, of lengths $m$ and $n$, lie...

Practice › Mathematics › IMO › IMO 1996 ›

IMO 1996 Problem 6

IMO 1996 Problem 6 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 10m33s Problem Let $p, q, n$ be three positive integers with $p+q<n$. Let $(x_0,x_1,\cdots ,x_n)$ be an $(n+1)$-tuple of integers satisfying the following conditions: (i) $x_0=x_n=0$; (ii) For each $i$ with $1 \le i \le n$, either $x_i-x_{i-1}=p$ or $x_i-x_{i-1}=-q$. Show that there exists indices $i<j$ with $(i,j) \ne (0,n)$, such that $x_i=x_j$. Exploration The...

Practice › Mathematics › IMO › IMO 1996 ›

IMO 1996 Problem 5

IMO 1996 Problem 5 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 13m43s Problem Let $ABCDEF$ be a convex hexagon such that $AB$ is parallel to $DE$, $BD$ is parallel to $EF$, and $CD$ is parallel to $FA$. Let $R_{A}$, $R_{C}$, $R_{E}$ denote the circumradii of triangles $FAB$, $BCD$, $DEF$, respectively, and let $P$ denote the perimeter of the hexagon. Prove that $R_{A}+R_{C}+R_{E} \ge \frac{P}{2}.$ Exploration The...

Practice › Mathematics › IMO › IMO 1996 ›

IMO 1996 Problem 4

IMO 1996 Problem 4 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 14m21s Problem The positive integers $a$ and $b$ are such that the numbers $15a+16b$ and $16a-15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares? Exploration Let $$x^2 = 15a + 16b, \qquad y^2 = 16a - 15b,$$ with $a,b...

Practice › Mathematics › IMO › IMO 1996 ›

IMO 1996 Problem 3

IMO 1996 Problem 3 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 14m02s Problem Let $S$ denote the set of nonnegative integers. Find all functions $f$ from $S$ to itself such that $f(m+f(n))=f(f(m))+f(n)$ $\forall m,n \in S.$ Exploration The functional equation is $$f(m+f(n))=f(f(m))+f(n)\quad \forall m,n\in S=\mathbb{Z}_{\ge 0}.$$ Substituting small values suggests strong rigidity. Setting $n=0$ yields a relation between $f(m+f(0))$ and $f(f(m))$, indicating a shift-type behavior controlled...

Practice › Mathematics › IMO › IMO 1996 ›

IMO 1996 Problem 2

IMO 1996 Problem 2 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 18m50s Problem Let $P$ be a point inside triangle $ABC$ such that $$ \angle APB-\angle ACB = \angle APC-\angle ABC $$ Let $D$, $E$ be the incenters of triangles $APB$, $APC$, respectively. Show that $AP$, $BD$, $CE$ meet at a point. Exploration A direct continuation of the previous angle-sum manipulation is unsafe because relations involving...

Practice › Mathematics › IMO › IMO 1996 ›

IMO 1996 Problem 1

IMO 1996 Problem 1 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 14m58s Problem We are given a positive integer $r$ and a rectangular board $ABCD$ with dimensions $|AB|=20$, $|BC|=12$. The rectangle is divided into a grid of $20 \times 12$ unit squares. The following moves are permitted on the board: one can move from one square to another only if the distance between the centers of...

Practice › Mathematics › IMO › IMO 1995 ›

IMO 1995 Problem 6

IMO 1995 Problem 6 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 6m45s Problem Let $p$ be an odd prime number. How many $p$-element subsets $A$ of ${1,2,\ldots,2p}$ are there, the sum of whose elements is divisible by $p$? Exploration For small primes the structure can be tested directly. When $p=3$, the $3$-element subsets of ${1,2,3,4,5,6}$ can be enumerated and the sums modulo $3$ checked. A complete...

Practice › Mathematics › IMO › IMO 1995 ›

IMO 1995 Problem 5

IMO 1995 Problem 5 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 13m35s Problem Let $ABCDEF$ be a convex hexagon with $AB=BC=CD$ and $DE=EF=FA$, such that $\angle BCD=\angle EFA=\frac{\pi}{3}$. Suppose $G$ and $H$ are points in the interior of the hexagon such that $\angle AGB=\angle DHE=\frac{2\pi}{3}$. Prove that $AG+GB+GH+DH+HE\ge CF$. Exploration The failure in the previous argument comes from a structural mismatch: the target inequality cannot be...

Practice › Mathematics › IMO › IMO 1995 ›

IMO 1995 Problem 4

IMO 1995 Problem 4 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 13m13s Problem The positive real numbers $x_0, x_1, x_2,.....x_{1994}, x_{1995}$ satisfy the relations $x_0=x_{1995}$ and $x_{i-1}+\frac{2}{x_{i-1}}=2{x_i}+\frac{1}{x_i}$ for $i=1,2,3,....1995$ Find the maximum value that $x_0$ can have. Exploration The recurrence is first rewritten correctly by clearing denominators: $$x_{i-1}+\frac{2}{x_{i-1}}=2x_i+\frac{1}{x_i}$$ Multiplying by $x_{i-1}x_i$ gives $$x_{i-1}^2x_i+2x_i=2x_{i-1}x_i^2+x_{i-1}.$$ Rearranging, $$x_{i-1}^2x_i-2x_{i-1}x_i^2+2x_i-x_{i-1}=0.$$ Grouping terms, $$x_{i-1}x_i(x_{i-1}-2x_i)-(x_{i-1}-2x_i)=0,$$ so $$(x_{i-1}-2x_i)(x_{i-1}x_i-1)=0.$$ Testing small instances confirms the structure...

Practice › Mathematics › IMO › IMO 1995 ›

IMO 1995 Problem 3

IMO 1995 Problem 3 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 13m37s Problem Determine all integers $n>3$ for which there exist $n$ points $A_1,\ldots,A_n$ in the plane, no three collinear, and real numbers $r_1,\ldots,r_n$ such that for $1\le i<j<k\le n$, the area of $\triangle A_iA_jA_k$ is $r_i+r_j+r_k$. Exploration Testing the condition for small values of $n$ starts with rewriting the requirement as a linear system in...

Practice › Mathematics › IMO › IMO 1994 ›

IMO 1994 Problem 6

IMO 1994 Problem 6 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 3m33s Problem Show that there exists a set $A$ of positive integers with the following property: For any infinite set $S$ of primes there exist two positive integers $m \in A$ and $n \not\in A$ each of which is a product of $k$ distinct elements of $S$ for some $k \ge 2$. Exploration For any...

Practice › Mathematics › IMO › IMO 1994 ›

IMO 1994 Problem 5

IMO 1994 Problem 5 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 13m25s Problem Let $S$ be the set of real numbers strictly greater than $-1$. Find all functions $f:S \to S$ satisfying the two conditions: $f(x+f(y)+xf(y)) = y+f(x)+yf(x)$ for all $x$ and $y$ in $S$; $\frac{f(x)}{x}$ is strictly increasing on each of the intervals $-1<x<0$ and $0<x$. Exploration Testing small structures suggests that the expression $x+f(y)+xf(y)$...

Practice › Mathematics › IMO › IMO 1995 ›

IMO 1995 Problem 2

IMO 1995 Problem 2 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 7m54s Problem Let $a, b, c$ be positive real numbers such that $abc = 1$. Prove that $$ \frac{1}{a^3(b+c)} + \frac{1}{b^3(c+a)} + \frac{1}{c^3(a+b)} \geq \frac{3}{2}. $$ Exploration The corrected target inequality is equivalent to proving a lower bound for a cyclic sum of fractions with denominators $a^3(b+c)$. The substitution $a=\frac1x$, $b=\frac1y$, $c=\frac1z$ with $xyz=1$ is...

Practice › Mathematics › IMO › IMO 1994 ›

IMO 1994 Problem 4

IMO 1994 Problem 4 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 13m10s Problem Find all ordered pairs $(m,n)$ where $m$ and $n$ are positive integers such that $\frac {n^3 + 1}{mn - 1}$ is an integer. Exploration The condition is $mn-1 \mid n^3+1,$ so there exists a positive integer $k$ such that $n^3+1 = k(mn-1).$ Small cases suggest structure appears only when $n^3+1$ has a factor...

Practice › Mathematics › IMO › IMO 1995 ›

IMO 1995 Problem 1

IMO 1995 Problem 1 Proposed by: - Verified: no Verdicts: PASS + UNKNOWN Solve time: 17m49s Problem Let $A,B,C,D$ be four distinct points on a line, in that order. The circles with diameters $AC$ and $BD$ intersect at $X$ and $Y$. The line $XY$ meets $BC$ at $Z$. Let $P$ be a point on the line $XY$ other than $Z$. The line $CP$ intersects the circle with diameter $AC$ at...

Practice › Mathematics › IMO › IMO 1994 ›

IMO 1994 Problem 3

IMO 1994 Problem 3 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 14m04s Problem For any positive integer $k$, let $f(k)$ be the number of elements in the set ${k + 1, k + 2,\dots, 2k}$ whose base 2 representation has precisely three $1$s. (a) Prove that, for each positive integer $m$, there exists at least one positive integer $k$ such that $f(k) = m$. (b) Determine...

Practice › Mathematics › IMO › IMO 1994 ›

IMO 1994 Problem 2

IMO 1994 Problem 2 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 3m49s Problem Let $ABC$ be an isosceles triangle with $AB = AC$. $M$ is the midpoint of $BC$ and $O$ is the point on the line $AM$ such that $OB$ is perpendicular to $AB$. $Q$ is an arbitrary point on $BC$ different from $B$ and $C$. $E$ lies on the line $AB$ and $F$ lies...

Practice › Mathematics › IMO › IMO 1994 ›

IMO 1994 Problem 1

IMO 1994 Problem 1 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 8m06s Problem Let $m$ and $n$ be two positive integers. Let $a_1$, $a_2$, $\ldots$, $a_m$ be $m$ different numbers from the set ${1, 2,\ldots, n}$ such that for any two indices $i$ and $j$ with $1\leq i \leq j \leq m$ and $a_i + a_j \leq n$, there exists an index $k$ such that $a_i...

Practice › Mathematics › IMO › IMO 1993 ›

IMO 1993 Problem 6

IMO 1993 Problem 6 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 5m48s Problem There are $n$ lamps $L_0, \ldots , L_{n-1}$ in a circle ($n > 1$), where we denote $L_{n+k} = L_k$. (A lamp at all times is either on or off.) Perform steps $s_0, s_1, \ldots$ as follows: at step $s_i$, if $L_{i-1}$ is lit, switch $L_i$ from on to off or vice versa,...

Practice › Mathematics › IMO › IMO 1993 ›

IMO 1993 Problem 5

IMO 1993 Problem 5 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 4m02s Problem Does there exist a function $f:\textbf{N}\rightarrow\textbf{N}$ such that $f(1)=2,f(f(n))=f(n)+n$ for all $n\in\textbf{N}$ and $f(n)<f(n+1)$ for all $n\in\textbf{N}$? Exploration The functional equation $f(f(n)) = f(n) + n$ forces a strong coupling between values at $n$ and at $f(n)$. Because $f$ is strictly increasing on $\mathbb{N}$, iterates of $f$ must grow without repetition, and the...

Practice › Mathematics › IMO › IMO 1993 ›

IMO 1993 Problem 4

IMO 1993 Problem 4 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 3m43s Problem For three points $P,Q,R$ in the plane, we define $m(PQR)$ as the minimum length of the three altitudes of $\triangle PQR$. (If the points are collinear, we set $m(PQR)=0$.) Prove that for points $A,B,C,X$ in the plane, $$ m(ABC)\le m(ABX)+m(AXC)+m(XBC). $$ Exploration The expression $m(PQR)$ is a geometric quantity defined as the minimum...

Practice › Mathematics › IMO › IMO 1993 ›

IMO 1993 Problem 3

IMO 1993 Problem 3 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 5m54s Problem On an infinite chessboard, a game is played as follows. At the start, $n^2$ pieces are arranged on the chessboard in an $n$ by $n$ block of adjoining squares, one piece in each square. A move in the game is a jump in a horizontal or vertical direction over an adjacent occupied square...

Practice › Mathematics › IMO › IMO 1993 ›

IMO 1993 Problem 2

IMO 1993 Problem 2 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 5m25s Problem Let $D$ be a point inside acute triangle $ABC$ such that $\angle ADB=\angle ACB+\pi/2$ and $AC\cdot BD=AD\cdot BC$. (a) Compute the ratio $(AB\cdot CD)/(AC\cdot BD).$ (b) Prove that the tangents at $C$ to the circumcircles of $\triangle ACD$ and $\triangle BCD$ are perpendicular. Exploration The two conditions suggest a strong metric and angle...

Practice › Mathematics › IMO › IMO 1993 ›

IMO 1993 Problem 1

IMO 1993 Problem 1 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 4m07s Problem Let $f(x)=x^n+5x^{n-1}+3$, where $n>1$ is an integer. Prove that $f(x)$ cannot be expressed as the product of two nonconstant polynomials with integer coefficients. Exploration The polynomial $f(x)=x^n+5x^{n-1}+3$ has integer coefficients and leading coefficient $1$. Any nontrivial factorization over $\mathbb{Z}[x]$ would correspond to a factorization over $\mathbb{Q}[x]$ by Gauss’s lemma, hence it suffices to...

Practice › Mathematics › IMO › IMO 1992 ›

IMO 1992 Problem 6

IMO 1992 Problem 6 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 5m34s Problem For each positive integer $n$, $S(n)$ is defined to be the greatest integer such that, for every positive integer $k \le S(n)$, $n^{2}$ can be written as the sum of $k$ positive squares. (a) Prove that $S(n) \le n^{2}-14$ for each $n \ge 4$. (b) Find an integer $n$ such that $S(n)=n^{2}-14$. (c)...

Practice › Mathematics › IMO › IMO 1992 ›

IMO 1992 Problem 5

IMO 1992 Problem 5 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 5m33s Problem Let $S$ be a finite set of points in three-dimensional space. Let $S_{x}$,$S_{y}$,$S_{z}$, be the sets consisting of the orthogonal projections of the points of $S$ onto the $yz$-plane, $zx$-plane, $xy$-plane, respectively. Prove that $$ |S|^{2} \le |S_{x}| \cdot |S_{y}| \cdot |S_{z}|, $$ where $|A|$ denotes the number of elements in the finite...

Practice › Mathematics › IMO › IMO 1992 ›

IMO 1992 Problem 4

IMO 1992 Problem 4 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 5m55s Problem In the plane let $C$ be a circle, $l$ a line tangent to the circle $C$, and $M$ a point on $l$. Find the locus of all points $P$ with the following property: there exists two points $Q$, $R$ on $l$ such that $M$ is the midpoint of $QR$ and $C$ is the...

Practice › Mathematics › IMO › IMO 1992 ›

IMO 1992 Problem 3

IMO 1992 Problem 3 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 17m30s Problem Consider nine points in space, no four of which are coplanar. Each pair of points is joined by an edge (that is, a line segment) and each edge is either colored blue or red or left uncolored. Find the smallest value of $n$ such that whenever exactly n edges are colored, the set...

Practice › Mathematics › IMO › IMO 1992 ›

IMO 1992 Problem 2

IMO 1992 Problem 2 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 5m37s Problem Let $\mathbb{R}$ denote the set of all real numbers. Find all functions $f:\mathbb{R} \to \mathbb{R}$ such that $$ f\left( x^{2}+f(y) \right)= y+(f(x))^{2} \hspace{0.5cm} \forall x,y \in \mathbb{R} $$ Exploration Substituting $y=0$ transforms the equation into $f(x^2+f(0)) = (f(x))^2$. The left-hand side depends on $x^2$, suggesting symmetry in $x$ and thus strong restrictions on...

Practice › Mathematics › IMO › IMO 1992 ›

IMO 1992 Problem 1

IMO 1992 Problem 1 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 11m47s Problem Find all integers $a$, $b$, $c$ satisfying $1 < a < b < c$ such that $(a - 1)(b -1)(c - 1)$ is a divisor of $abc - 1$. Exploration The original reduction to $xyz \mid xy+yz+zx+x+y+z$ is correct after setting $x=a-1$, $y=b-1$, $z=c-1$ with $1\le x<y<z$. A direct inequality comparison between $xyz$...

Practice › Mathematics › IMO › IMO 1991 ›

IMO 1991 Problem 6

IMO 1991 Problem 6 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 6m37s Problem An infinite sequence $,x_{0},x_{1},x_{2},\ldots ,$ of real numbers is said to be bounded if there is a constant $,C,$ such that $, \vert x_{i} \vert \leq C,$ for every $,i\geq 0$. Given any real number $,a > 1,,$ construct a bounded infinite sequence $x_{0},x_{1},x_{2},\ldots ,$ such that $$ \vert x_{i} - x_{j} \vert...

Practice › Mathematics › IMO › IMO 1991 ›

IMO 1991 Problem 5

IMO 1991 Problem 5 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 3m47s Problem Let $,ABC,$ be a triangle and $,P,$ an interior point of $,ABC,$. Show that at least one of the angles $,\angle PAB,;\angle PBC,;\angle PCA,$ is less than or equal to $30^{\circ }$. Exploration Let $ABC$ be a triangle with an interior point $P$. The goal is to force at least one of the...

Practice › Mathematics › IMO › IMO 1991 ›

IMO 1991 Problem 4

IMO 1991 Problem 4 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 8m29s Problem Suppose $,G,$ is a connected graph with $,k,$ edges. Prove that it is possible to label the edges $1,2,\ldots ,k,$ in such a way that at each vertex which belongs to two or more edges, the greatest common divisor of the integers labeling those edges is equal to 1. Exploration The earlier attempt...

Practice › Mathematics › IMO › IMO 1991 ›

IMO 1991 Problem 3

IMO 1991 Problem 3 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 5m54s Problem Let $S = {1,2,3,\cdots ,280}$. Find the smallest integer $n$ such that each $n$-element subset of $S$ contains five numbers which are pairwise relatively prime. Exploration The problem asks for the smallest integer $n$ such that every $n$-element subset of $S={1,2,\dots,280}$ contains five integers that are pairwise coprime. Pairwise coprime means that among...

Practice › Mathematics › IMO › IMO 1991 ›

IMO 1991 Problem 2

IMO 1991 Problem 2 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 5m54s Problem Let $,n > 6,$ be an integer and $,a_{1},a_{2},\cdots ,a_{k},$ be all the natural numbers less than $n$ and relatively prime to $n$. If $$ a_{2} - a_{1} = a_{3} - a_{2} = \cdots = a_{k} - a_{k - 1} > 0, $$ prove that $,n,$ must be either a prime number or...

Practice › Mathematics › IMO › IMO 1991 ›

IMO 1991 Problem 1

IMO 1991 Problem 1 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 10m50s Problem Given a triangle $,ABC,,$ let $,I,$ be the center of its inscribed circle. The internal bisectors of the angles $,A,B,C,$ meet the opposite sides in $,A^{\prime },B^{\prime },C^{\prime },$ respectively. Prove that $$ \frac {1}{4} < \frac {AI\cdot BI\cdot CI}{AA^{\prime }\cdot BB^{\prime }\cdot CC^{\prime }} \leq \frac {8}{27}. $$ Exploration The earlier failure...

Practice › Mathematics › IMO › IMO 1990 ›

IMO 1990 Problem 6

IMO 1990 Problem 6 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 9m54s Problem Prove that there exists a convex 1990-gon with the following two properties : a.) All angles are equal. b.) The lengths of the 1990 sides are the numbers $1^2$, $2^2$, $3^2$, $\cdots$, $1990^2$ in some order. Exploration The earlier approach fails because cyclicity was incorrectly used as a substitute for equal angles. A...

Practice › Mathematics › IMO › IMO 1990 ›

IMO 1990 Problem 5

IMO 1990 Problem 5 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 3m59s Problem Given an initial integer $n_0 > 1$, two players, ${\mathcal A}$ and ${\mathcal B}$, choose integers $n_1$, $n_2$, $n_3$, $\ldots$ alternately according to the following rules : I.) Knowing $n_{2k}$, $\mathcal{A}$ chooses any integer $n_{2k + 1}$ such that $$ n_{2k} \leq n_{2k + 1} \leq n_{2k}^2. $$ II.) Knowing $n_{2k + 1}$,...

Practice › Mathematics › IMO › IMO 1990 ›

IMO 1990 Problem 4

IMO 1990 Problem 4 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 5m31s Problem Let ${\mathbb Q}^ +$ be the set of positive rational numbers. Construct a function $f : {\mathbb Q}^ + \rightarrow {\mathbb Q}^ +$ such that $$ f(xf(y)) = \frac {f(x)}{y} $$ for all $x$, $y$ in ${\mathbb Q}^ +$. Exploration The functional equation is $$f(x f(y)) = \frac{f(x)}{y} \quad \text{for all } x,y...

Practice › Mathematics › IMO › IMO 1990 ›

IMO 1990 Problem 3

IMO 1990 Problem 3 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 14m06s Problem Determine all integers $n > 1$ such that $$ \frac {2^n + 1}{n^2} $$ is an integer. Exploration Checking small values gives $n=2,3,4,5$ producing $\frac{5}{4},1,\frac{17}{16},\frac{33}{25}$, so only $n=3$ works in this range. Any correct proof must exclude all larger $n$ without assuming a special form for $n$. The condition $n^2 \mid 2^n+1$ implies...

Practice › Mathematics › IMO › IMO 1990 ›

IMO 1990 Problem 2

IMO 1990 Problem 2 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 5m11s Problem Let $n \geq 3$ and consider a set $E$ of $2n - 1$ distinct points on a circle. Suppose that exactly $k$ of these points are to be colored black. Such a coloring is good if there is at least one pair of black points such that the interior of one of the...

Practice › Mathematics › IMO › IMO 1990 ›

IMO 1990 Problem 1

IMO 1990 Problem 1 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 4m03s Problem Chords $AB$ and $CD$ of a circle intersect at a point $E$ inside the circle. Let $M$ be an interior point of the segment $EB$. The tangent line at $E$ to the circle through $D$, $E$, and $M$ intersects the lines $BC$ and $AC$ at $F$ and $G$, respectively. If $$ \frac {AM}{AB}...

Practice › Mathematics › IMO › IMO 1989 ›

IMO 1989 Problem 6

IMO 1989 Problem 6 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 4m10s Problem A permutation ${x_1, x_2, \ldots, x_{2n}}$ of the set ${1,2, \ldots, 2n}$ where $n$ is a positive integer, is said to have property $T$ if $|x_i - x_{i + 1}| = n$ for at least one $i$ in ${1,2, \ldots, 2n - 1}.$ Show that, for each $n$, there are more permutations with...

Practice › Mathematics › IMO › IMO 1989 ›

IMO 1989 Problem 4

IMO 1989 Problem 4 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 15m12s Problem Let $ABCD$ be a convex quadrilateral such that the sides $AB, AD, BC$ satisfy $AB = AD + BC.$ There exists a point $P$ inside the quadrilateral at a distance $h$ from the line $CD$ such that $AP = h + AD$ and $BP = h + BC.$ Show that: $$ \frac {1}{\sqrt...

Practice › Mathematics › IMO › IMO 1989 ›

IMO 1989 Problem 3

IMO 1989 Problem 3 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 5m32s Problem Let $n$ and $k$ be positive integers and let $S$ be a set of $n$ points in the plane such that i.) no three points of $S$ are collinear, and ii.) for every point $P$ of $S$ there are at least $k$ points of $S$ equidistant from $P.$ Prove that: $$ k <...

Practice › Mathematics › IMO › IMO 1989 ›

IMO 1989 Problem 2

IMO 1989 Problem 2 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 4m12s Problem $ABC$ is a triangle, the bisector of angle $A$ meets the circumcircle of triangle $ABC$ in $A_1$, points $B_1$ and $C_1$ are defined similarly. Let $AA_1$ meet the lines that bisect the two external angles at $B$ and $C$ in $A_0$. Define $B_0$ and $C_0$ similarly. Prove that the area of triangle $A_0B_0C_0...

Practice › Mathematics › IMO › IMO 1989 ›

IMO 1989 Problem 1

IMO 1989 Problem 1 Proposed by: - Verified: no Verdicts: FAIL + PASS Solve time: 18m36s Problem Prove that in the set ${1,2, \ldots, 1989}$ can be expressed as the disjoint union of subsets $A_i, {i = 1,2, \ldots, 117}$ such that i.) each $A_i$ contains 17 elements ii.) the sum of all the elements in each $A_i$ is the same. Exploration The previous construction failed because it broke the...

Practice › Mathematics › IMO › IMO 1988 ›

IMO 1988 Problem 6

IMO 1988 Problem 6 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 6m58s Problem Let $a$ and $b$ be two positive integers such that $a \cdot b + 1$ divides $a^{2} + b^{2}$. Show that $\frac {a^{2} + b^{2}}{a \cdot b + 1}$ is a perfect square. Exploration Testing small cases gives $k=1$ for $(a,b)=(1,1)$. No other small pairs $(a,b)$ with $a,b\le 10$ produce integer values of...

Practice › Mathematics › IMO › IMO 1988 ›

IMO 1988 Problem 5

IMO 1988 Problem 5 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 11m05s Problem In a right-angled triangle $ABC$ let $AD$ be the altitude drawn to the hypotenuse and let the straight line joining the incentres of the triangles $ABD, ACD$ intersect the sides $AB, AC$ at the points $K,L$ respectively. If $E$ and $E_1$ denote the areas of triangles $ABC$ and $AKL$ respectively, show that $$...

Practice › Mathematics › IMO › IMO 1988 ›

IMO 1988 Problem 4

IMO 1988 Problem 4 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 4m05s Problem Show that the solution set of the inequality $$ \sum^{70}_{k = 1} \frac {k}{x - k} \geq \frac {5}{4} $$ is a union of disjoint intervals, the sum of whose length is 1988. Exploration The expression $$\sum_{k=1}^{70}\frac{k}{x-k}$$ has simple poles at each integer from $1$ to $70$, and is strictly monotone on every...

Practice › Mathematics › IMO › IMO 1988 ›

IMO 1988 Problem 3

IMO 1988 Problem 3 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 14m49s Problem A function $f$ defined on the positive integers (and taking positive integers values) is given by: $\begin{matrix} f(1) = 1, f(3) = 3 \ f(2 \cdot n) = f(n) \ f(4 \cdot n + 1) = 2 \cdot f(2 \cdot n + 1) - f(n) \ f(4 \cdot n + 3) = 3...

Practice › Mathematics › IMO › IMO 1988 ›

IMO 1988 Problem 2

IMO 1988 Problem 2 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 5m51s Problem Let $n$ be an even positive integer. Let $A_1, A_2, \ldots, A_{n + 1}$ be sets having $n$ elements each such that any two of them have exactly one element in common while every element of their union belongs to at least two of the given sets. For which $n$ can one assign...

Practice › Mathematics › IMO › IMO 1988 ›

IMO 1988 Problem 1

IMO 1988 Problem 1 Proposed by: - Verified: no Verdicts: UNKNOWN + FAIL Solve time: 16m36s Problem Consider 2 concentric circle radii $R$ and $r$ ($R > r$) with centre $O.$ Fix $P$ on the small circle and consider the variable chord $PA$ of the small circle. Points $B$ and $C$ lie on the large circle; $B,P,C$ are collinear and $BC$ is perpendicular to $AP.$ i.) For which values of...

Practice › Mathematics › IMO › IMO 1987 ›

IMO 1987 Problem 6

IMO 1987 Problem 6 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 11m26s Problem Let $n$ be an integer greater than or equal to 2. Prove that if $k^2 + k + n$ is prime for all integers $k$ such that $0 \leq k \leq \sqrt{n/3}$, then $k^2 + k + n$ is prime for all integers $k$ such that $0 \leq k \leq n - 2$....

Practice › Mathematics › IMO › IMO 1987 ›

IMO 1987 Problem 5

IMO 1987 Problem 5 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 6m10s Problem Let $n$ be an integer greater than or equal to 3. Prove that there is a set of $n$ points in the plane such that the distance between any two points is irrational and each set of three points determines a non-degenerate triangle with rational area. Exploration A configuration with rational areas for...

Practice › Mathematics › IMO › IMO 1987 ›

IMO 1987 Problem 4

IMO 1987 Problem 4 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 5m45s Problem Prove that there is no function $f$ from the set of non-negative integers into itself such that $f(f(n)) = n + 1987$ for every $n$. Exploration Assume there exists a function $f : \mathbb{Z} {\ge 0} \to \mathbb{Z} {\ge 0}$ satisfying $f(f(n)) = n + 1987$ for every non-negative integer $n$. Iterating once...

Practice › Mathematics › IMO › IMO 1987 ›

IMO 1987 Problem 3

IMO 1987 Problem 3 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 4m00s Problem Let $x_1 , x_2 , \ldots , x_n$ be real numbers satisfying $x_1^2 + x_2^2 + \cdots + x_n^2 = 1$. Prove that for every integer $k \ge 2$ there are integers $a_1, a_2, \ldots a_n$, not all 0, such that $| a_i | \le k-1$ for all $i$ and $|a_1x_1 + a_2x_2...

Practice › Mathematics › IMO › IMO 1987 ›

IMO 1987 Problem 2

IMO 1987 Problem 2 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 8m22s Problem In an acute-angled triangle $ABC$ the interior bisector of the angle $A$ intersects $BC$ at $L$ and intersects the circumcircle of $ABC$ again at $N$. From point $L$ perpendiculars are drawn to $AB$ and $AC$, the feet of these perpendiculars being $K$ and $M$ respectively. Prove that the quadrilateral $AKNM$ and the triangle...

Practice › Mathematics › IMO › IMO 1987 ›

IMO 1987 Problem 1

IMO 1987 Problem 1 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 6m25s Problem Let $p_n (k)$ be the number of permutations of the set ${ 1, \ldots , n } , ; n \ge 1$, which have exactly $k$ fixed points. Prove that $\sum_{k=0}^{n} k \cdot p_n (k) = n!$. (Remark: A permutation $f$ of a set $S$ is a one-to-one mapping of $S$ onto itself....

Practice › Mathematics › IMO › IMO 1986 ›

IMO 1986 Problem 6

IMO 1986 Problem 6 Proposed by: - Verified: no Verdicts: FAIL + PASS Solve time: 8m02s Problem Given a finite set of points in the plane, each with integer coordinates, is it always possible to color the points red or white so that for any straight line $L$ parallel to one of the coordinate axes the difference (in absolute value) between the numbers of white and red points on $L$...

Practice › Mathematics › IMO › IMO 1986 ›

IMO 1986 Problem 5

IMO 1986 Problem 5 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 10m14s Problem Find all functions $f$ defined on the non-negative reals and taking non-negative real values such that: $f(2)=0,f(x)\ne0$ for $0\le x<2$, and $f(xf(y))f(y)=f(x+y)$ for all $x,y$. Exploration Substituting $y=2$ into the functional equation yields $f(xf(2))f(2)=f(x+2).$ Since $f(2)=0$, this forces $f(x+2)=0$ for all $x\ge 0$, hence $f(x)=0$ for all $x\ge 2$. This immediately isolates the...

Practice › Mathematics › IMO › IMO 1986 ›

IMO 1986 Problem 4

IMO 1986 Problem 4 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 8m44s Problem Let $A,B$ be adjacent vertices of a regular $n$-gon ($n\ge5$) with center $O$. A triangle $XYZ$, which is congruent to and initially coincides with $OAB$, moves in the plane in such a way that $Y$ and $Z$ each trace out the whole boundary of the polygon, with $X$ remaining inside the polygon. Find...

Practice › Mathematics › IMO › IMO 1986 ›

IMO 1986 Problem 3

IMO 1986 Problem 3 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 14m37s Problem To each vertex of a regular pentagon an integer is assigned, so that the sum of all five numbers is positive. If three consecutive vertices are assigned the numbers $x,y,z$ respectively, and $y<0$, then the following operation is allowed: $x,y,z$ are replaced by $x+y,-y,z+y$ respectively. Such an operation is performed repeatedly as long...

Practice › Mathematics › IMO › IMO 1986 ›

IMO 1986 Problem 2

IMO 1986 Problem 2 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 4m15s Problem Given a point $P_0$ in the plane of the triangle $A_1A_2A_3$. Define $A_s=A_{s-3}$ for all $s\ge4$. Construct a set of points $P_1,P_2,P_3,\ldots$ such that $P_{k+1}$ is the image of $P_k$ under a rotation center $A_{k+1}$ through an angle $120^o$ clockwise for $k=0,1,2,\ldots$. Prove that if $P_{1986}=P_0$, then the triangle $A_1A_2A_3$ is equilateral. Exploration...

Practice › Mathematics › IMO › IMO 1986 ›

IMO 1986 Problem 1

IMO 1986 Problem 1 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 15m13s Problem Let $d$ be any positive integer not equal to $2, 5$ or $13$. Show that one can find distinct $a,b$ in the set ${2,5,13,d}$ such that $ab-1$ is not a perfect square. Exploration Assume, for contradiction, that all six expressions $2\cdot 5-1,;2\cdot 13-1,;5\cdot 13-1,;2d-1,;5d-1,;13d-1$ are perfect squares. The first three are already known...

Practice › Mathematics › IMO › IMO 1985 ›

IMO 1985 Problem 6

IMO 1985 Problem 6 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 12m28s Problem For every real number $x_1$, construct the sequence $x_1,x_2,\ldots$ by setting: $x_{n + 1} = x_n(x_n + {1\over n}).$ Prove that there exists exactly one value of $x_1$ which gives $0 < x_n < x_{n + 1} < 1$ for all $n$. Exploration Testing small initial values clarifies the mechanism. For $n=1$, the...

Practice › Mathematics › IMO › IMO 1985 ›

IMO 1985 Problem 5

IMO 1985 Problem 5 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 5m24s Problem A circle with center $O$ passes through the vertices $A$ and $C$ of the triangle $ABC$ and intersects the segments $AB$ and $BC$ again at distinct points $K$ and $N$ respectively. Let $M$ be the point of intersection of the circumcircles of triangles $ABC$ and $KBN$ (apart from $B$). Prove that $\angle OMB...

Practice › Mathematics › IMO › IMO 1985 ›

IMO 1985 Problem 4

IMO 1985 Problem 4 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 9m00s Problem Given a set $M$ of $1985$ distinct positive integers, none of which has a prime divisor greater than $23$, prove that $M$ contains a subset of $4$ elements whose product is the $4$th power of an integer. Exploration The original approach fails because it tries to force a 4-term zero sum inside a...

Practice › Mathematics › IMO › IMO 1984 ›

IMO 1984 Problem 5

IMO 1984 Problem 5 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 10m10s Problem Let $d$ be the sum of the lengths of all the diagonals of a plane convex polygon with $n$ vertices (where $n>3$). Let $p$ be its perimeter. Prove that: $$ n-3<{2d\over p}<\Bigl[{n\over2}\Bigr]\cdot\Bigl[{n+1\over 2}\Bigr]-2, $$ where $[x]$ denotes the greatest integer not exceeding $x$. Exploration Testing small cases gives a sanity check on the...

Practice › Mathematics › IMO › IMO 1984 ›

IMO 1984 Problem 4

IMO 1984 Problem 4 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 3m56s Problem Let $ABCD$ be a convex quadrilateral with the line $CD$ being tangent to the circle on diameter $AB$. Prove that the line $AB$ is tangent to the circle on diameter $CD$ if and only if the lines $BC$ and $AD$ are parallel. Exploration The configuration contains two circles defined by diameters, hence both...

Practice › Mathematics › IMO › IMO 1984 ›

IMO 1984 Problem 3

IMO 1984 Problem 3 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 5m56s Problem Given points $O$ and $A$ in the plane. Every point in the plane is colored with one of a finite number of colors. Given a point $X$ in the plane, the circle $C(X)$ has center $O$ and radius $OX+{\angle AOX\over OX}$, where $\angle AOX$ is measured in radians in the range $[0,2\pi)$. Prove...

Practice › Mathematics › IMO › IMO 1984 ›

IMO 1984 Problem 2

IMO 1984 Problem 2 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 5m30s Problem Find one pair of positive integers $a,b$ such that $ab(a+b)$ is not divisible by $7$, but $(a+b)^7-a^7-b^7$ is divisible by $7^7$. Exploration The expression $$(a+b)^7-a^7-b^7$$ is naturally expanded using the binomial theorem. All intermediate binomial coefficients $\binom{7}{k}$ for $1 \le k \le 6$ are multiples of $7$, since $7$ is prime. Hence every...

Practice › Mathematics › IMO › IMO 1984 ›

IMO 1984 Problem 1

IMO 1984 Problem 1 Proposed by: - Verified: no Verdicts: PASS + FAIL Solve time: 13m35s Problem Prove that $0\le yz+zx+xy-2xyz\le{7\over27}$, where $x,y$ and $z$ are non-negative real numbers satisfying $x+y+z=1$. Exploration The expression is $$F(x,y,z)=xy+yz+zx-2xyz,\qquad x+y+z=1,\quad x,y,z\ge 0.$$ Testing extreme configurations gives immediate structure. At a vertex such as $(1,0,0)$, the value is $F=0$. At a midpoint such as $(\tfrac12,\tfrac12,0)$, the value is $F=\tfrac14$. At the fully symmetric point...

Practice › Mathematics › IMO › IMO 1983 ›

IMO 1983 Problem 6

IMO 1983 Problem 6 Proposed by: - Verified: no Verdicts: UNKNOWN + FAIL Solve time: 21m09s Problem Let $a$, $b$ and $c$ be the lengths of the sides of a triangle. Prove that $$ a^{2}b(a - b) + b^{2}c(b - c) + c^{2}a(c - a)\ge 0. $$ Determine when equality occurs. Exploration The reviewer’s critique isolates a structural mistake: any proof must avoid importing missing reverse-direction terms. A correct approach...

Practice › Mathematics › IMO › IMO 1983 ›

IMO 1983 Problem 5

IMO 1983 Problem 5 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 3m56s Problem Is it possible to choose $1983$ distinct positive integers, all less than or equal to $10^5$, no three of which are consecutive terms of an arithmetic progression? Exploration A set containing no three terms of an arithmetic progression is constrained by the absence of configurations of the form $a-d, a, a+d$ with $d...

Practice › Mathematics › IMO › IMO 1983 ›

IMO 1983 Problem 4

IMO 1983 Problem 4 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 4m01s Problem Let $ABC$ be an equilateral triangle and $\mathcal{E}$ the set of all points contained in the three segments $AB$, $BC$, and $CA$ (including $A$, $B$, and $C$). Determine whether, for every partition of $\mathcal{E}$ into two disjoint subsets, at least one of the two subsets contains the vertices of a right-angled triangle. Exploration...

Practice › Mathematics › IMO › IMO 1983 ›

IMO 1983 Problem 3

IMO 1983 Problem 3 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 3m51s Problem Let $a,b$ and $c$ be positive integers, no two of which have a common divisor greater than $1$. Show that $2abc-ab-bc-ca$ is the largest integer which cannot be expressed in the form $xbc+yca+zab$, where $x,y,z$ are non-negative integers. Exploration The expression $xbc+yca+zab$ is a non-negative integer linear combination of the three pairwise products...

Practice › Mathematics › IMO › IMO 1983 ›

IMO 1983 Problem 2

IMO 1983 Problem 2 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 4m02s Problem Let $A$ be one of the two distinct points of intersection of two unequal coplanar circles $C_1$ and $C_2$ with centers $O_1$ and $O_2$ respectively. One of the common tangents to the circles touches $C_1$ at $P_1$ and $C_2$ at $P_2$, while the other touches $C_1$ at $Q_1$ and $C_2$ at $Q_2$. Let...

Practice › Mathematics › IMO › IMO 1983 ›

IMO 1983 Problem 1

IMO 1983 Problem 1 Proposed by: - Verified: no Verdicts: FAIL + UNKNOWN Solve time: 8m46s Problem Find all functions $f$ defined on the set of positive reals which take positive real values and satisfy: $f(xf(y))=yf(x)$ for all $x,y$; and $f(x)\to0$ as $x\to\infty$. Exploration Fixing $x=1$ in the functional equation yields $f(f(y))=y f(1)$, so the behavior of $f$ is controlled by its value at $1$. Testing small values conceptually, take...

Practice › Mathematics › IMO › IMO 1982 ›

IMO 1982 Problem 6

IMO 1982 Problem 6 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 9m50s Problem Let $S$ be a square with sides length $100$. Let $L$ be a path within $S$ which does not meet itself and which is composed of line segments $A_0A_1,A_1A_2,A_2A_3,\ldots,A_{n-1}A_n$ with $A_0=A_n$. Suppose that for every point $P$ on the boundary of $S$ there is a point of $L$ at a distance from $P$...

Practice › Mathematics › IMO › IMO 1982 ›

IMO 1982 Problem 5

IMO 1982 Problem 5 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 13m39s Problem The diagonals $AC$ and $CE$ of the regular hexagon $ABCDEF$ are divided by inner points $M$ and $N$ respectively, so that $$ {AM\over AC}={CN\over CE}=r. $$ Determine $r$ if $B,M$ and $N$ are collinear. Exploration A regular hexagon admits a convenient coordinate realization in which opposite vertices lie on coordinate axes and successive...

Practice › Mathematics › IMO › IMO 1982 ›

IMO 1982 Problem 4

IMO 1982 Problem 4 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 4m00s Problem Prove that if $n$ is a positive integer such that the equation $$ x^3-3xy^2+y^3=n $$ has a solution in integers $x,y$, then it has at least three such solutions. Show that the equation has no solutions in integers for $n=2891$. Exploration Write $\omega = e^{2\pi i/3}$. Then $\omega^3 = 1$ and $1 +...

Practice › Mathematics › IMO › IMO 1982 ›

IMO 1982 Problem 3

IMO 1982 Problem 3 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 6m09s Problem Consider infinite sequences ${x_n}$ of positive reals such that $x_0=1$ and $x_0\ge x_1\ge x_2\ge\ldots$. a) Prove that for every such sequence there is an $n\ge1$ such that: $$ {x_0^2\over x_1}+{x_1^2\over x_2}+\ldots+{x_{n-1}^2\over x_n}\ge3.999. $$ b) Find such a sequence such that for all $n$: $$ {x_0^2\over x_1}+{x_1^2\over x_2}+\ldots+{x_{n-1}^2\over x_n}<4. $$ Exploration The expression $$\frac{x_{k-1}^2}{x_k}$$...

Practice › Mathematics › IMO › IMO 1982 ›

IMO 1982 Problem 2

IMO 1982 Problem 2 Proposed by: - Verified: no Verdicts: FAIL + UNKNOWN Solve time: 22m25s Problem A non-isosceles triangle $A_{1}A_{2}A_{3}$ has sides $a_{1}$, $a_{2}$, $a_{3}$ with the side $a_{i}$ lying opposite to the vertex $A_{i}$. Let $M_{i}$ be the midpoint of the side $a_{i}$, and let $T_{i}$ be the point where the inscribed circle of triangle $A_{1}A_{2}A_{3}$ touches the side $a_{i}$. Denote by $S_{i}$ the reflection of the point...

Practice › Mathematics › IMO › IMO 1982 ›

IMO 1982 Problem 1

IMO 1982 Problem 1 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 15m34s Problem The function $f(n)$ is defined on the positive integers and takes non-negative integer values. $f(2)=0,f(3)>0,f(9999)=3333$ and for all $m,n:$ $$ f(m+n)-f(m)-f(n)=0 \text{ or } 1. $$ Determine $f(1982)$. Exploration The previous approach failed because it attempted to propagate “local errors” globally without any invariant controlling how the defect $f(m+n)-f(m)-f(n)\in{0,1}$ behaves under different decompositions...

Practice › Mathematics › IMO › IMO 1981 ›

IMO 1981 Problem 6

IMO 1981 Problem 6 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 7m05s Problem The function $\displaystyle f(x,y)$ satisfies (1) $\displaystyle f(0,y)=y+1,$ (2) $\displaystyle f(x+1,0)=f(x,1),$ (3) $\displaystyle f(x+1,y+1)=f(x,f(x+1,y)),$ for all non-negative integers $\displaystyle x,y$. Determine $\displaystyle f(4,1981)$. Exploration The defining relations suggest a two-dimensional recursion where increasing the first argument modifies the entire dependence on the second argument, while the second argument evolves through repeated application of...

Practice › Mathematics › IMO › IMO 1981 ›

IMO 1981 Problem 5

IMO 1981 Problem 5 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 3m47s Problem Three congruent circles have a common point $\displaystyle O$ and lie inside a given triangle. Each circle touches a pair of sides of the triangle. Prove that the incenter and the circumcenter of the triangle and the point $\displaystyle O$ are collinear. Exploration The configuration involves three congruent circles inside a triangle, each...

Practice › Mathematics › IMO › IMO 1981 ›

IMO 1981 Problem 4

IMO 1981 Problem 4 Proposed by: - Verified: no Verdicts: UNKNOWN + FAIL Solve time: 8m32s Problem (a) For which values of $\displaystyle n>2$ is there a set of $\displaystyle n$ consecutive positive integers such that the largest number in the set is a divisor of the least common multiple of the remaining $\displaystyle n-1$ numbers? (b) For which values of $\displaystyle n>2$ is there exactly one set having the...

Practice › Mathematics › IMO › IMO 1981 ›

IMO 1981 Problem 3

IMO 1981 Problem 3 Proposed by: - Verified: no Verdicts: FAIL + UNKNOWN Solve time: 8m08s Problem Determine the maximum value of $\displaystyle m^2 + n^2$, where $\displaystyle m$ and $\displaystyle n$ are integers satisfying $m, n \in { 1,2, \ldots , 1981 }$ and $\displaystyle ( n^2 - mn - m^2 )^2 = 1$. Exploration The condition $(n^2 - mn - m^2)^2 = 1$ forces $n^2 - mn -...

Practice › Mathematics › IMO › IMO 1981 ›

IMO 1981 Problem 2

IMO 1981 Problem 2 Proposed by: - Verified: no Verdicts: UNKNOWN + PASS Solve time: 11m36s Problem Let $\displaystyle 1 \le r \le n$ and consider all subsets of $\displaystyle r$ elements of the set ${ 1, 2, \ldots , n }$. Each of these subsets has a smallest member. Let $\displaystyle F(n,r)$ denote the arithmetic mean of these smallest numbers; prove that $F(n,r) = \frac{n+1}{r+1}.$ Exploration Fix integers $1...

Practice › Mathematics › IMO › IMO 1981 ›

IMO 1981 Problem 1

IMO 1981 Problem 1 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 13m46s Problem $\displaystyle P$ is a point inside a given triangle $\displaystyle ABC$. $\displaystyle D, E, F$ are the feet of the perpendiculars from $\displaystyle P$ to the lines $\displaystyle BC, CA, AB$, respectively. Find all $\displaystyle P$ for which $\frac{BC}{PD} + \frac{CA}{PE} + \frac{AB}{PF}$ is least. Exploration Let $a=BC$, $b=CA$, $c=AB$. The expression becomes...

Practice › Mathematics › IMO › IMO 1979 ›

IMO 1979 Problem 6

IMO 1979 Problem 6 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 5m42s Problem Let $A$ and $E$ be opposite vertices of an octagon. A frog starts at vertex $A.$ From any vertex except $E$ it jumps to one of the two adjacent vertices. When it reaches $E$ it stops. Let $a_n$ be the number of distinct paths of exactly $n$ jumps ending at $E$. Prove that:$$...

Practice › Mathematics › IMO › IMO 1979 ›

IMO 1979 Problem 5

IMO 1979 Problem 5 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 9m17s Problem Determine all real numbers a for which there exists non-negative reals $x_{1}, \ldots, x_{5}$ which satisfy the relations $\sum_{k=1}^{5} kx_{k}=a,$ $\sum_{k=1}^{5} k^{3}x_{k}=a^{2},$ $\sum_{k=1}^{5} k^{5}x_{k}=a^{3}.$ Exploration The system involves three weighted power sums over the set ${1,2,3,4,5}$ with nonnegative weights $x_1,\dots,x_5$. The constraints prescribe that the first, third, and fifth power moments are $a$,...

Practice › Mathematics › IMO › IMO 1979 ›

IMO 1979 Problem 4

IMO 1979 Problem 4 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 3m54s Problem We consider a point $P$ in a plane $p$ and a point $Q \not\in p$. Determine all the points $R$ from $p$ for which$$ \frac{QP+PR}{QR} $$is maximum. Exploration Fix the plane $p$ and points $P \in p$ and $Q \notin p$. For a variable point $R \in p$, the quantity $$\frac{QP+PR}{QR}$$ depends only...

Practice › Mathematics › IMO › IMO 1979 ›

IMO 1979 Problem 3

IMO 1979 Problem 3 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 4m04s Problem Two circles in a plane intersect. $A$ is one of the points of intersection. Starting simultaneously from $A$ two points move with constant speed, each travelling along its own circle in the same sense. The two points return to $A$ simultaneously after one revolution. Prove that there is a fixed point $P$ in...

Practice › Mathematics › IMO › IMO 1979 ›

IMO 1979 Problem 2

IMO 1979 Problem 2 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 6m37s Problem We consider a prism which has the upper and inferior basis the pentagons: $A_{1}A_{2}A_{3}A_{4}A_{5}$ and $B_{1}B_{2}B_{3}B_{4}B_{5}$. Each of the sides of the two pentagons and the segments $A_{i}B_{j}$ with $i,j=1,\ldots$,5 is colored in red or blue. In every triangle which has all sides colored there exists one red side and one blue side....

Practice › Mathematics › IMO › IMO 1979 ›

IMO 1979 Problem 1

IMO 1979 Problem 1 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 9m29s Problem If $p$ and $q$ are natural numbers so that$$ \frac{p}{q}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+ \ldots -\frac{1}{1318}+\frac{1}{1319}, $$prove that $p$ is divisible by $1979$. Exploration The alternating sum is examined for small truncations to detect structure. For $n=1$, the sum is $1$. For $n=2$, it is $1-\frac12=\frac12$. For $n=3$, it is $\frac12+\frac13=\frac56$. For $n=4$, it is $\frac56-\frac14=\frac{7}{12}$. For...

Practice › Mathematics › IMO › IMO 1978 ›

IMO 1978 Problem 6

IMO 1978 Problem 6 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 12m29s Problem An international society has its members from six different countries. The list of members contain $1978$ names, numbered $1, 2, \dots, 1978$. Prove that there is at least one member whose number is the sum of the numbers of two members from his own country, or twice as large as the number of...

Practice › Mathematics › IMO › IMO 1978 ›

IMO 1978 Problem 5

IMO 1978 Problem 5 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 8m21s Problem Let $f$ be an injective function from ${1,2,3,\ldots}$ in itself. Prove that for any $n$ we have: $\sum_{k=1}^{n} f(k)k^{-2} \geq \sum_{k=1}^{n} k^{-1}.$ Exploration The expression involves a weighted sum of distinct positive integers $f(1),\dots,f(n)$ with weights $1/k^2$, which decrease strictly with $k$. The right-hand side depends only on the first $n$ harmonic terms,...

Practice › Mathematics › IMO › IMO 1978 ›

IMO 1978 Problem 4

IMO 1978 Problem 4 Proposed by: - Verified: no Verdicts: FAIL + UNKNOWN Solve time: 6m32s Problem In a triangle $ABC$ we have $AB = AC.$ A circle which is internally tangent with the circumscribed circle of the triangle is also tangent to the sides $AB, AC$ in the points $P,$ respectively $Q.$ Prove that the midpoint of $PQ$ is the center of the inscribed circle of the triangle $ABC.$...

Practice › Mathematics › IMO › IMO 1978 ›

IMO 1978 Problem 3

IMO 1978 Problem 3 Proposed by: - Verified: no Verdicts: UNKNOWN + FAIL Solve time: 6m40s Problem Let $0<f(1)<f(2)<f(3)<\ldots$ a sequence with all its terms positive$.$ The $n^{\text{th}}$ positive integer which doesn't belong to the sequence is $f(f(n))+1.$ Find $f(240).$ Exploration The sequence $f(1)<f(2)<f(3)<\cdots$ is strictly increasing and consists of positive integers, hence it defines an increasing subset $S\subset \mathbb{N}$. The complementary set $T=\mathbb{N}\setminus S$ is also strictly increasing when...

Practice › Mathematics › IMO › IMO 1978 ›

IMO 1978 Problem 2

IMO 1978 Problem 2 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 13m31s Problem We consider a fixed point $P$ in the interior of a fixed sphere$.$ We construct three segments $PA, PB,PC$, perpendicular two by two$,$ with the vertexes $A, B, C$ on the sphere$.$ We consider the vertex $Q$ which is opposite to $P$ in the parallelepiped (with right angles) with $PA, PB, PC$ as...

Practice › Mathematics › IMO › IMO 1978 ›

IMO 1978 Problem 1

IMO 1978 Problem 1 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 16m39s Problem Let $m$ and $n$ be positive integers such that $1 \le m < n$. In their decimal representations, the last three digits of $1978^m$ are equal, respectively, to the last three digits of $1978^n$. Find $m$ and $n$ such that $m + n$ has its least value. Exploration The original failure comes entirely...

Practice › Mathematics › IMO › IMO 1977 ›

IMO 1977 Problem 6

IMO 1977 Problem 6 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 8m36s Problem Let $\mathbb{N}$ be the set of positive integers. Let $f$ be a function defined on $\mathbb{N}$, which satisfies the inequality $f(n + 1) > f(f(n))$ for all $n \in \mathbb{N}$. Prove that for any $n$ we have $f(n) = n.$ Exploration Testing small indices does not reveal any oscillatory behavior compatible with the...

Practice › Mathematics › IMO › IMO 1977 ›

IMO 1977 Problem 5

IMO 1977 Problem 5 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 9m21s Problem Let $a,b$ be two natural numbers. When we divide $a^2+b^2$ by $a+b$, we the the remainder $r$ and the quotient $q.$ Determine all pairs $(a, b)$ for which $q^2 + r = 1977.$ Exploration The division condition is written as $a^2+b^2 = q(a+b) + r$ with $0 \le r < a+b$, and the...

Practice › Mathematics › IMO › IMO 1977 ›

IMO 1977 Problem 4

IMO 1977 Problem 4 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 4m07s Problem Let $a,b,A,B$ be given reals. We consider the function defined by$$ f(x) = 1 - a \cdot \cos(x) - b \cdot \sin(x) - A \cdot \cos(2x) - B \cdot \sin(2x). $$Prove that if for any real number $x$ we have $f(x) \geq 0$ then $a^2 + b^2 \leq 2$ and $A^2 + B^2...

Practice › Mathematics › IMO › IMO 1977 ›

IMO 1977 Problem 3

IMO 1977 Problem 3 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 4m16s Problem Let $n$ be a given number greater than 2. We consider the set $V_n$ of all the integers of the form $1 + kn$ with $k = 1, 2, \ldots$ A number $m$ from $V_n$ is called indecomposable in $V_n$ if there are not two numbers $p$ and $q$ from $V_n$ so that...

Practice › Mathematics › IMO › IMO 1977 ›

IMO 1977 Problem 2

IMO 1977 Problem 2 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 17m06s Problem In a finite sequence of real numbers the sum of any seven successive terms is negative and the sum of any eleven successive terms is positive. Determine the maximum number of terms in the sequence. Exploration The inequality system is local with two window lengths, $7$ and $11$, so every term participates in...

Practice › Mathematics › IMO › IMO 1977 ›

IMO 1977 Problem 1

IMO 1977 Problem 1 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 14m04s Problem In the interior of a square $ABCD$ we construct the equilateral triangles $ABK, BCL, CDM, DAN.$ Prove that the midpoints of the four segments $KL, LM, MN, NK$ and the midpoints of the eight segments $AK, BK, BL, CL, CM, DM, DN, AN$ are the 12 vertices of a regular dodecagon. Exploration A...

Practice › Mathematics › IMO › IMO 1976 ›

IMO 1976 Problem 6

IMO 1976 Problem 6 Proposed by: - Verified: no Verdicts: FAIL + UNKNOWN Solve time: 7m15s Problem For all positive integral $n$, $u_{n+1}=u_n(u_{n-1}^2-2)-u_1$, $u_0=2$, and $u_1=2\frac12$. Prove that $$ 3\log_2[u_n]=2^n-(-1)^n, $$ where $[x]$ is the integral part of $x$. Exploration The recurrence couples each term with the previous two terms in a nonlinear way, and the conclusion concerns the integer part of a logarithm, suggesting that the central structure is...

Practice › Mathematics › IMO › IMO 1976 ›

IMO 1976 Problem 5

IMO 1976 Problem 5 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 5m25s Problem Let a set of $p$ equations be given, $$ \begin{array}{ccccccc} a_{11}x_1&+&\cdots&+&a_{1q}x_q&=&0,\ a_{21}x_1&+&\cdots&+&a_{2q}x_q&=&0,\ &&&\vdots&&&\ a_{p1}x_1&+&\cdots&+&a_{pq}x_q&=&0,\ \end{array} $$ with coefficients $a_{ij}$ satisfying $a_{ij}=-1$, $0$, or $+1$ for all $i=1,\dots, p$, and $j=1,\dots, q$. Prove that if $q=2p$, there exists a solution $x_1, \dots, x_q$ of this system such that all $x_j$ ($j=1,\dots, q$) are integers...

Practice › Mathematics › IMO › IMO 1976 ›

IMO 1976 Problem 4

IMO 1976 Problem 4 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 5m33s Problem Find the largest number obtainable as the product of positive integers whose sum is $1976$. Exploration A product of positive integers with fixed sum increases when the summands are replaced by a partition that favors factors close to the maximizer of the function $x \mapsto x^{1/x}$. Testing small decompositions shows that $3$ contributes...

Practice › Mathematics › IMO › IMO 1976 ›

IMO 1976 Problem 3

IMO 1976 Problem 3 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 4m11s Problem A box whose shape is a parallelepiped can be completely filled with cubes of side $1.$ If we put in it the maximum possible number of cubes, each of volume $2$, with the sides parallel to those of the box, then exactly $40$ percent from the volume of the box is occupied. Determine...

Practice › Mathematics › IMO › IMO 1976 ›

IMO 1976 Problem 2

IMO 1976 Problem 2 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 4m04s Problem Let $P_{1}(x) = x^{2} - 2$ and $P_{j}(x) = P_{1}(P_{j - 1}(x))$ for $j= 2,\ldots$ Prove that for any positive integer n the roots of the equation $P_{n}(x) = x$ are all real and distinct. Exploration The polynomials are defined by iteration of $P_1(x)=x^2-2$, so that $P_n$ is the $n$-fold composition of $x^2-2$...

Practice › Mathematics › IMO › IMO 1976 ›

IMO 1976 Problem 1

IMO 1976 Problem 1 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 4m06s Problem In a convex quadrilateral (in the plane) with the area of $32 \text{ cm}^{2}$ the sum of two opposite sides and a diagonal is $16 \text{ cm}$. Determine all the possible values that the other diagonal can have. Exploration Let a convex quadrilateral $ABCD$ be given. One diagonal, say $AC$, appears in the...

Practice › Mathematics › IMO › IMO 1975 ›

IMO 1975 Problem 6

IMO 1975 Problem 6 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 6m58s Problem Find all polynomials $P$, in two variables, with the following properties: (i) for a positive integer $n$ and all real $t, x, y$ $$ P(tx, ty) = t^n P(x, y) $$ (that is, $P$ is homogeneous of degree $n$), (ii) for all real $a, b, c,$ $$ P(b+c, a) + P(c+a, b) +...

Practice › Mathematics › IMO › IMO 1975 ›

IMO 1975 Problem 5

IMO 1975 Problem 5 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 7m27s Problem Determine, with proof, whether or not one can find 1975 points on the circumference of a circle with unit radius such that the distance between any two of them is a rational number. Exploration The flawed argument tried to force a cyclotomic structure via Mann’s theorem. The reviewer correctly identified that this is...

Practice › Mathematics › IMO › IMO 1975 ›

IMO 1975 Problem 4

IMO 1975 Problem 4 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 11m31s Problem When $4444^{4444}$ is written in decimal notation, the sum of its digits is $A$. Let $B$ be the sum of the digits of $A$. Find the sum of the digits of $B$. ($A$ and $B$ are written in decimal notation.) Problem Understanding We are given the number $$N = 4444^{4444}.$$ Let $A =...

Practice › Mathematics › IMO › IMO 1975 ›

IMO 1975 Problem 3

IMO 1975 Problem 3 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 11m35s Problem On the sides of an arbitrary triangle $ABC$, triangles $ABR, BCP, CAQ$ are constructed externally with $\angle CBP = \angle CAQ = 45^\circ, \angle BCP = \angle ACQ = 30^\circ, \angle ABR = \angle BAR = 15^\circ$. Prove that $\angle QRP = 90^\circ$ and $QR = RP$. Exploration Directed-angle accumulation across a multi-centered...

Practice › Mathematics › IMO › IMO 1975 ›

IMO 1975 Problem 2

IMO 1975 Problem 2 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 9m00s Problem Let $a_1, a_2, a_3, \cdots$ be an infinite increasing sequence of positive integers. Prove that for every $p \geq 1$ there are infinitely many $a_m$ which can be written in the form $$ a_m = xa_p + ya_q $$ with $x, y$ positive integers and $q > p$. Exploration The statement concerns an...

Practice › Mathematics › IMO › IMO 1975 ›

IMO 1975 Problem 1

IMO 1975 Problem 1 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 5m39s Problem Let $x_i, y_i~(i = 1, 2, \ldots, n)$ be real numbers such that $$ x_1 \geq x_2 \geq \cdots \geq x_n \text{ and } y_1 \geq y_2 \geq \cdots \geq y_n. $$ Prove that, if $z_1, z_2, \cdots, z_n$ is any permutation of $y_1, y_2, \cdots, y_n$, then $$ \sum_{i=1}^n (x_i - y_i)^2...

Practice › Mathematics › IMO › IMO 1974 ›

IMO 1974 Problem 6

IMO 1974 Problem 6 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 14m57s Problem Let $P$ be a non-constant polynomial with integer coefficients. If $n(P)$ is the number of distinct integers $k$ such that $(P(k))^2 = 1$, prove that $n(P) - \deg(P) \leq 2$, where $\deg(P)$ denotes the degree of the polynomial $P$. Exploration The earlier approach fails because it invents a hierarchy of sets based on...

Practice › Mathematics › IMO › IMO 1974 ›

IMO 1974 Problem 5

IMO 1974 Problem 5 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 7m27s Problem Determine all possible values of $$ S = \frac{a}{a+b+d} + \frac{b}{a+b+c} + \frac{c}{b+c+d} + \frac{d}{a+c+d} $$ where $a, b, c, d$ are arbitrary positive numbers. Exploration Direct computation of extremal configurations shows that the expression approaches $2$ when two opposite variables are small and the other two are large, and approaches $1$ when...

Practice › Mathematics › IMO › IMO 1974 ›

IMO 1974 Problem 4

IMO 1974 Problem 4 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 13m33s Problem Consider decompositions of an $8 \times 8$ chessboard into p non-overlapping rectangles subject to the following conditions: (i) Each rectangle has as many white squares as black squares. (ii) If $a_i$ is the number of white squares in the $i$-th rectangle, then $a_1 < a_2 < \cdots < a_p$. Find the maximum value...

Practice › Mathematics › IMO › IMO 1974 ›

IMO 1974 Problem 3

IMO 1974 Problem 3 Proposed by: - Verified: no Verdicts: PASS + FAIL Solve time: 11m12s Problem Prove that the number $\textstyle \sum_{k = 0}^n \tbinom{2n+1}{2k+1} 2^{3k}$ is not divisible by 5 for any integer $n \geq 0$. Exploration Let $$S_n=\sum_{k=0}^{n}\binom{2n+1}{2k+1}2^{3k}.$$ The first five values are $$S_0=1,$$ $$S_1=\binom31+\binom33\cdot 8=3+8=11,$$ $$S_2=\binom51+\binom53\cdot 8+\binom55\cdot 64 =5+80+64=149,$$ $$S_3=\binom71+\binom73\cdot 8+\binom75\cdot 64+\binom77\cdot 512 =7+280+1344+512=2143,$$ $$S_4=9+84\cdot 8+126\cdot 64+36\cdot 512+4096=30857.$$ Reducing modulo $5$, $$1,\ 11,\ 149,\ 2143,\ 30857...

Practice › Mathematics › IMO › IMO 1974 ›

IMO 1974 Problem 2

IMO 1974 Problem 2 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 18m19s Problem In the triangle $ABC$, prove that there is a point $D$ on side $AB$ such that $CD$ is the geometric mean of $AD$ and $DB$ if and only if $$ \sin{A} \sin{B} \leq \sin^2 \frac{C}{2}. $$ Problem Understanding In triangle $ABC$, we seek a point $D$ on the side $AB$ such that $$CD^{2}=AD\cdot...

Practice › Mathematics › IMO › IMO 1974 ›

IMO 1974 Problem 1

IMO 1974 Problem 1 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 9m35s Problem Three players $A, B$ and $C$ play the following game: On each of three cards an integer is written. These three numbers $p, q, r$ satisfy $0 < p < q < r$. The three cards are shuffled and one is dealt to each player. Each then receives the number of counters indicated...

Practice › Mathematics › IMO › IMO 1970 ›

IMO 1970 Problem 6

IMO 1970 Problem 6 Proposed by: - Verified: no Verdicts: FAIL + UNKNOWN Solve time: 11m18s Problem In a plane there are $100$ points, no three of which are collinear. Consider all possible triangles having these point as vertices. Prove that no more than $70 %$ of these triangles are acute-angled. Exploration A direct “vertex–angle ≥ 60°” charging argument cannot be repaired by any local bound of the form “at...

Practice › Mathematics › IMO › IMO 1970 ›

IMO 1970 Problem 5

IMO 1970 Problem 5 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 11m47s Problem In the tetrahedron $ABCD$, angle $BDC$ is a right angle. Suppose that the foot $H$ of the perpendicular from $D$ to the plane $ABC$ in the tetrahedron is the intersection of the altitudes of $\triangle ABC$. Prove that $( AB+BC+CA )^2 \leq 6( AD^2 + BD^2 + CD^2 )$. For what tetrahedra does...

Practice › Mathematics › IMO › IMO 1970 ›

IMO 1970 Problem 4

IMO 1970 Problem 4 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 15m36s Problem Find the set of all positive integers $n$ with the property that the set ${ n, n+1, n+2, n+3, n+4, n+5 }$ can be partitioned into two sets such that the product of the numbers in one set equals the product of the numbers in the other set. Exploration The flawed solution failed...

Practice › Mathematics › IMO › IMO 1970 ›

IMO 1970 Problem 3

IMO 1970 Problem 3 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 6m39s Problem The real numbers $a_0, a_1, \ldots, a_n, \ldots$ satisfy the condition: $1 = a_{0} \leq a_{1} \leq \cdots \leq a_{n} \leq \cdots$. The numbers $b_{1}, b_{2}, \ldots, b_n, \ldots$ are defined by $b_n = \sum_{k=1}^{n} \left( 1 - \frac{a_{k-1}}{a_{k}} \right)\dfrac{1}{\sqrt{a_k}}$ (a) Prove that $0 \leq b_n < 2$ for all $n$. (b) given...

Practice › Mathematics › IMO › IMO 1970 ›

IMO 1970 Problem 2

IMO 1970 Problem 2 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 7m24s Problem Let $a, b$, and $n$ be integers greater than 1, and let $a$ and $b$ be the bases of two number systems. $A_{n-1}$ and $A_{n}$ are numbers in the system with base $a$ and $B_{n-1}$ and $B_{n}$ are numbers in the system with base $b$; these are related as follows: $A_{n} = x_{n}x_{n-1}\cdots...

Practice › Mathematics › IMO › IMO 1970 ›

IMO 1970 Problem 1

IMO 1970 Problem 1 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 18m34s Problem Let $M$ be a point on the side $AB$ of $\triangle ABC$. Let $r_1, r_2$, and $r$ be the inscribed circles of triangles $AMC, BMC$, and $ABC$. Let $q_1, q_2$, and $q$ be the radii of the exscribed circles of the same triangles that lie in $\angle ACB$. Prove that $\frac{r_1}{q_1} \cdot \frac{r_2}{q_2}...

Practice › Mathematics › IMO › IMO 1969 ›

IMO 1969 Problem 6

IMO 1969 Problem 6 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 4m32s Problem Prove that for all real numbers $x_1, x_2, y_1, y_2, z_1, z_2$, with $x_1 > 0, x_2 > 0, x_1y_1 - z_1^2 > 0, x_2y_2 - z_2^2 > 0$, the inequality $$ \frac{8}{(x_1 + x_2)(y_1 + y_2) - (z_1 + z_2)^2} \leq \frac{1}{x_1y_1 - z_1^2} + \frac{1}{x_2y_2 - z_2^2} $$ is satisfied. Give...

Practice › Mathematics › IMO › IMO 1969 ›

IMO 1969 Problem 5

IMO 1969 Problem 5 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 5m46s Problem Given $n > 4$ points in the plane such that no three are collinear. Prove that there are at least $\tbinom{n - 3}{2}$ convex quadrilaterals whose vertices are four of the given points. Exploration A convex quadrilateral is determined by four points in convex position, meaning all four lie on the boundary of...

Practice › Mathematics › IMO › IMO 1969 ›

IMO 1969 Problem 4

IMO 1969 Problem 4 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 4m37s Problem A semicircular arc $\gamma$ is drawn on $AB$ as diameter. $C$ is a point on $\gamma$ other than $A$ and $B$, and $D$ is the foot of the perpendicular from $C$ to $AB$. We consider three circles, $\gamma_1, \gamma_2, \gamma_3$, all tangent to the line $AB$. Of these, $\gamma_1$ is inscribed in $\Delta...

Practice › Mathematics › IMO › IMO 1969 ›

IMO 1969 Problem 3

IMO 1969 Problem 3 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 4m24s Problem For each value of $k = 1, 2, 3, 4, 5$, find necessary and sufficient conditions on the number $a > 0$ so that there exists a tetrahedron with k edges of length $a$, and the remaining $6 - k$ edges of length 1. Exploration A tetrahedron has six edges, so for each...

Practice › Mathematics › IMO › IMO 1969 ›

IMO 1969 Problem 2

IMO 1969 Problem 2 Proposed by: - Verified: no Verdicts: PASS + FAIL Solve time: 5m39s Problem Let $a_1, a_2, \cdots, a_n$ be real constants, $x$ a real variable, and $$ f(x) = \cos(a_1 + x) + \frac{1}{2}\cos(a_2 + x) + \frac{1}{4}\cos(a_3 + x) + \cdots + \frac{1}{2^{n - 1}}\cos(a_n + x). $$ Given that $f(x_1) = f(x_2) = 0$, prove that $x_2 - x_1 = m\pi$ for some integer...

Practice › Mathematics › IMO › IMO 1969 ›

IMO 1969 Problem 1

IMO 1969 Problem 1 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 4m22s Problem Prove that there are infinitely many natural numbers $a$ with the following property: the number $z = n^4 + a$ is not prime for any natural number $n$. Exploration The requirement is to construct infinitely many natural numbers $a$ such that for every natural number $n$, the integer $n^4 + a$ is composite....

Practice › Mathematics › IMO › IMO 1968 ›

IMO 1968 Problem 6

IMO 1968 Problem 6 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 8m57s Problem For every natural number $n$, evaluate the sum $$ \sum_{k = 0}^\infty\bigg[\frac{n + 2^k}{2^{k + 1}}\bigg] = \Big[\frac{n + 1}{2}\Big] + \Big[\frac{n + 2}{4}\Big] + \cdots + \bigg[\frac{n + 2^k}{2^{k + 1}}\bigg] + \cdots $$ (The symbol $[x]$ denotes the greatest integer not exceeding $x$.) Exploration The expression is $$S(n)=\sum_{k=0}^\infty \left\lfloor \frac{n+2^k}{2^{k+1}} \right\rfloor.$$...

Practice › Mathematics › IMO › IMO 1968 ›

IMO 1968 Problem 5

IMO 1968 Problem 5 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 4m05s Problem Let $f$ be a real-valued function defined for all real numbers $x$ such that, for some positive constant $a$, the equation $$ f(x + a) = \frac{1}{2} + \sqrt{f(x) - (f(x))^2} $$ holds for all $x$. (a) Prove that the function $f$ is periodic (i.e., there exists a positive number $b$ such that...

Practice › Mathematics › IMO › IMO 1968 ›

IMO 1968 Problem 4

IMO 1968 Problem 4 Proposed by: - Verified: no Verdicts: PASS + UNKNOWN Solve time: 18m05s Problem Prove that in every tetrahedron there is a vertex such that the three edges meeting there have lengths which are the sides of a triangle. Exploration Assume a tetrahedron with vertices $A,B,C,D$ satisfies the opposite of the claim, meaning that at every vertex the three incident edges fail to form the sides of...

Practice › Mathematics › IMO › IMO 1968 ›

IMO 1968 Problem 3

IMO 1968 Problem 3 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 9m13s Problem Consider the system of equations $$ ax_1^2 + bx_1 + c = x_2 $$ $$ ax_2^2 + bx_2 + c = x_3 $$ $$ \cdots $$ $$ ax_{n-1}^2 + bx_{n-1} + c = x_n $$ $$ ax_n^2 + bx_n + c = x_1 $$ with unknowns $x_1, x_2, \cdots, x_n$ where $a, b,...

Practice › Mathematics › IMO › IMO 1968 ›

IMO 1968 Problem 2

IMO 1968 Problem 2 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 4m57s Problem Find all natural numbers $x$ such that the product of their digits (in decimal notation) is equal to $x^2 - 10x - 22$. Exploration The expression on the right side is quadratic in $x$, while the left side is a product of decimal digits, hence grows at most exponentially in the number of...

Practice › Mathematics › IMO › IMO 1968 ›

IMO 1968 Problem 1

IMO 1968 Problem 1 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 4m22s Problem Prove that there is one and only one triangle whose side lengths are consecutive integers, and one of whose angles is twice as large as another. Exploration Let the triangle have consecutive integer side lengths. There exist integers $n \ge 2$ such that the sides are $n-1$, $n$, and $n+1$. Let the angle...

Practice › Mathematics › IMO › IMO 1967 ›

IMO 1967 Problem 6

IMO 1967 Problem 6 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 5m29s Problem In a sports contest, there were $m$ medals awarded on $n$ successive days ($n>1$). On the first day, one medal and $1/7$ of the remaining $m - 1$ medals were awarded. On the second day, two medals and $1/7$ of the now remaining medals were awarded; and so on. On the $n$-th and...

Practice › Mathematics › IMO › IMO 1967 ›

IMO 1967 Problem 5

IMO 1967 Problem 5 Proposed by: - Verified: no Verdicts: PASS + FAIL Solve time: 15m08s Problem Consider the sequence ${ c_n }$, where $$ c_1 = a_1 + a_2 + \cdots + a_8 $$ $$ c_2 = a_1^2 + a_2^2 + \cdots + a_8^2 $$ $$ \cdots $$ $$ c_n = a_1^n + a_2^n + \cdots + a_8^n $$ $$ \cdots $$ in which $a_1$, $a_2$, $\cdots$, $a_8$ are...

Practice › Mathematics › IMO › IMO 1967 ›

IMO 1967 Problem 4

IMO 1967 Problem 4 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 5m43s Problem Let $A_0 B_0 C_0$ and $A_1 B_1 C_1$ be any two acute-angled triangles. Consider all triangles $ABC$ that are similar to $\triangle A_1 B_1 C_1$ (so that vertices $A_1$, $B_1$, $C_1$ correspond to vertices $A$, $B$, $C$, respectively) and circumscribed about triangle $A_0 B_0 C_0$ (where $A_0$ lies on $BC$, $B_0$ on $CA$,...

Practice › Mathematics › IMO › IMO 1967 ›

IMO 1967 Problem 3

IMO 1967 Problem 3 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 6m57s Problem Let $k$, $m$, $n$ be natural numbers such that $m + k + 1$ is a prime greater than $n + 1$. Let $c_s = s(s + 1)$. Prove that the product $$ (c_{m+1} - c_k)(c_{m+2}- c_k)\cdots (c_{m+n}- c_k) $$ is divisible by the product $c_1 c_2\cdots c_n$. Exploration The expression factorises as...

Practice › Mathematics › IMO › IMO 1967 ›

IMO 1967 Problem 2

IMO 1967 Problem 2 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 4m21s Problem Prove that if one and only one edge of a tetrahedron is greater than $1$, then its volume is $\leq 1/8$. Exploration Let $AB$ be the unique edge of the tetrahedron exceeding $1$, while all other edges do not exceed $1$. The geometric intuition is that the four vertices cannot simultaneously spread far...

Practice › Mathematics › IMO › IMO 1967 ›

IMO 1967 Problem 1

IMO 1967 Problem 1 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 14m33s Problem Let $ABCD$ be a parallelogram with side lengths $AB = a$, $AD = 1$, and with $\angle BAD = \alpha$. If $\triangle ABD$ is acute, prove that the four circles of radius $1$ with centers $A$, $B$, $C$, $D$ cover the parallelogram if and only if $$ a\leq \cos \alpha + \sqrt{3} \sin...

Practice › Mathematics › IMO › IMO 1966 ›

IMO 1966 Problem 6

IMO 1966 Problem 6 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 6m43s Problem Let $ABC$ be a triangle, and let $P$, $Q$, $R$ be three points in the interiors of the sides $BC$, $CA$, $AB$ of this triangle. Prove that the area of at least one of the three triangles $AQR$, $BRP$, $CPQ$ is less than or equal to one quarter of the area of triangle...

Practice › Mathematics › IMO › IMO 1966 ›

IMO 1966 Problem 5

IMO 1966 Problem 5 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 8m43s Problem Solve the system of equations $$ |a_1-a_2|x_2+|a_1-a_3|x_3+|a_1-a_4|x_4=1 $$ $$ |a_2-a_1|x_1+|a_2-a_3|x_3+|a_2-a_4|x_4=1 $$ $$ |a_3-a_1|x_1+|a_3-a_2|x_2+|a_3-a_4|x_4=1 $$ $$ |a_4-a_1|x_1+|a_4-a_2|x_2+|a_4-a_3|x_3=1 $$ where $a_1, a_2, a_3, a_4$ are four different real numbers. Exploration Let $$d_{ij}=|a_i-a_j|.$$ The system can be written as $$\sum_{j\ne i} d_{ij}x_j=1 \qquad (i=1,2,3,4).$$ Since the coefficients are symmetric, the first idea is to subtract equations....

Practice › Mathematics › IMO › IMO 1966 ›

IMO 1966 Problem 4

IMO 1966 Problem 4 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 5m33s Problem Prove that for every natural number $n$, and for every real number $x \neq \frac{k\pi}{2^t}$ ($t=0,1, \dots, n$; $k$ any integer) $$ \frac{1}{\sin{2x}}+\frac{1}{\sin{4x}}+\dots+\frac{1}{\sin{2^nx}}=\cot{x}-\cot{2^nx} $$ Exploration The identity resembles a telescoping sum. The summands contain reciprocals of $\sin(2^k x)$, while the right-hand side contains cotangents at consecutive powers of $2$. The first task is...

Practice › Mathematics › IMO › IMO 1966 ›

IMO 1966 Problem 3

IMO 1966 Problem 3 Proposed by: - Verified: no Verdicts: FAIL + PASS Solve time: 6m20s Problem Prove that the sum of the distances of the vertices of a regular tetrahedron from the center of its circumscribed sphere is less than the sum of the distances of these vertices from any other point in space. Exploration The problem concerns a regular tetrahedron with vertices $A,B,C,D$ and circumcenter $O$. Since the...

Practice › Mathematics › IMO › IMO 1966 ›

IMO 1966 Problem 2

IMO 1966 Problem 2 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 6m50s Problem Let $a,b,c$ be the lengths of the sides of a triangle, and $\alpha, \beta, \gamma$ respectively, the angles opposite these sides. Prove that if $$ a+b=\tan{\frac{\gamma}{2}}(a\tan{\alpha}+b\tan{\beta}) $$ the triangle is isosceles. Exploration The condition mixes side lengths and tangents of angles. Since the angles and sides belong to the same triangle, the Law...

Practice › Mathematics › IMO › IMO 1966 ›

IMO 1966 Problem 1

IMO 1966 Problem 1 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 8m58s Problem In a mathematical contest, three problems, $A,B,C$ were posed. Among the participants there were 25 students who solved at least one problem each. Of all the contestants who did not solve problem $A$, the number who solved $B$ was twice the number who solved $C$. The number of students who solved only problem...

Practice › Mathematics › IMO › IMO 1965 ›

IMO 1965 Problem 6

IMO 1965 Problem 6 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 5m55s Problem In a plane a set of $n$ points ($n\geq 3$) is given. Each pair of points is connected by a segment. Let $d$ be the length of the longest of these segments. We define a diameter of the set to be any connecting segment of length $d$. Prove that the number of diameters...

Practice › Mathematics › IMO › IMO 1965 ›

IMO 1965 Problem 5

IMO 1965 Problem 5 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 7m42s Problem Consider $\triangle OAB$ with acute angle $AOB$. Through a point $M \neq O$ perpendiculars are drawn to $OA$ and $OB$, the feet of which are $P$ and $Q$ respectively. The point of intersection of the altitudes of $\triangle OPQ$ is $H$. What is the locus of $H$ if $M$ is permitted to range...

Practice › Mathematics › IMO › IMO 1965 ›

IMO 1965 Problem 4

IMO 1965 Problem 4 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 6m07s Problem Find all sets of four real numbers $x_1$, $x_2$, $x_3$, $x_4$ such that the sum of any one and the product of the other three is equal to $2$. Exploration Let $$x_1+x_2x_3x_4=2,\qquad x_2+x_1x_3x_4=2,$$ $$x_3+x_1x_2x_4=2,\qquad x_4+x_1x_2x_3=2.$$ The condition is completely symmetric in the four variables. Subtracting two equations immediately eliminates the constant $2$. For...

Practice › Mathematics › IMO › IMO 1965 ›

IMO 1965 Problem 3

IMO 1965 Problem 3 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 6m24s Problem Given the tetrahedron $ABCD$ whose edges $AB$ and $CD$ have lengths $a$ and $b$ respectively. The distance between the skew lines $AB$ and $CD$ is $d$, and the angle between them is $\omega$. Tetrahedron $ABCD$ is divided into two solids by plane $\varepsilon$, parallel to lines $AB$ and $CD$. The ratio of the...

Practice › Mathematics › IMO › IMO 1965 ›

IMO 1965 Problem 2

IMO 1965 Problem 2 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 5m48s Problem Consider the system of equations $$ a_{11}x_1 + a_{12}x_2 + a_{13}x_3 = 0 $$ $$ a_{21}x_1 + a_{22}x_2 + a_{23}x_3 = 0 $$ $$ a_{31}x_1 + a_{32}x_2 + a_{33}x_3 = 0 $$ with unknowns $x_1$, $x_2$, $x_3$. The coefficients satisfy the conditions: (a) $a_{11}$, $a_{22}$, $a_{33}$ are positive numbers; (b) the remaining coefficients...

Practice › Mathematics › IMO › IMO 1965 ›

IMO 1965 Problem 1

IMO 1965 Problem 1 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 8m00s Problem Determine all values $x$ in the interval $0\leq x\leq 2\pi$ which satisfy the inequality $$ 2\cos x \leq \left| \sqrt{1+\sin 2x} - \sqrt{1-\sin 2x } \right| \leq \sqrt{2}. $$ Exploration The expression involves $\sin 2x$ inside two square roots of complementary forms, suggesting the standard algebraic identity $$\sqrt{1+u}-\sqrt{1-u}=\frac{2u}{\sqrt{1+u}+\sqrt{1-u}}.$$ This converts the absolute value...

Practice › Mathematics › IMO › IMO 1964 ›

IMO 1964 Problem 6

IMO 1964 Problem 6 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 4m05s Problem In tetrahedron $ABCD$, vertex $D$ is connected with $D_0$, the centroid of $\triangle ABC$. Lines parallel to $DD_0$ are drawn through $A,B$ and $C$. These lines intersect the planes $BCD, CAD$ and $ABD$ in points $A_1, B_1,$ and $C_1$, respectively. Prove that the volume of $ABCD$ is one third the volume of $A_1B_1C_1D_0$....

Practice › Mathematics › IMO › IMO 1964 ›

IMO 1964 Problem 5

IMO 1964 Problem 5 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 4m06s Problem Suppose five points in a plane are situated so that no two of the straight lines joining them are parallel, perpendicular, or coincident. From each point perpendiculars are drawn to all the lines joining the other four points. Determine the maximum number of intersections that these perpendiculars can have. Exploration Five given points...

Practice › Mathematics › IMO › IMO 1964 ›

IMO 1964 Problem 4

IMO 1964 Problem 4 Proposed by: - Verified: no Verdicts: UNKNOWN + FAIL Solve time: 20m27s Problem Seventeen people correspond by mail with one another - each one with all the rest. In their letters only three different topics are discussed. Each pair of correspondents deals with only one of these topics. Prove that there are at least three people who write to each other about the same topic. Something...

Practice › Mathematics › IMO › IMO 1964 ›

IMO 1964 Problem 3

IMO 1964 Problem 3 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 10m26s Problem A circle is inscribed in a triangle $ABC$ with sides $a,b,c$. Tangents to the circle parallel to the sides of the triangle are contructed. Each of these tangents cuts off a triangle from $\triangle ABC$. In each of these triangles, a circle is inscribed. Find the sum of the areas of all four...

Practice › Mathematics › IMO › IMO 1964 ›

IMO 1964 Problem 2

IMO 1964 Problem 2 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 8m53s Problem Suppose $a, b, c$ are the sides of a triangle. Prove that $$ a^2(b+c-a)+b^2(c+a-b)+c^2(a+b-c)\le{3abc}. $$ Exploration The expression is symmetric and homogeneous of degree three, suggesting a normalization or a classical symmetric inequality. Expanding each term gives a combination of cubic and mixed products, which suggests Schur-type inequalities or rearrangements involving $a^3$, $b^3$,...

Practice › Mathematics › IMO › IMO 1964 ›

IMO 1964 Problem 1

IMO 1964 Problem 1 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 7m13s Problem (a) Find all positive integers $n$ for which $2^n-1$ is divisible by $7$. (b) Prove that there is no positive integer $n$ for which $2^n+1$ is divisible by $7$. Exploration The problem concerns divisibility of powers of $2$ by $7$, so the natural setting is arithmetic modulo $7$. Since divisibility by $7$ is...

Practice › Mathematics › IMO › IMO 1963 ›

IMO 1963 Problem 6

IMO 1963 Problem 6 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 16m15s Problem Five students, $A,B,C,D,E$, took part in a contest. One prediction was that the contestants would finish in the order $ABCDE$. This prediction was very poor. In fact no contestant finished in the position predicted, and no two contestants predicted to finish consecutively actually did so. A second prediction had the contestants finishing in...

Practice › Mathematics › IMO › IMO 1963 ›

IMO 1963 Problem 5

IMO 1963 Problem 5 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 19m23s Problem Prove that $\cos{\frac{\pi}{7}}-\cos{\frac{2\pi}{7}}+\cos{\frac{3\pi}{7}}=\frac{1}{2}$. Exploration Direct substitution with high-precision numerical evaluation confirms that $\cos\frac{\pi}{7}-\cos\frac{2\pi}{7}+\cos\frac{3\pi}{7}$ is extremely close to $0.5$, and perturbing each angle slightly destroys the cancellation pattern, indicating a rigid algebraic structure rather than a numerical coincidence. The previous approach failed because it attempted to import a cubic factor of $T_7(x)+1$ without proving...

Practice › Mathematics › IMO › IMO 1963 ›

IMO 1963 Problem 4

IMO 1963 Problem 4 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 9m04s Problem Find all solutions $x_1,x_2,x_3,x_4,x_5$ of the system $$ \begin{eqnarray*} x_5+x_2&=&yx_1\ x_1+x_3&=&yx_2\ x_2+x_4&=&yx_3\ x_3+x_5&=&yx_4\ x_4+x_1&=&yx_5,\end{eqnarray*} $$ where $y$ is a parameter. Exploration The system is a cyclic linear homogeneous system with five variables and a real parameter $y$. The structure suggests a circulant matrix, so complex Fourier modes $x_j=\omega^{kj}$ diagonalize it. This reduces the...

Practice › Mathematics › IMO › IMO 1963 ›

IMO 1963 Problem 3

IMO 1963 Problem 3 Proposed by: - Verified: no Verdicts: FAIL + UNKNOWN Solve time: 14m22s Problem In an $n$-gon all of whose interior angles are equal, the lengths of consecutive sides satisfy the relation $a_1\ge a_2\ge \cdots \ge a_n$. Prove that $a_1=a_2=\cdots = a_n$. Exploration The first issue is whether the hypothesis “consecutive sides satisfy $a_1 \ge a_2 \ge \cdots \ge a_n$” is purely linear or implicitly cyclic. In...

Practice › Mathematics › IMO › IMO 1963 ›

IMO 1963 Problem 2

IMO 1963 Problem 2 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 4m09s Problem Point $A$ and segment $BC$ are given. Determine the locus of points in space which are the vertices of right angles with one side passing through $A$, and the other side intersecting the segment $BC$. Exploration For a point $P$ in space, the condition that the angle with vertex at $P$ has one...

Practice › Mathematics › IMO › IMO 1963 ›

IMO 1963 Problem 1

IMO 1963 Problem 1 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 10m55s Problem Find all real roots of the equation $\sqrt{x^2-p}+2\sqrt{x^2-1}=x$, where $p$ is a real parameter. Exploration The equation $$\sqrt{x^2-p}+2\sqrt{x^2-1}=x$$ forces $x\ge 0$, and the second radical forces $x^2\ge 1$, hence $x\ge 1$. Testing boundary behavior gives a first consistency check. At $x=1$, the equation reduces to $\sqrt{1-p}=1$, so $p=0$. For slightly larger $x$, the...

Practice › Mathematics › IMO › IMO 1959 ›

IMO 1959 Problem 6

IMO 1959 Problem 6 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 15m55s Problem Two planes, $P$ and $Q$, intersect along the line $p$. The point $A$ is in the plane $P$, and the point ${C}$ is in the plane $Q$; neither of these points lies on the straight line $p$. Construct an isosceles trapezoid $ABCD$ (with $AB$ parallel to $DC$) in which a circle can be...

Practice › Mathematics › IMO › IMO 1959 ›

IMO 1959 Problem 5

IMO 1959 Problem 5 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 14m21s Problem An arbitrary point $M$ is selected in the interior of the segment $AB$. The squares $AMCD$ and $MBEF$ are constructed on the same side of $AB$, with the segments $AM$ and $MB$ as their respective bases. The circles about these squares, with respective centers $P$ and $Q$, intersect at $M$ and also at...

Practice › Mathematics › IMO › IMO 1959 ›

IMO 1959 Problem 4

IMO 1959 Problem 4 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 7m25s Problem Construct a right triangle with a given hypotenuse $c$ such that the median drawn to the hypotenuse is the geometric mean of the two legs of the triangle. Exploration The previous approach failed because it tried to manufacture several derived lengths from $c$ without a coherent straightedge-and-compass dependency chain and without a well-defined...

Practice › Mathematics › IMO › IMO 1959 ›

IMO 1959 Problem 3

IMO 1959 Problem 3 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 4m43s Problem Let $a,b,c$ be real numbers. Consider the quadratic equation in $\cos{x}$ : $a\cos ^{2}x + b\cos{x} + c = 0.$ Using the numbers $a,b,c$, form a quadratic equation in $\cos{2x}$, whose roots are the same as those of the original equation. Compare the equations in $\cos{x}$ and $\cos{2x}$ for $a=4, b=2, c=-1$. Exploration...

Practice › Mathematics › IMO › IMO 1959 ›

IMO 1959 Problem 2

IMO 1959 Problem 2 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 4m19s Problem For what real values of $x$ is $\sqrt{x+\sqrt{2x-1}}+\sqrt{x-\sqrt{2x-1}}=A,$ given (a) $A = \sqrt{2}$, (b) $A=1$, (c) $A=2$, where only non-negative real numbers are admitted for square roots? Exploration Let $$S=\sqrt{x+\sqrt{2x-1}}+\sqrt{x-\sqrt{2x-1}}.$$ The domain requires $2x-1\ge 0$, hence $x\ge \tfrac12$. Both radicals are real when additionally $x\pm \sqrt{2x-1}\ge 0$. Since $x\ge \tfrac12$, the quantity $x-\sqrt{2x-1}$...

Practice › Mathematics › IMO › IMO 1972 ›

IMO 1972 Problem 5

IMO 1972 Problem 5 Proposed by: - Verified: no Verdicts: FAIL + FAIL Solve time: 14m14s Problem Let $f$ and $g$ be real-valued functions defined for all real values of $x$ and $y$, and satisfying the equation $$ f(x + y) + f(x - y) = 2f(x)g(y) $$ for all $x, y$. Prove that if $f(x)$ is not identically zero, and if $|f(x)| \leq 1$ for all $x$, then $|g(y)|...

Practice › Mathematics › IMO › IMO 1962 ›

IMO 1962 Problem 7

IMO 1962 Problem 7 Proposed by: - Verified: no Verdicts: PASS + FAIL Solve time: 15m54s Problem Prove that a tetrahedron $SABC$ has five spheres each tangent to all six lines determined by its edges if and only if it is regular. Exploration The previous solution failed at the structural point where it tried to characterize tetrahedra admitting a sphere tangent to all six edge-lines. Before rebuilding the argument, it...

Practice › Mathematics › IMO › IMO 1962 ›

IMO 1962 Problem 5

IMO 1962 Problem 5 Proposed by: - Verified: no Verdicts: UNKNOWN + UNKNOWN Solve time: 12m07s Problem On a circle $k$ three points $A$, $B$, $C$ are given. Construct a fourth point $D$ on $k$ such that a circle can be inscribed in quadrilateral $ABCD$. Exploration The problem is a construction problem on a fixed circumcircle. Since the quadrilateral must admit an incircle, the first condition to examine is the...

Practice › Mathematics › IMO › IMO 1962 ›

IMO 1962 Problem 4

IMO 1962 Problem 4 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 21m14s Problem Solve the equation $\cos^2 x + \cos^2 2x + \cos^2 3x = 1$. Exploration Consider the equation $\cos^2 x + \cos^2 2x + \cos^2 3x = 1.$ Using standard trigonometric identities, we have $\cos 2x = 2\cos^2 x - 1$ and $\cos 3x = 4\cos^3 x - 3\cos x$. Squaring these, we obtain...

Practice › Mathematics › IMO › IMO 1962 ›

IMO 1962 Problem 2

IMO 1962 Problem 2 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 25m30s Problem Find all real numbers $x$ satisfying $$\sqrt{3-x} - \sqrt{x+1} > \frac{1}{2}.$$ Problem Understanding We seek all real numbers $x$ satisfying $$\sqrt{3-x}-\sqrt{x+1}>\frac12.$$ Because square roots are involved, the first step is to determine the domain. After that, the natural strategy is to study the related equation $$\sqrt{3-x}-\sqrt{x+1}=\frac12,$$ since the inequality is expected to change...

Practice › Mathematics › IMO › IMO 1962 ›

IMO 1962 Problem 1

IMO 1962 Problem 1 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 7m17s Problem Find the smallest natural number $n$ with the following properties: (a) Its decimal representation ends in 6. (b) If the final digit 6 is moved to the front, the resulting number is four times as large. Exploration Let $n$ be a natural number ending in 6 such that moving the 6 to the...

Practice › Mathematics › IMO › IMO 1961 ›

IMO 1961 Problem 5

IMO 1961 Problem 5 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 39m11s Problem Construct a triangle ABC if the following elements are given: $AC = b, AB = c$, and $\angle AMB = \omega \left(\omega < 90^{\circ}\right)$ where M is the midpoint of BC . Prove that the construction has a solution if and only if $b \tan{\frac{\omega}{2}} \le c < b$ In what case does...

Practice › Mathematics › IMO › IMO 1961 ›

IMO 1961 Problem 4

IMO 1961 Problem 4 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 19m03s Problem In the interior of triangle $P_1 P_2 P_3$ a point P is given. Let $Q_1,Q_2,Q_3$ be the intersections of $PP_1, PP_2,PP_3$ with the opposing edges of triangle $ABC$. Prove that among the ratios $\frac{PP_1}{PQ_1},\frac{PP_2}{PQ_2},\frac{PP_3}{PQ_3}$ there exists one not larger than 2 and one not smaller than 2. Exploration The problem is a pure...

Practice › Mathematics › IMO › IMO 1961 ›

IMO 1961 Problem 2

IMO 1961 Problem 2 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 11m24s Problem Let a , b , and c be the lengths of a triangle whose area is S . Prove that $a^2 + b^2 + c^2 \ge 4S\sqrt{3}$ In what case does equality hold? Exploration The statement asks for a lower bound on $a^2+b^2+c^2$ in terms of the area $S$ of a triangle. Since...

Practice › Mathematics › IMO › IMO 1961 ›

IMO 1961 Problem 1

IMO 1961 Problem 1 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 4m48s Problem ( Hungary ) Solve the system of equations: $\begin{matrix} \quad x + y + z !!! &= a ; , \ x^2 +y^2+z^2 !!! &=b^2 \ \qquad \qquad xy !!! &= z^2 \end{matrix}$ where $a$ and $b$ are constants. Give the conditions that $a$ and $b$ must satisfy so that $x, y, z$...

Practice › Mathematics › IMO › IMO 1960 ›

IMO 1960 Problem 7

IMO 1960 Problem 7 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 5m59s Problem A regular cone is inscribed in a sphere. Around the sphere a cylinder is circumscribed so that its base lies in the same plane as the base of the cone. Let $V_1$ be the volume of the cone and $V_2$ that of the cylinder. (a) Prove that $V_1 = V_2$ is impossible. (b)...

Practice › Mathematics › IMO › IMO 1960 ›

IMO 1960 Problem 4

IMO 1960 Problem 4 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 33m08s Problem Construct a triangle $ABC$ given the lengths of the altitudes $h_a$ and $h_b$ (from $A$ and $B$ respectively) and the length of the median $m_a$ from $A$. Exploration The previous solution failed at two genuinely important points. The first was the incorrect geometric construction of $\sqrt{m_a^2-h_a^2}$. The second was the unsupported claim that...

Practice › Mathematics › IMO › IMO 1960 ›

IMO 1960 Problem 3

IMO 1960 Problem 3 Proposed by: - Verified: yes Verdicts: PASS + PASS Solve time: 17m36s Problem A right-angled triangle $ABC$ is given for which the hypotenuse $BC$ has length $a$ and is divided into $n$ equal segments ($n$ odd). Let $\alpha$ be the angle subtended at $A$ by the segment containing the midpoint of $BC$. If $h$ denotes the altitude from $A$, prove that $$\tan\alpha = \frac{4nh}{(n^2-1)a}.$$ Exploration The...

Practice › Mathematics › IMO Shortlist › IMO 1992 Shortlist ›

IMO 1992 SL 5

IMO 1992 SL 5 Origin: COL Problem Let ABCD be a convex quadrilateral such that AC = BD. Equilateral triangles are constructed on the sides of the quadrilat- eral. Let O1, O2, O3, O4 be the centers of the triangles constructed on AB, BC, CD, DA respectively. Show that O1O3 is perpendicular to O2O4. Solution Denote by K, L, M, and N the midpoints of the sides AB, BC, CD,...

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 28

IMO 1990 SL 28 Origin: USS Problem Prove that on the coordinate plane it is impossible to draw a closed broken line such that (i) the coordinates of each vertex are rational; (ii) the length each of its edges is 1; (iii) the line has an odd number of vertices. Solution Let us ﬁrst prove the following lemma. Lemma. Let (b′/a′, d′/c′) and (b′′/a′′, d′′/c′′) be two points with rational...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 5

IMO 1979 SL 5 Origin: CZS Problem Let n \geq2 be an integer. Find the maximal cardinality of a set M of pairs (j, k) of integers, 1 \leqj < k \leqn, with the following property: If (j, k) \inM, then (k, m) ̸\inM for any m. Solution Let A = {x | (x, y) \inM} and B = {y | (x, y) \inM. Then A and B are disjoint...

Practice › Mathematics › IMO Shortlist › IMO 1972 Shortlist ›

IMO 1972 SL 10

IMO 1972 SL 10 Origin: NET Problem Prove that for each n \geq4 every cyclic quadrilateral can be decomposed into n cyclic quadrilaterals. Solution Consider ﬁrst a triangle. It can be decomposed into k = 3 cyclic quadri- laterals by perpendiculars from some interior point of it to the sides; also, it can be decomposed into a cyclic quadrilateral and a triangle, and it follows by induction that this decomposition...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL N7

IMO 2004 SL N7 Origin: BUL | Category: Number Theory Problem Let p be an odd prime and n a positive integer. In the coordinate plane, eight distinct points with integer coordinates lie on a circle with diameter of length pn. Prove that there exists a triangle with vertices at three of the given points such that the squares of its side lengths are integers divisible by pn+1. Solution We...

Practice › Mathematics › IMO Shortlist › IMO 1975 Shortlist ›

IMO 1975 SL 6

IMO 1975 SL 6 Origin: USS Problem Let A be the sum of the digits of the number 1616 and B the sum of the digits of the number A. Find the sum of the digits of the number B without calculating 1616. Solution Let us denote by C the sum of digits of B. We know that 1616 \equivA \equiv B \equivC (mod 9). Since 1616 = 264 =...

Practice › Mathematics › IMO Shortlist › IMO 1976 Shortlist ›

IMO 1976 SL 4

IMO 1976 SL 4 Problem (GBR 1a)IMO6 For all positive integral n, un+1 = un(u2 n−1 −2) −u1, u0 = 2, and u1 = 5/2. Prove that 3 log2[un] = 2n −(−1)n, where [x] is the integral part of x.

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 10

IMO 1990 SL 10 Origin: ICE Problem A plane cuts a right circular cone into two parts. The plane is tangent to the circumference of the base of the cone and passes through the midpoint of the altitude. Find the ratio of the volume of the smaller part to the volume of the whole cone. Solution Let r be the radius of the base and h the height of the...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 3

IMO 1979 SL 3 Origin: BUL Problem Find all polynomials f(x) with real coeﬃcients for which f(x)f(2x2) = f(2x3 + x). Solution An obvious solution is f(x) = 0. We now look for nonzero solutions. We note that plugging in x = 0 we get f(0)2 = f(0); hence f(0) = 0 or f(0) = 1. If f(0) = 0, then f is of the form f(x) = xkg(x), where...

Practice › Mathematics › IMO Shortlist › IMO 1976 Shortlist ›

IMO 1976 SL 1

IMO 1976 SL 1 Origin: BUL Problem Let ABC be a triangle with bisectors AA1, BB1, CC1 (A1 \in BC, etc.) and M their common point. Consider the triangles MB1A, MC1A, MC1B, MA1B, MA1C, MB1C, and their inscribed circles. Prove that if four of these six inscribed circles have equal radii, then AB = BC = CA. Solution Let r denote the common inradius. Some two of the four triangles...

Practice › Mathematics › IMO Shortlist › IMO 1982 Shortlist ›

IMO 1982 SL 13

IMO 1982 SL 13 Problem C1 (NET 1)IMO2 A scalene triangle A1A2A3 is given with sides a1, a2, a3 (ai is the side opposite to Ai). For all i = 1, 2, 3, Mi is the midpoint of side ai, Ti is the point where the incircle touches side ai, and the reﬂection of Ti in the interior bisector of Ai yields the point Si. Prove that the lines M1S1,...

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL C3

IMO 2003 SL C3 Origin: LIT | Category: Combinatorics Problem Let n \geq5 be a given integer. Determine the largest integer k for which there exists a polygon with n vertices (convex or not, with non-self-intersecting boundary) having k internal right angles. Solution Suppose that k of the angles of an n-gon are right. Since the other n −k angles are less than 360◦and the sum of the angles is...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 16

IMO 1979 SL 16 Origin: ISR Problem Let K denote the set {a, b, c, d, e}. F is a collection of 16 diﬀerent subsets of K, and it is known that any three members of F have at least one element in common. Show that all 16 members of F have exactly one element in common. Solution Obviously, no two elements of F can be complements of each other....

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 26

IMO 1989 SL 26 Origin: KOR Problem Let n be a positive integer and let a, b be given real numbers. Determine the range of x0 for which n i=0 xi = a and n i=0 x2 i = b, where x0, x1, . . . , xn are real variables. Solution By the Cauchy–Schwarz inequality, n i=1 xi \leqn n i=1 x2 i . Since...

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL C2

IMO 2003 SL C2 Origin: GEO | Category: Combinatorics Problem Let D1, . . . , Dn be closed disks in the plane. (A closed disk is a region bounded by a circle, taken jointly with this circle.) Suppose that every point in the plane is contained in at most 2003 disks Di. Prove that there exists disk Dk that intersects at most 7 \cdot 2003 −1 other disks Di....

Practice › Mathematics › IMO Shortlist › IMO 1978 Shortlist ›

IMO 1978 SL 12

IMO 1978 SL 12 Origin: USA Problem In a triangle ABC we have AB = AC. A circle is tangent internally to the circumcircle of ABC and also to the sides AB, AC, at P, Q respectively. Prove that the midpoint of PQ is the center of the incircle of ABC. Solution Let O be the center of the smaller circle, T its contact point with the circumcircle of ABC,...

Practice › Mathematics › IMO Shortlist › IMO 1994 Shortlist ›

IMO 1994 SL A5

IMO 1994 SL A5 Origin: POL | Category: Algebra Problem Let f(x) = x2+1 2x for x ̸= 0. Deﬁne f (0)(x) = x and f (n)(x) = f(f (n−1)(x)) for all positive integers n and x ̸= 0. Prove that for all nonnegative integers n and x ̸= −1, 0, or 1, f (n)(x) f (n+1)(x) = 1 + f x+1 x−1 2n. Solution If f (n)(x) =...

Practice › Mathematics › IMO Shortlist › IMO 1987 Shortlist ›

IMO 1987 SL 6

IMO 1987 SL 6 Origin: GRE Problem Show that if a, b, c are the lengths of the sides of a triangle and if 2S = a + b + c, then an b + c + bn c + a + cn a + b \geq 2 n−2 Sn−1, n \geq1. Solution Suppose w.l.o.g. that a \geqb \geqc. Then 1/(b + c) \geq1/(a + c) \geq1/(a + b). Chebyshev’s...

Practice › Mathematics › IMO Shortlist › IMO 1970 Shortlist ›

IMO 1970 SL 5

IMO 1970 SL 5 Origin: CZS Problem Let M be an interior point of the tetrahedron ABCD. Prove that −−\to MA vol(MBCD) + −−\to MB vol(MACD) +−−\to MC vol(MABD) + −−\to MD vol(MABC) = 0 (vol(PQRS) denotes the volume of the tetrahedron PQRS). Solution Denote respectively by A1, B1, C1 and D1 the points of intersection of the lines AM, BM, CM, and DM with the opposite sides of the...

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL A4

IMO 2001 SL A4 Origin: LIT | Category: Algebra Problem Find all functions f : R \toR satisfying f(xy)(f(x) −f(y)) = (x −y)f(x)f(y) for all x, y. Solution Let (∗) denote the given functional equation. Substituting y = 1 we get f(x)2 = xf(x)f(1). If f(1) = 0, then f(x) = 0 for all x, which is the trivial solution. Suppose f(1) = C ̸= 0. Let G = {y...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL C7

IMO 1996 SL C7 Origin: IRE | Category: Combinatorics Problem Let $U$ be a finite set and let $f$, $g$ be bijective functions from $U$ onto itself. Let $$S={w\in U:f(f(w))=g(g(w))},$$ $$T={w\in U:f(g(w))=g(f(w))},$$ and suppose that $U=S\cup T$. Prove that for $w\in U$, $f(w)\in S$ if and only if $g(w)\in S$. Solution For convenience, we shall write $f^2$, $fg$, $\ldots$ for the functions $f\circ f$, $f\circ g$, $\ldots$. We need two...

Practice › Mathematics › IMO Shortlist › IMO 1968 Shortlist ›

IMO 1968 SL 6

IMO 1968 SL 6 Origin: HUN Problem If $a_i$ $(i = 1, 2, \ldots, n)$ are distinct non-zero real numbers, prove that the equation $$\frac{a_1}{a_1 - x} + \frac{a_2}{a_2 - x} + \cdots + \frac{a_n}{a_n - x} = n$$ has at least $n - 1$ real roots. Solution We define $$f(x) = \frac{a_1}{a_1 - x} + \frac{a_2}{a_2 - x} + \cdots + \frac{a_n}{a_n - x}.$$ Let us assume w.l.o.g. $$a_1...

Practice › Mathematics › IMO Shortlist › IMO 1981 Shortlist ›

IMO 1981 SL 2

IMO 1981 SL 2 Origin: BUL Problem A sphere S is tangent to the edges AB, BC, CD, DA of a tetrahe- dron ABCD at the points E, F, G, H respectively. The points E, F, G, H are the vertices of a square. Prove that if the sphere is tangent to the edge AC, then it is also tangent to the edge BD. Solution Lemma. Let E, F, G,...

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL C5

IMO 1999 SL C5 Origin: BLR | Category: Combinatorics Problem Let n be an even positive integer. We say that two dif- ferent cells of an n \times n board are neighboring if they have a common side. Find the minimal number of cells on the n\timesn board that must be marked so that every cell (marked or not marked) has a marked neighboring cell. Solution Let n = 2k....

Practice › Mathematics › IMO Shortlist › IMO 1968 Shortlist ›

IMO 1968 SL 14

IMO 1968 SL 14 Origin: BUL Problem A line in the plane of a triangle $ABC$ intersects the sides $AB$ and $AC$ respectively at points $X$ and $Y$ such that $BX = CY$. Find the locus of the center of the circumcircle of triangle $XAY$. Solution Let $O$, $D$, $E$ be the circumcenter of $\triangle ABC$ and the midpoints of $AB$ and $AC$, and given arbitrary $X \in AB$ and...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 7

IMO 1979 SL 7 Origin: FRG Problem Given that 1 −1 2 + 1 3 −1 4 + \cdot \cdot \cdot − 1318 + 1319 = p q , where p and q are natural numbers having no common factor, prove that p is divisible by 1979. Solution We denote the sum mentioned above by S. We have the following equali- ties: S = 1 −1 2 + 1 3...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 24

IMO 1991 SL 24 Origin: IND Problem An odd integer n \geq3 is said to be “nice” if there is at least one permutation a1, a2, . . . , an of 1, 2, . . ., n such that the n sums a1 −a2 + a3 −\cdot \cdot \cdot −an−1 + an, a2 −a3 + a4 −\cdot \cdot \cdot −an + a1, a3 −a4 + a5 −\cdot \cdot \cdot...

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 15

IMO 1990 SL 15 Origin: MEX Problem Determine for which positive integers k the set X = {1990, 1990 + 1, 1990 + 2, . . . , 1990 + k} can be partitioned into two disjoint subsets A and B such that the sum of the elements of A is equal to the sum of the elements of B. Solution Let S(Z) denote the sum of all the elements...

Practice › Mathematics › IMO Shortlist › IMO 1984 Shortlist ›

IMO 1984 SL 10

IMO 1984 SL 10 Origin: GBR Problem Prove that the product of ﬁve consecutive positive integers cannot be the square of an integer. Solution Suppose that the product of some ﬁve consecutive numbers is a square. It is easily seen that among them at least one, say n, is divisible neither by 2 nor 3. Since n is coprime to the remaining four numbers, it is itself a square of...

Practice › Mathematics › IMO Shortlist › IMO 1986 Shortlist ›

IMO 1986 SL 13

IMO 1986 SL 13 Origin: FRG Problem A particle moves from (0, 0) to (n, n) directed by a fair coin. For each head it moves one step east and for each tail it moves one step north. At (n, y), y < n, it stays there if a head comes up and at (x, n), x < n, it stays there if a tail comes up. Let k be...

Practice › Mathematics › IMO Shortlist › IMO 1976 Shortlist ›

IMO 1976 SL 9

IMO 1976 SL 9 Origin: FIN Problem Let P1(x) = x2 −2, Pj(x) = P1(Pj−1(x)), j = 2, 3, . . . . Show that for arbitrary n the roots of the equation Pn(x) = x are real and diﬀerent from one another. Solution The equation Pn(x) = x is of degree 2n, and has at most 2n distinct roots. If x > 2, then by simple induction Pn(x) >...

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL C1

IMO 2002 SL C1 Origin: COL | Category: Combinatorics Problem Let n be a positive integer. Each point (x, y) in the plane, where x and y are nonnegative integers with x + y \leqn, is colored red or blue, subject to the following condition: If a point (x, y) is red, then so are all points (x′, y′) with x′ \leqx and y′ \leqy. Let A be the number...

Practice › Mathematics › IMO Shortlist › IMO 1985 Shortlist ›

IMO 1985 SL 11

IMO 1985 SL 11 Problem 3a.(USS 3) Find a method by which one can compute the coeﬃcients of P(x) = x6 + a1x5 + \cdot \cdot \cdot + a6 from the roots of P(x) = 0 by performing not more than 15 additions and 15 multiplications. Solution Let −x1, . . . , −x6 be the roots of the polynomial. Let sk,i (k \leqi \leq6) denote the sum of all...

Practice › Mathematics › IMO Shortlist › IMO 1983 Shortlist ›

IMO 1983 SL 20

IMO 1983 SL 20 Origin: ROM Problem Solve the system of equations x1|x1| = x2|x2| + (x1 −a)|x1 −a|, x2|x2| = x3|x3| + (x2 −a)|x2 −a|, \cdot \cdot \cdot xn|xn| = x1|x1| + (xn −a)|xn −a|, in the set of real numbers, where a > 0. Solution If (x1, x2, . . . , xn) satisﬁes the system with parameter a, then (−x1, −x2, . . . , −xn) satisﬁes...

Practice › Mathematics › IMO Shortlist › IMO 1974 Shortlist ›

IMO 1974 SL 12

IMO 1974 SL 12 Problem II 6 (USS 1) In a certain language words are formed using an alphabet of three letters. Some words of two or more letters are not allowed, and any two such distinct words are of diﬀerent lengths. Prove that one can form a word of arbitrary length that does not contain any nonallowed word. Solution We say that a word is good if it doesn’t...

Practice › Mathematics › IMO Shortlist › IMO 1970 Shortlist ›

IMO 1970 SL 4

IMO 1970 SL 4 Origin: CZS Problem For what natural numbers n can the product of some of the numbers n, n + 1, n + 2, n + 3, n + 4, n + 5 be equal to the product of the remaining ones? Solution Suppose that n is such a natural number. If a prime number p divides any of the numbers n, n+1, . . ., n+5,...

Practice › Mathematics › IMO Shortlist › IMO 1981 Shortlist ›

IMO 1981 SL 19

IMO 1981 SL 19 Origin: YUG Problem A ﬁnite set of unit circles is given in a plane such that the area of their union U is S. Prove that there exists a subset of mutually disjoint circles such that the area of their union is greater that 2S 9 . Solution Consider the partition of plane \pi into regular hexagons, each having in- radius 2. Fix one of these...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL G5

IMO 2004 SL G5 Origin: SMN | Category: Geometry Problem Let A1A2 . . . An be a regular n-gon. The points B1, . . . , Bn−1 are deﬁned as follows: (i) If i = 1 or i = n −1, then Bi is the midpoint of the side AiAi+1. (ii) If i ̸= 1, i ̸= n −1, and S is the intersection point of A1Ai+1 and AnAi,...

Practice › Mathematics › IMO Shortlist › IMO 1971 Shortlist ›

IMO 1971 SL 5

IMO 1971 SL 5 Origin: HUN Problem Let a, b, c, d, e be real numbers. Prove that the expression (a−b)(a−c)(a−d)(a−e)+(b−a)(b−c)(b−d)(b−e)+(c−a)(c−b)(c−d)(c−e) (d −a)(d −b)(d −c)(d −e) + (e −a)(e −b)(e −c)(e −d) is nonnegative. Solution Without loss of generality, we may assume that a \geqb \geqc \geqd \geqe. Then a −b = −(b −a) \geq0, a −c \geqb −c \geq0, a −d \geqb −d \geq0 and a −e \geqb −e...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL G1

IMO 2004 SL G1 Origin: ROM | Category: Geometry Problem Let ABC be an acute-angled triangle with AB ̸= AC. The circle with diameter BC intersects the sides AB and AC at M and N, respectively. Denote by O the midpoint of BC. The bisectors of the angles BAC and MON intersect at R. Prove that the circumcircles of the triangles BMR and CNR have a common point lying on...

Practice › Mathematics › IMO Shortlist › IMO 1982 Shortlist ›

IMO 1982 SL 2

IMO 1982 SL 2 Problem A2 (YUG 1) Let K be a convex polygon in the plane and suppose that K is positioned in the coordinate system in such a way that area (K \capQi) = 1 4 area K (i = 1, 2, 3, 4, ), where the Qi denote the quadrants of the plane. Prove that if K contains no nonzero lattice point, then the area of K...

Practice › Mathematics › IMO Shortlist › IMO 1972 Shortlist ›

IMO 1972 SL 7

IMO 1972 SL 7 Origin: GBR Problem (a) A plane \pi passes through the vertex O of the regular tetrahedron OPQR. We deﬁne p, q, r to be the signed distances of P, Q, R from \pi measured along a directed normal to \pi. Prove that p2 + q2 + r2 + (q −r)2 + (r −p)2 + (p −q)2 = 2a2, where a is the length of an edge...

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 24

IMO 1998 SL 24 Origin: SWE Problem Cards numbered 1 to 9 are arranged at random in a row. In a move, one may choose any block of consecutive cards whose numbers are in ascending or descending order, and switch the block around. For example, 916532748 may be changed to 913562748. Prove that in at most 12 moves, one can arrange the 9 cards so that their numbers are in...

Practice › Mathematics › IMO Shortlist › IMO 1985 Shortlist ›

IMO 1985 SL 19

IMO 1985 SL 19 Origin: ISR Problem For which integers n \geq3 does there exist a regular n-gon in the plane such that all its vertices have integer coordinates in a rectangular coordinate system? Solution Suppose that for some n > 6 there is a regular n-gon with vertices having integer coordinates, and that A1A2 . . . An is the smallest such n-gon, of side length a. If O...

Practice › Mathematics › IMO Shortlist › IMO 1973 Shortlist ›

IMO 1973 SL 2

IMO 1973 SL 2 Origin: CZS Problem Given a circle K, ﬁnd the locus of vertices A of parallelograms ABCD with diagonals AC \leqBD, such that BD is inside K. Solution Let D′ be the reﬂection of D across A. Since BCAD′ is then a parallel- ogram, the condition BD \geqAC is equivalent to BD \geqBD′, which is in turn equivalent to \angleBAD \geq\angleBAD′, i.e. to \angleBAD \geq90◦. Thus the...

Practice › Mathematics › IMO Shortlist › IMO 1970 Shortlist ›

IMO 1970 SL 2

IMO 1970 SL 2 Origin: ROM Problem Let a and b be the bases of two number systems and let An = x1x2 . . . xn (a), An+1 = x0x1x2 . . . xn (a), Bn = x1x2 . . . xn(b), Bn+1 = x0x1x2 . . . xn(b), be numbers in the number systems with respective bases a and b, so that x0, x1, x2, . . ....

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 16

IMO 1989 SL 16 Origin: ISR Problem The set {a0, a1, . . . , an} of real numbers satisﬁes the following conditions: (i) a0 = an = 0; (ii) for 1 \leqk \leqn −1, ak = c + n−1 i=k ai−k(ai + ai+1). Prove that c \leq 4n. Solution Deﬁne Sk = k i=0 ai (k = 0, 1, . . . , n) and S−1 = 0....

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 15

IMO 1998 SL 15 Origin: AUS Problem Determine all pairs (a, b) of real numbers such that a\lfloorbn\rfloor= b\lflooran\rfloor for all positive integers n. (Note that \lfloorx\rfloordenotes the greatest integer less than or equal to x.) Solution The condition is obviously satisﬁed if a = 0 or b = 0 or a = b or a, b are both integers. We claim that these are the only solutions. Suppose that...

Practice › Mathematics › IMO Shortlist › IMO 1997 Shortlist ›

IMO 1997 SL 23

IMO 1997 SL 23 Origin: GBR Problem Let ABCD be a convex quadrilateral and O the intersection of its diagonals AC and BD. If OA sin \angleA + OC sin \angleC = OB sin \angleB + OD sin \angleD, prove that ABCD is cyclic. Solution Let K, L, M, and N be the projections of O onto the lines AB, BC, CD, and DA, and let \alpha1, \alpha2, \alpha3, \alpha4,...

Practice › Mathematics › IMO Shortlist › IMO 1993 Shortlist ›

IMO 1993 SL 15

IMO 1993 SL 15 Origin: MCD Problem For three points A, B, C in the plane we deﬁne m(ABC) to be the smallest length of the three altitudes of the triangle ABC, where in the case of A, B, C collinear, m(ABC) = 0. Let A, B, C be given points in the plane. Prove that for any point X in the plane, m(ABC) \leqm(ABX) + m(AXC) + m(XBC). Solution...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 7

IMO 1989 SL 7 Origin: FIN Problem Show that any two points lying inside a regular n-gon E can be joined by two circular arcs lying inside E and meeting at an angle of at least 1 −2 n \pi. Solution Assume that P1 and P2 are points inside E, and that the line P1P2 inter- sects the perimeter of E at Q1 and Q2. If we prove...

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 13

IMO 1990 SL 13 Origin: IRE Problem An eccentric mathematician has a ladder with n rungs that he always ascends and descends in the following way: When he ascends, each step he takes covers a rungs of the ladder, and when he descends, each step he takes covers b rungs of the ladder, where a and b are ﬁxed positive integers. By a sequence of ascending and descending steps he...

Practice › Mathematics › IMO Shortlist › IMO 1997 Shortlist ›

IMO 1997 SL 17

IMO 1997 SL 17 Origin: CZE Problem Find all pairs of integers x, y \geq1 satisfying the equation xy2 = yx. Solution We note ﬁrst that x and y must be powers of the same positive integer. Indeed, if x = p\alpha1 1 \cdot \cdot \cdot p\alphak k and y = p\beta1 1 \cdot \cdot \cdot p\betak k (some of \alphai and \betai may be 0, but not both for...

Practice › Mathematics › IMO Shortlist › IMO 1970 Shortlist ›

IMO 1970 SL 9

IMO 1970 SL 9 Origin: GDR Problem Let u1, u2, . . . , un, v1, v2, . . . , vn be real numbers. Prove that 1 + n i=1 (ui + vi)2 \leq4 1 + n i=1 u2 i 1 + n i=1 v2 i . In what case does equality hold? Solution Let us set a = n i=1 u2 i and b =...

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL G2

IMO 2003 SL G2 Origin: GRE | Category: Geometry Problem Three distinct points A, B, C are ﬁxed on a line in this order. Let \Gamma be a circle passing through A and C whose center does not lie on the line AC. Denote by P the intersection of the tangents to \Gamma at A and C. Suppose \Gamma meets the segment PB at Q. Prove that the intersection of...

Practice › Mathematics › IMO Shortlist › IMO 1983 Shortlist ›

IMO 1983 SL 5

IMO 1983 SL 5 Origin: BRA Problem Consider the set of all strictly decreasing sequences of n natural numbers having the property that in each sequence no term divides any other term of the sequence. Let A = (aj) and B = (bj) be any two such sequences. We say that A precedes B if for some k, ak < bk and ai = bi for i < k. Find...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 12

IMO 1988 SL 12 Origin: GRE Problem In a triangle ABC, choose any points K \inBC, L \inAC, M \inAB, N \inLM, R \inMK, and F \inKL. If E1, E2, E3, E4, E5, E6, and E denote the areas of the triangles AMR, CKR, BKF, ALF, BNM, CLN, and ABC respectively, show that E \geq8 E1E2E3E4E5E6. Remark. Points K, L, M, N, R, F lie on segments BC, AC, AB,...

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL A1

IMO 2000 SL A1 Origin: USA | Category: Algebra Problem Let a, b, c be positive real numbers with product 1. Prove that a −1 + 1 b b −1 + 1 c c −1 + 1 a \leq1. Solution Elementary computation gives a −1 + 1 b b −1 + 1 c = ab−a+ a c −b+ 1 −1...

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL N2

IMO 2000 SL N2 Origin: FRA | Category: Number Theory Problem For a positive integer n, let d(n) be the number of all positive divisors of n. Find all positive integers n such that d(n)3 = 4n. Solution Let n = p\alpha1 1 p\alpha2 2 \cdot \cdot \cdot p\alphak k be the factorization of n onto primes (p1 < p2 < \cdot \cdot \cdot < pk). Since 4n is a...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL G4

IMO 2004 SL G4 Origin: POL | Category: Geometry Problem In a convex quadrilateral ABCD the diagonal BD does not bisect the angles ABC and CDA. The point P lies inside ABCD and satisﬁes \anglePBC = \angleDBA and \anglePDC = \angleBDA. Prove that ABCD is a cyclic quadrilateral if and only if AP = CP. Solution Assume ﬁrst that the points A, B, C, D are concyclic. Let the lines...

Practice › Mathematics › IMO Shortlist › IMO 1968 Shortlist ›

IMO 1968 SL 12

IMO 1968 SL 12 Origin: POL Problem If $a$ and $b$ are arbitrary positive real numbers and $m$ an integer, prove that $$\left(1 + \frac{a}{b}\right)^m + \left(1 + \frac{b}{a}\right)^m \geq 2^{m+1}.$$ Solution The given inequality is equivalent to $$\frac{(a + b)^m}{b^m} + \frac{(a + b)^m}{a^m} \geq 2^{m+1},$$ which can be rewritten as $$\left(\frac{1}{a^m} + \frac{1}{b^m}\right) \geq \frac{2^{m+1}}{(a + b)^m}.$$ Since $f(x) = \frac{1}{x^m}$ is a convex function for every $m...

Practice › Mathematics › IMO Shortlist › IMO 1992 Shortlist ›

IMO 1992 SL 16

IMO 1992 SL 16 Origin: KOR Problem Prove that N = 5125−1 525−1 is a composite number. Solution Observe that x4 + x3 + x2 + x + 1 = (x2 + 3x + 1)2 −5x(x + 1)2. Thus for x = 525 we have N = x4 + x3 + x2 + x + 1 = (x2 + 3x + 1 −513(x + 1))(x2 + 3x + 1 +...

Practice › Mathematics › IMO Shortlist › IMO 1973 Shortlist ›

IMO 1973 SL 6

IMO 1973 SL 6 Origin: POL Problem Does there exist a ﬁnite set M of points in space, not all in the same plane, such that for each two points A, B \inM there exist two other points C, D \inM such that lines AB and CD are parallel? Solution Yes. Take for M the set of vertices of a cube ABCDEFGH and two points I, J symmetric to the...

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL N1

IMO 1995 SL N1 Origin: ROM | Category: Number Theory Problem Let k be a positive integer. Prove that there are inﬁnitely many perfect squares of the form n2k −7, where n is a positive integer. Solution We show by induction on k that there exists a positive integer ak for which a2 k \equiv−7 (mod 2k). The statement of the problem follows, since every ak + r2k (r =...

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL N6

IMO 2000 SL N6 Origin: ROM | Category: Number Theory Problem Show that the set of positive integers that cannot be repre- sented as a sum of distinct perfect squares is ﬁnite. Solution Suppose that a natural number N satisﬁes N = a2 1 + \cdot \cdot \cdot + a2 k, 2N = b2 1 +\cdot \cdot \cdot+b2 l , where ai, bj are natural numbers such that none of...

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL N2

IMO 2001 SL N2 Origin: COL | Category: Number Theory Problem Consider the system x + y = z + u, 2xy = zu. Find the greatest value of the real constant m such that m \leqx/y for every positive integer solution x, y, z, u of the system with x \geqy. Solution We shall ﬁnd the general solution to the system. Squaring both sides of the ﬁrst equation and...

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL G5

IMO 1999 SL G5 Origin: FRA | Category: Geometry Problem Let ABC be a triangle, Ωits incircle and Ωa, Ωb, Ωc three circles three circles orthogonal to Ωpassing through B and C, A and C, and A and B respectively. The circles Ωa, Ωb meet again in C′; in the same way we obtain the points B′ and A′. Prove that the radius of the circumcircle of A′B′C′ is half...

Practice › Mathematics › IMO Shortlist › IMO 1968 Shortlist ›

IMO 1968 SL 8

IMO 1968 SL 8 Origin: ROM Problem Given an oriented line $\Delta$ and a fixed point $A$ on it, consider all trapezoids $ABCD$ one of whose bases $AB$ lies on $\Delta$, in the positive direction. Let $E, F$ be the midpoints of $AB$ and $CD$ respectively. Find the loci of vertices $B, C, D$ of trapezoids that satisfy the following: (i) $|AB| \leq a$ ($a$ fixed); (ii) $|EF| = l$...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL N3

IMO 2004 SL N3 Origin: IRN | Category: Number Theory Problem A function f from the set of positive integers N into itself is such that for all m, n \inN the number (m2 + n)2 is divisible by f 2(m) + f(n). Prove that f(n) = n for each n \inN. Solution For m = n = 1 we obtain that f(1)2 + f(1) divides (12 + 1)2 =...

Practice › Mathematics › IMO Shortlist › IMO 1993 Shortlist ›

IMO 1993 SL 18

IMO 1993 SL 18 Origin: POL Problem Let Sn be the number of sequences (a1, a2, . . . , an), where ai \in {0, 1}, in which no six consecutive blocks are equal. Prove that Sn \to\infty as n \to\infty. Solution Let Bn be the set of sequences with the stated property (Sn = |Bn|). We shall prove by induction on n that Sn \geq3 2Sn−1 for every n....

Practice › Mathematics › IMO Shortlist › IMO 1994 Shortlist ›

IMO 1994 SL N6

IMO 1994 SL N6 Origin: LAT | Category: Number Theory Problem Let x1 and x2 be relatively prime positive integers. For n \geq2, deﬁne xn+1 = xnxn−1 + 1. (a) Prove that for every i > 1, there exists j > i such that xi i divides xj j. (b) Is it true that x1 must divide xj j for some j > 1? Solution (a) Let p be a...

Practice › Mathematics › IMO Shortlist › IMO 1978 Shortlist ›

IMO 1978 SL 14

IMO 1978 SL 14 Origin: VIE Problem Prove that it is possible to place 2n(2n + 1) parallelepipedic (rectangular) pieces of soap of dimensions 1 \times 2 \times (n + 1) in a cubic box with edge 2n + 1 if and only if n is even or n = 1. Remark. It is assumed that the edges of the pieces of soap are parallel to the edges of the...

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL A7

IMO 2000 SL A7 Origin: RUS | Category: Algebra Problem For a polynomial P of degree 2000 with distinct real co- eﬃcients let M(P) be the set of all polynomials that can be produced from P by permutation of its coeﬃcients. A polynomial P will be called n-independent if P(n) = 0 and we can get from any Q in M(P) a poly- nomial Q1 such that Q1(n) = 0...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 26

IMO 1988 SL 26 Origin: GBR Problem A function f deﬁned on the positive integers (and taking positive integer values) is given by f(1) = 1, f(3) = 3, f(2n) = f(n), f(4n + 1) = 2f(2n + 1) −f(n), f(4n + 3) = 3f(2n + 1) −2f(n), for all positive integers n. Determine with proof the number of positive integers less than or equal to 1988 for which f(n)...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 10

IMO 1988 SL 10 Origin: GDR Problem Let N = {1, 2, . . ., n}, n \geq2. A collection F = {A1, . . . , At} of subsets Ai \subseteqN, i = 1, . . . , t, is said to be separating if for every pair {x, y} \subseteqN, there is a set Ai \inF such that Ai \cap{x, y} contains just one element. A collection F...

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL G8

IMO 2000 SL G8 Origin: RUS | Category: Geometry Problem A1A2A3 is an acute-angled triangle. The foot of the altitude from Ai is Ki, and the incircle touches the side opposite Ai at Li. The line K1K2 is reﬂected in the line L1L2. Similarly, the line K2K3 is reﬂected in L2L3, and K3K1 is reﬂected in L3L1. Show that the three new lines form a triangle with vertices on the...

Practice › Mathematics › IMO Shortlist › IMO 1976 Shortlist ›

IMO 1976 SL 3

IMO 1976 SL 3 Origin: CZS Problem In a convex quadrangle with area 32 cm2, the sum of the lengths of two nonadjacent edges and of the length of one diagonal is equal to 16 cm. (a) What is the length of the other diagonal? (b) What are the lengths of the edges of the quadrangle if the perimeter is a minimum? (c) Is it possible to choose the edges...

Practice › Mathematics › IMO Shortlist › IMO 1985 Shortlist ›

IMO 1985 SL 7

IMO 1985 SL 7 Problem 1a.(CZS 3) The positive integers x1, . . . , xn, n \geq3, satisfy x1 < x2 < \cdot \cdot \cdot < xn < 2x1. Set P = x1x2 \cdot \cdot \cdot xn. Prove that if p is a prime number, k a positive integer, and P is divisible by pk, then P pk \geqn!. Solution Let ki \geq0 be the largest integer such that...

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 20

IMO 1990 SL 20 Origin: POL Problem Prove that every integer k greater than 1 has a multiple that is less than k4 and can be written in the decimal system with at most four diﬀerent digits. Solution Let n be the unique integer such that 2n−1 \leqk < 2n. Let S(n) be the set of numbers less than 10n that are written with only the digits {0, 1} in...

Practice › Mathematics › IMO Shortlist › IMO 1972 Shortlist ›

IMO 1972 SL 3

IMO 1972 SL 3 Origin: CZS Problem Let x1, x2, . . . , xn be real numbers satisfying x1+x2+\cdot \cdot \cdot+xn = 0. Let m be the least and M the greatest among them. Prove that x2 1 + x2 2 + \cdot \cdot \cdot + x2 n \leq−nmM. Solution We have for each k = 1, 2, . . . , n that m \leqxk \leqM, which gives...

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 26

IMO 1990 SL 26 Origin: USA Problem Let P be a cubic polynomial with rational coeﬃcients, and let q1, q2, q3, . . . be a sequence of rational numbers such that qn = P(qn+1) for all n \geq1. Prove that there exists k \geq1 such that for all n \geq1, qn+k = qn. Solution We note that |P(x)/x| \to\infty. Hence, there exists an integer number M such that M...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL N4

IMO 2004 SL N4 Origin: POL | Category: Number Theory Problem Let k be a ﬁxed integer greater than 1, and let m = 4k2 −5. Show that there exist positive integers a and b such that the sequence (xn) deﬁned by x0 = a, x1 = b, xn+2 = xn+1 + xn for n = 0, 1, 2, . . . has all of its terms relatively prime to...

Practice › Mathematics › IMO Shortlist › IMO 1985 Shortlist ›

IMO 1985 SL 22

IMO 1985 SL 22 Origin: USS Problem A circle with center O passes through points A and C and intersects the sides AB and BC of the triangle ABC at points K and N, respectively. The circumscribed circles of the triangles ABC and KBN intersect at two distinct points B and M. Prove that \angleOMB = 90◦. Solution Assume that \triangleABC is acute (the case of an obtuse \triangleABC is...

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 9

IMO 1998 SL 9 Origin: MON Problem Let a1, a2, . . . , an be positive real numbers such that a1 + a2 + \cdot \cdot \cdot + an < 1. Prove that a1a2 \cdot \cdot \cdot an[1 −(a1 + a2 + \cdot \cdot \cdot + an)] (a1 + a2 + \cdot \cdot \cdot + an)(1 −a1)(1 −a2) \cdot \cdot \cdot (1 −an) \leq nn+1 . Solution Set an+1...

Practice › Mathematics › IMO Shortlist › IMO 1968 Shortlist ›

IMO 1968 SL 11

IMO 1968 SL 11 Origin: ROM Problem Find all solutions $(x_1, x_2, \ldots, x_n)$ of the equation $$1 + \frac{1}{x_1} + \frac{x_1 + 1}{x_1x_2} + \frac{(x_1 + 1)(x_2 + 1)}{x_1x_2x_3} + \cdots + \frac{(x_1 + 1) \cdots (x_{n-1} + 1)}{x_1x_2 \cdots x_n} = 0.$$ Solution Introducing $y_i = \frac{1}{x_i}$, we transform our equation to $$\begin{aligned} 0 &= 1 + y_1 + (1 + y_1)y_2 + \cdots + (1 + y_1)...

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL C5

IMO 2002 SL C5 Origin: BRA | Category: Combinatorics Problem Let r \geq2 be a ﬁxed positive integer, and let F be an inﬁnite family of sets, each of size r, no two of which are disjoint. Prove that there exists a set of size r −1 that meets each set in F. Solution Assume to the contrary that no set of size less than r meets all sets in...

Practice › Mathematics › IMO Shortlist › IMO 1984 Shortlist ›

IMO 1984 SL 17

IMO 1984 SL 17 Origin: FRG Problem In a permutation (x1, x2, . . . , xn) of the set 1, 2, . . . , n we call a pair (xi, xj) discordant if i < j and xi > xj. Let d(n, k) be the number of such permutations with exactly k discordant pairs. Find d(n, 2) and d(n, 3). Solution For any m = 0, 1, ....

Practice › Mathematics › IMO Shortlist › IMO 1994 Shortlist ›

IMO 1994 SL A2

IMO 1994 SL A2 Origin: FRA | Category: Algebra Problem Let m and n be positive integers. The set A = {a1, a2, . . . , am} is a subset of {1, 2, . . . , n}. Whenever ai + aj \leqn, 1 \leqi \leqj \leqm, ai + aj also belongs to A. Prove that a1 + a2 + \cdot \cdot \cdot + am m \geqn + 1...

Practice › Mathematics › IMO Shortlist › IMO 1978 Shortlist ›

IMO 1978 SL 9

IMO 1978 SL 9 Origin: GBR Problem Let {f(n)} be a strictly increasing sequence of positive integers: 0 < f(1) < f(2) < f(3) < \cdot \cdot \cdot . Of the positive integers not belonging to the sequence, the nth in order of magnitude is f(f(n)) + 1. Determine f(240). Solution Since the nth missing number (gap) is f(f(n))+1 and f(f(n)) is a member of the sequence, there are exactly...

Practice › Mathematics › IMO Shortlist › IMO 1993 Shortlist ›

IMO 1993 SL 24

IMO 1993 SL 24 Origin: USA Problem Prove that a b + 2c + 3d + b c + 2d + 3a + c d + 2a + 3b + d a + 2b + 3c \geq2 for all positive real numbers a, b, c, d. Solution By the Cauchy–Schwarz inequality, if x1, x2, . . . , xn and y1, y2, . . . , yn are positive numbers,...

Practice › Mathematics › IMO Shortlist › IMO 1997 Shortlist ›

IMO 1997 SL 18

IMO 1997 SL 18 Origin: GBR Problem The altitudes through the vertices A, B, C of an acute-angled triangle ABC meet the opposite sides at D, E, F, respectively. The line through D parallel to EF meets the lines AC and AB at Q and R, re- spectively. The line EF meets BC at P. Prove that the circumcircle of the triangle PQR passes through the midpoint of BC. Solution...

Practice › Mathematics › IMO Shortlist › IMO 1971 Shortlist ›

IMO 1971 SL 14

IMO 1971 SL 14 Origin: USS Problem A broken line A1A2 . . . An is drawn in a 50\times50 square, so that the distance from any point of the square to the broken line is less than Solution Denote by V the ﬁgure made by a circle of radius 1 whose center moves along the broken line. From the condition of the problem, V contains the whole 50 \times...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL A1

IMO 2004 SL A1 Origin: KOR | Category: Algebra Problem Let n \geq3 be an integer and t1, t2, . . . , tn positive real numbers such that n2 + 1 > (t1 + t2 + \cdot \cdot \cdot + tn) 1 t1 1 t2 \cdot \cdot \cdot + 1 tn . Show that ti, tj, tk are the side lengths of a triangle for all i,...

Practice › Mathematics › IMO Shortlist › IMO 1987 Shortlist ›

IMO 1987 SL 17

IMO 1987 SL 17 Origin: ROM Problem Prove that there exists a four-coloring of the set M = {1, 2, . . ., 1987} such that any arithmetic progression with 10 terms in the set M is not monochromatic. Alternative formulation. Let M = {1, 2, . . ., 1987}. Prove that there is a function f : M \to{1, 2, 3, 4} that is not constant on every set...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 21

IMO 1991 SL 21 Origin: HKG Problem Let f(x) be a monic polynomial of degree 1991 with integer coeﬃcients. Deﬁne g(x) = f 2(x) −9. Show that the number of distinct integer solutions of g(x) = 0 cannot exceed 1995. Solution The polynomial g(x) factorizes as g(x) = f(x)2 −9 = (f(x)−3)(f(x)+3). If one of the equations f(x) + 3 = 0 and f(x) −3 = 0 has no integer...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 20

IMO 1988 SL 20 Origin: MON Problem Find the least natural number n such that if the set {1, 2, . . ., n} is arbitrarily divided into two nonintersecting subsets, then one of the subsets contains three distinct numbers such that the product of two of them equals the third. Solution Suppose that An = {1, 2, . . ., n} is partitioned into Bn and Cn, and that...

Practice › Mathematics › IMO Shortlist › IMO 1987 Shortlist ›

IMO 1987 SL 2

IMO 1987 SL 2 Origin: USA Problem At a party attended by n married couples, each person talks to everyone else at the party except his or her spouse. The conversations involve sets of persons or cliques C1, C2, . . . , Ck with the following prop- erty: no couple are members of the same clique, but for every other pair of persons there is exactly one clique to...

Practice › Mathematics › IMO Shortlist › IMO 1986 Shortlist ›

IMO 1986 SL 7

IMO 1986 SL 7 Origin: FRA Problem Let real numbers x1, x2, . . . , xn satisfy 0 < x1 < x2 < \cdot \cdot \cdot < xn < 1 and set x0 = 0, xn+1 = 1. Suppose that these numbers satisfy the following system of equations: n+1 j=0, j̸=i xi −xj = 0 where i = 1, 2, . . ., n. (1) Prove that xn+1−i...

Practice › Mathematics › IMO Shortlist › IMO 1973 Shortlist ›

IMO 1973 SL 15

IMO 1973 SL 15 Origin: CUB Problem Prove that for all n \inN the following is true: 2n n k=1 sin k\pi 2n + 1 = \sqrt 2n + 1. Solution If z = cos \theta + i sin \theta, then z −z−1 = 2i sin \theta. Now put z = cos \pi 2n+1 + i sin \pi 2n+1. Using de Moivre’s formula we transform the required equality into A...

Practice › Mathematics › IMO Shortlist › IMO 1986 Shortlist ›

IMO 1986 SL 17

IMO 1986 SL 17 Origin: CHN Problem Let A, B, C be ﬁxed points in the plane. A man starts from a certain point P0 and walks directly to A. At A he turns his di- rection by 60◦to the left and walks to P1 such that P0A = AP1. Af- ter he does the same action 1986 times successively around the points A, B, C, A, B, C, ....

Practice › Mathematics › IMO Shortlist › IMO 1997 Shortlist ›

IMO 1997 SL 25

IMO 1997 SL 25 Origin: POL Problem The bisectors of angles A, B, C of a triangle ABC meet its cir- cumcircle again at the points K, L, M, respectively. Let R be an internal point on the side AB. The points P and Q are deﬁned by the following conditions: RP is parallel to AK, and BP is perpendicular to BL; RQ is parallel to BL, and AQ is...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 24

IMO 1979 SL 24 Origin: USA Problem A circle O with center O on base BC of an isosceles triangle ABC is tangent to the equal sides AB, AC. If point P on AB and point Q on AC are selected such that PB \times CQ = (BC/2)2, prove that line segment PQ is tangent to circle O, and prove the converse. Solution Clearly O is the midpoint of BC....

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 18

IMO 1991 SL 18 Origin: BUL Problem Find the highest degree k of 1991 for which 1991k divides the number 199019911992 + 199219911990. Solution For integers a > 0, n > 0 and \alpha \geq0, we shall write a\alpha \paralleln when a\alpha | n and a\alpha+1 ∤n. Lemma. For every odd number a \geq3 and an integer n \geq0 it holds that an+1 \parallel(a + 1)an −1 and an+1 \parallel(a...

Practice › Mathematics › IMO Shortlist › IMO 1993 Shortlist ›

IMO 1993 SL 4

IMO 1993 SL 4 Origin: SPA Problem In the triangle ABC, let D, E be points on the side BC such that \angleBAD = \angleCAE. If M, N are, respectively, the points of tangency with BC of the incircles of the triangles ABD and ACE, show that MB + MD = NC + NE . Solution We note that p does not exceed one half of the circumradius of \triangleA′B′C′....

Practice › Mathematics › IMO Shortlist › IMO 1986 Shortlist ›

IMO 1986 SL 6

IMO 1986 SL 6 Origin: NET Problem Find four positive integers each not exceeding 70000 and each having more than 100 divisors. Solution There are ﬁve such numbers: 69300 = 22 \cdot 32 \cdot 52 \cdot 7 \cdot 11 : 3 \cdot 3 \cdot 3 \cdot 2 \cdot 2 = 108 divisors; 50400 = 25 \cdot 32 \cdot 52 \cdot 7 : 6 \cdot 3 \cdot 3 \cdot 2 =...

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL G3

IMO 2003 SL G3 Origin: IND | Category: Geometry Problem Let ABC be a triangle and let P be a point in its interior. Denote by D, E, F the feet of the perpendiculars from P to the lines BC, CA, and AB, respectively. Suppose that AP 2 + PD2 = BP 2 + PE2 = CP 2 + PF 2. Denote by IA, IB, IC the excenters of the...

Practice › Mathematics › IMO Shortlist › IMO 1993 Shortlist ›

IMO 1993 SL 25

IMO 1993 SL 25 Origin: VIE Problem Solve the following system of equations, in which a is a given number satisfying |a| > 1: x2 1 = ax2 + 1, x2 2 = ax3 + 1, \cdot \cdot \cdot \cdot \cdot \cdot x2 999 = ax1000 + 1, x2 1000 = ax1 + 1. Solution We need only consider the case a > 1 (since the case a < −1...

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL A2

IMO 1995 SL A2 Origin: SWE | Category: Algebra Problem Let a and b be nonnegative integers such that ab \geqc2, where c is an integer. Prove that there is a number n and integers x1, x2, . . . , xn, y1, y2, . . . , yn such that n i=1 x2 i = a, n i=1 y2 i = b, and n i=1 xiyi...

Practice › Mathematics › IMO Shortlist › IMO 1976 Shortlist ›

IMO 1976 SL 2

IMO 1976 SL 2 Origin: BUL Problem Let a0, a1, . . . , an, an+1 be a sequence of real numbers satisfying the following conditions: a0 = an+1 = 0, |ak−1 −2ak + ak+1| \leq1 (k = 1, 2, . . ., n). Prove that |ak| \leqk(n+1−k) (k = 0, 1, . . ., n + 1). Solution Let us put bi = i(n + 1 −i)/2, and let...

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL A5

IMO 2000 SL A5 Origin: BLR | Category: Algebra Problem Let n \geq2 be a positive integer and \lambda a positive real number. Initially there are n ﬂeas on a horizontal line, not all at the same point. We deﬁne a move of choosing two ﬂeas at some points A and B, with A to the left of B, and letting the ﬂea from A jump over the ﬂea from...

Practice › Mathematics › IMO Shortlist › IMO 1993 Shortlist ›

IMO 1993 SL 10

IMO 1993 SL 10 Origin: IND Problem A natural number n is said to have the property P if whenever n divides an −1 for some integer a, n2 also necessarily divides an −1. (a) Show that every prime number has property P. (b) Show that there are inﬁnitely many composite numbers n that possess property P. Solution (a) Let n = p be a prime and let p |...

Practice › Mathematics › IMO Shortlist › IMO 1972 Shortlist ›

IMO 1972 SL 1

IMO 1972 SL 1 Origin: BUL Problem Let f and ϕ be real functions deﬁned on the set R satisfying the functional equation f(x + y) + f(x −y) = 2ϕ(y)f(x), (1) for arbitrary real x, y (give examples of such functions). Prove that if f(x) is not identically 0 and |f(x)| \leq1 for all x, then |ϕ(x)| \leq1 for all x. Solution Suppose that f(x0) ̸= 0 and for...

Practice › Mathematics › IMO Shortlist › IMO 1970 Shortlist ›

IMO 1970 SL 8

IMO 1970 SL 8 Origin: POL Problem Given a point M on the side AB of the triangle ABC, let r1 and r2 be the radii of the inscribed circles of the triangles ACM and BCM respectively and let \rho1 and \rho2 be the radii of the excircles of the triangles ACM and BCM at the sides AM and BM respectively. Let r and \rho denote the radii of the...

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL C4

IMO 2001 SL C4 Origin: NZL | Category: Combinatorics Problem A set of three nonnegative integers {x, y, z} with x < y < z is called historic if {z −y, y −x} = {1776, 2001}. Show that the set of all nonnegative integers can be written as the union of disjoint historic sets. Solution For convenience let us write a = 1776, b = 2001, 0 < a <...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL C3

IMO 2004 SL C3 Origin: AUS | Category: Combinatorics Problem The following operation is allowed on a ﬁnite graph: Choose an arbitrary cycle of length 4 (if there is any), choose an arbitrary edge in that cycle, and delete it from the graph. For a ﬁxed integer n \geq4, ﬁnd the least number of edges of a graph that can be obtained by repeated ap- plications of this operation from...

Practice › Mathematics › IMO Shortlist › IMO 1993 Shortlist ›

IMO 1993 SL 5

IMO 1993 SL 5 Origin: FIN Problem On an inﬁnite chessboard, a solitaire game is played as follows: At the start, we have n2 pieces occupying n2 squares that form a square of side n. The only allowed move is a jump horizontally or vertically over an occupied square to an unoccupied one, and the piece that has been jumped over is removed. For what positive integers n can the...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 3

IMO 1988 SL 3 Origin: CAN Problem The triangle ABC is inscribed in a circle. The interior bi- sectors of the angles A, B, and C meet the circle again at A′, B′, and C′ respectively. Prove that the area of triangle A′B′C′ is greater than or equal to the area of triangle ABC. Solution Let R be the circumradius, r the inradius, s the semiperimeter, ∆the area of ABC...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 6

IMO 1989 SL 6 Origin: CZS Problem For a triangle ABC, let k be its circumcircle with radius r. The bisectors of the inner angles A, B, and C of the triangle intersect respec- tively the circle k again at points A′, B′, and C′. Prove the inequality 16Q3 \geq27r4P, where Q and P are the areas of the triangles A′B′C′ and ABC respec- tively. Solution Let us denote the...

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL A3

IMO 1995 SL A3 Origin: UKR | Category: Algebra Problem Let n be an integer, n \geq3. Let a1, a2, . . . , an be real numbers such that 2 \leqai \leq3 for i = 1, 2, . . ., n. If s = a1 + a2 + \cdot \cdot \cdot + an, prove that a2 1 + a2 2 −a2 a1 + a2 −a3 a2 2 + a2...

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL G4

IMO 2002 SL G4 Origin: RUS | Category: Geometry Problem Circles S1 and S2 intersect at points P and Q. Distinct points A1 and B1 (not at P or Q) are selected on S1. The lines A1P and B1P meet S2 again at A2 and B2 respectively, and the lines A1B1 and A2B2 meet at C. Prove that as A1 and B1 vary, the circumcenters of triangles A1A2C all lie...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 15

IMO 1991 SL 15 Origin: USS Problem Let an be the last nonzero digit in the decimal representation of the number n!. Does the sequence a1, a2, . . . , an, . . . become periodic after a ﬁnite number of terms? Solution Assume that the sequence has the period T . We can ﬁnd integers k > m > 0, as large as we like, such that 10k...

Practice › Mathematics › IMO Shortlist › IMO 1997 Shortlist ›

IMO 1997 SL 6

IMO 1997 SL 6 Origin: IRE Problem (a) Let n be a positive integer. Prove that there exist distinct positive integers x, y, z such that xn−1 + yn = zn+1. (b) Let a, b, c be positive integers such that a and b are relatively prime and c is relatively prime either to a or to b. Prove that there exist inﬁnitely many triples (x, y, z) of distinct...

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL G4

IMO 1999 SL G4 Origin: GBR | Category: Geometry Problem For a triangle T = ABC we take the point X on the side (AB) such that AX/XB = 4/5, the point Y on the segment (CX) such that CY = 2Y X, and, if possible, the point Z on the ray (CA such that ∡CXZ = 180◦−∡ABC. We denote by \Sigma the set of all triangles T for which...

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL G7

IMO 2002 SL G7 Origin: BUL | Category: Geometry Problem The incircle Ωof the acute-angled triangle ABC is tangent to BC at K. Let AD be an altitude of triangle ABC and let M be the midpoint of AD. If N is the other common point of Ωand KM, prove that Ωand the circumcircle of triangle BCN are tangent at N. Solution Let k be the circle through B, C...

Practice › Mathematics › IMO Shortlist › IMO 1971 Shortlist ›

IMO 1971 SL 4

IMO 1971 SL 4 Origin: GBR Problem We are given two mutually tangent circles in the plane, with radii r1, r2. A line intersects these circles in four points, determining three segments of equal length. Find this length as a function of r1 and r2 and the condition for the solvability of the problem. Solution In the coordinate system in which the x-axis passes through the centers of the circles...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 1

IMO 1979 SL 1 Origin: BEL Problem Prove that in the Euclidean plane every regular polygon having an even number of sides can be dissected into lozenges. (A lozenge is a quadrilateral whose four sides are all of equal length). Solution We prove more generally, by induction on n, that any 2n-gon with equal edges and opposite edges parallel to each other can be dissected. For n = 2 the...

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL G1

IMO 2002 SL G1 Origin: FRA | Category: Geometry Problem Let B be a point on a circle S1, and let A be a point distinct from B on the tangent at B to S1. Let C be a point not on S1 such that the line segment AC meets S1 at two distinct points. Let S2 be the circle touching AC at C and touching S1 at a point...

Practice › Mathematics › IMO Shortlist › IMO 1975 Shortlist ›

IMO 1975 SL 2

IMO 1975 SL 2 Origin: CZS Problem Let x1 \geqx2 \geq\cdot \cdot \cdot \geqxn and y1 \geqy2 \geq\cdot \cdot \cdot \geqyn be two n-tuples of numbers. Prove that n i=1 (xi −yi)2 \leq n i=1 (xi −zi)2 is true when z1, z2, . . . , zn denote y1, y2, . . . , yn taken in another order. Solution Since there are ﬁnitely many arrangements of the...

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL C1

IMO 1999 SL C1 Origin: IND | Category: Combinatorics Problem Let n \geq1 be an integer. A path from (0, 0) to (n, n) in the xy plane is a chain of consecutive unit moves either to the right (move denoted by E) or upwards (move denoted by N), all the moves being made inside the half-plane x \geqy. A step in a path is the occurrence of two consecutive...

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL N1

IMO 2002 SL N1 Origin: UZB | Category: Number Theory Problem What is the smallest positive integer t such that there exist integers x1, x2, . . . , xt with x3 1 + x3 2 + \cdot \cdot \cdot + x3 t = 20022002? Solution Consider the given equation modulo 9. Since each cube is congruent to either −1, 0 or 1, whereas 20022002 \equiv42002 = 4 \cdot 64667...

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL N3

IMO 2002 SL N3 Origin: MON | Category: Number Theory Problem Let p1, p2, . . . , pn be distinct primes greater than 3. Show that 2p1p2\cdot\cdot\cdotpn + 1 has at least 4n divisors. Solution We observe that if a, b are coprime odd numbers, then gcd(2a+1, 2b+1) = 3. In fact, this g.c.d. divides gcd(22a−1, 22b−1) = 2gcd(2a,2b)−1 = 22−1 = 3, while 3 obviously divides both 2a...

Practice › Mathematics › IMO Shortlist › IMO 1987 Shortlist ›

IMO 1987 SL 22

IMO 1987 SL 22 Origin: VIE Problem Does there exist a function f : N \toN, such that f(f(n)) = n + 1987 for every natural number n? Solution Suppose that there exists such function f. Then we obtain f(n + 1987) = f(f(f(n))) = f(n) + 1987 for all n \inN, and from here, by induction, f(n + 1987t) = f(n) + 1987t for all n, t \inN. Further,...

Practice › Mathematics › IMO Shortlist › IMO 1992 Shortlist ›

IMO 1992 SL 6

IMO 1992 SL 6 Origin: IND Problem Find all functions f : R \toR such that f(x2 + f(y)) = y + f(x)2 for all x, y in R. Solution It is easy to see that f is injective and surjective. From f(x2 + f(y)) = f((−x)2 + f(y)) it follows that f(x)2 = (f(−x))2, which implies f(−x) = −f(x) because f is injective. Furthermore, there exists z \inR such...

Practice › Mathematics › IMO Shortlist › IMO 1985 Shortlist ›

IMO 1985 SL 6

IMO 1985 SL 6 Origin: POL Problem Let xn = 2 + 3 + . . . + n\sqrtn. Prove that xn+1 −xn < 1 n!, n = 2, 3, . . . . Alternatives Solution Let us set xn,i = i i + i+1 i + 1 + \cdot \cdot \cdot + n\sqrtn, yn,i = xi−1 n+1,i + xi−2 n+1,ixn,i + \cdot \cdot \cdot + xi−1 n,i ....

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL C5

IMO 2001 SL C5 Origin: FIN | Category: Combinatorics Problem Find all ﬁnite sequences (x0, x1, . . . , xn) such that for every j, 0 \leqj \leqn, xj equals the number of times j appears in the sequence. Solution Let (x0, x1, . . . , xn) be any such sequence: its terms are clearly nonnegative integers. Also, x0 = 0 yields a contradiction, so x0 > 0....

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL C3

IMO 2000 SL C3 Origin: COL | Category: Combinatorics Problem Let n \geq4 be a ﬁxed positive integer. Given a set S = {P1, P2, . . . , Pn} of points in the plane such that no three are collinear and no four concyclic, let at, 1 \leqt \leqn, be the number of circles PiPjPk that contain Pt in their interior, and let m(S) = a1 + a2 +...

Practice › Mathematics › IMO Shortlist › IMO 1970 Shortlist ›

IMO 1970 SL 12

IMO 1970 SL 12 Origin: USS Problem We are given 100 points in the plane, no three of which are on the same line. Consider all triangles that have all vertices chosen from the 100 given points. Prove that at most 70% of these triangles are acute angled. Solution Lemma. Five points are given in the plane such that no three of them are collinear. Then there are at least...

Practice › Mathematics › IMO Shortlist › IMO 1970 Shortlist ›

IMO 1970 SL 6

IMO 1970 SL 6 Origin: FRA Problem In the triangle ABC let B′ and C′ be the midpoints of the sides AC and AB respectively and H the foot of the altitude passing through the vertex A. Prove that the circumcircles of the triangles AB′C′, BC′H, and B′CH have a common point I and that the line HI passes through the midpoint of the segment B′C′. Solution Let F be...

Practice › Mathematics › IMO Shortlist › IMO 1971 Shortlist ›

IMO 1971 SL 8

IMO 1971 SL 8 Origin: NET Problem Determine whether there exist distinct real numbers a, b, c, t for which: (i) the equation ax2 + btx + c = 0 has two distinct real roots x1, x2, (ii) the equation bx2 + ctx + a = 0 has two distinct real roots x2, x3, (iii) the equation cx2 + atx + b = 0 has two distinct real roots x3,...

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL G7

IMO 2000 SL G7 Origin: IRN | Category: Geometry Problem Ten gangsters are standing on a ﬂat surface, and the distances between them are all distinct. At twelve o’clock, when the church bells start chiming, each of them fatally shoots the one among the other nine gangsters who is the nearest. At least how many gangsters will be killed? Solution The problem can be reformulated in the following way: Given...

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL C2

IMO 1999 SL C2 Origin: CAN | Category: Combinatorics Problem (a) If a 5 \times n rectangle can be tiled using n pieces like those shown in the diagram, prove that n is even. (b) Show that there are more than 2\cdot3k−1 ways to tile a ﬁxed 5\times2k rect- angle (k \geq3) with 2k pieces. (Symmetric constructions are considered to be diﬀerent.) Solution (a) Color the ﬁrst, third, and ﬁfth...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 26

IMO 1979 SL 26 Origin: YUG Problem Prove that the functional equations f(x + y) = f(x) + f(y), and f(x + y + xy) = f(x) + f(y) + f(xy) (x, y \inR) are equivalent. Solution Let us assume that f(x + y) = f(x) + f(y) for all reals. In this case we trivially apply the equation to get f(x + y + xy) = f(x + y)...

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL C5

IMO 2003 SL C5 Origin: ROM | Category: Combinatorics Problem Every point with integer coordinates in the plane is the center of a disk with radius 1/1000. (a) Prove that there exists an equilateral triangle whose vertices lie in diﬀerent disks. (b) Prove that every equilateral triangle with vertices in diﬀerent disks has side length greater than 96. Solution (a) By the pigeonhole principle there are two diﬀerent integers x1,...

Practice › Mathematics › IMO Shortlist › IMO 1993 Shortlist ›

IMO 1993 SL 6

IMO 1993 SL 6 Origin: GER Problem Let N = {1, 2, 3, . . .}. Determine whether there exists a strictly increasing function f : N \toN with the following properties: f(1) = 2; (1) f(f(n)) = f(n) + n (n \inN). (2) Solution Notice that for \alpha = 1+ \sqrt , \alpha2n = \alphan+n for all n \inN. We shall show that f(n) = / \alphan + 1...

Practice › Mathematics › IMO Shortlist › IMO 1981 Shortlist ›

IMO 1981 SL 18

IMO 1981 SL 18 Origin: USS Problem Several equal spherical planets are given in outer space. On the surface of each planet there is a set of points that is invisible from any of the remaining planets. Prove that the sum of the areas of all these sets is equal to the area of the surface of one planet. Solution Let C be the convex hull of the set of...

Practice › Mathematics › IMO Shortlist › IMO 1994 Shortlist ›

IMO 1994 SL C4

IMO 1994 SL C4 Origin: EST | Category: Combinatorics Problem There are n + 1 ﬁxed positions in a row, labeled 0 to n in increasing order from right to left. Cards numbered 0 to n are shuﬄed and dealt, one in each position. The object of the game is to have card i in the ith position for 0 \leqi \leqn. If this has not been achieved, the following...

Practice › Mathematics › IMO Shortlist › IMO 1982 Shortlist ›

IMO 1982 SL 1

IMO 1982 SL 1 Problem A1 (GBR 3)IMO1 The function f(n) is deﬁned for all positive integers n and takes on nonnegative integer values. Also, for all m, n, f(m + n) −f(m) −f(n) = 0 or 1; f(2) = 0, f(3) > 0, and f(9999) = 3333. Determine f(1982). Solution From f(1) + f(1) \leqf(2) = 0 we obtain f(1) = 0. Since 0 < f(3) \leq f(1) +...

Practice › Mathematics › IMO Shortlist › IMO 1986 Shortlist ›

IMO 1986 SL 12

IMO 1986 SL 12 Origin: GDR Problem To each vertex Pi (i = 1, . . . , 5) of a pentagon an integer xi is assigned, the sum s = xi being positive. The following operation is allowed, provided at least one of the xi’s is negative: Choose a negative xi, replace it by −xi, and add the former value of xi to the integers assigned to the...

Practice › Mathematics › IMO Shortlist › IMO 1993 Shortlist ›

IMO 1993 SL 21

IMO 1993 SL 21 Origin: GBR Problem A circle S is said to cut a circle \Sigma diametrally if their common chord is a diameter of \Sigma. Let SA, SB, SC be three circles with distinct centers A, B, C respectively. Prove that A, B, C are collinear if and only if there is no unique circle S that cuts each of SA, SB, SC diametrally. Prove further that if...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 10

IMO 1989 SL 10 Origin: GRE Problem Let g : C \toC, w \inC, a \inC, w3 = 1 (w ̸= 1). Show that there is one and only one function f : C \toC such that f(z) + f(wz + a) = g(z), z \inC. Find the function f. Solution Plugging in wz + a instead of z into the functional equation, we obtain f(wz + a) + f(w2z...

Practice › Mathematics › IMO Shortlist › IMO 1992 Shortlist ›

IMO 1992 SL 8

IMO 1992 SL 8 Origin: IND Problem Show that in the plane there exists a convex polygon of 1992 sides satisfying the following conditions: (i) its side lengths are 1, 2, 3, . . ., 1992 in some order; (ii) the polygon is circumscribable about a circle. Alternative formulation. Does there exist a 1992-gon with side lengths 1, 2, 3, . . ., 1992 circumscribed about a circle? Answer the...

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 11

IMO 1990 SL 11 Problem (IND 3′)IMO1 Given a circle with two chords AB, CD that meet at E, let M be a point of chord AB other than E. Draw the circle through D, E, and M. The tangent line to the circle DEM at E meets the lines BC, AC at F, G, respectively. Given AM AB = \lambda, ﬁnd GE EF . Solution Assume B(A, E, M,...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL N2

IMO 1996 SL N2 Origin: RUS | Category: Number Theory Problem The positive integers $a$ and $b$ are such that the numbers $15a + 16b$ and $16a - 15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares? Solution Let $15a + 16b = x^2$ and $16a - 15b = y^2$, where $x, y \in...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 8

IMO 1991 SL 8 Origin: NET Problem Let S be a set of n points in the plane. No three points of S are collinear. Prove that there exists a set P containing 2n −5 points satisfying the following condition: In the interior of every triangle whose three vertices are elements of S lies a point that is an element of P. Solution Let P1(x1, y1), P2(x2, y2), . ....

Practice › Mathematics › IMO Shortlist › IMO 1986 Shortlist ›

IMO 1986 SL 9

IMO 1986 SL 9 Origin: GDR Problem Prove or disprove: Given a ﬁnite set of points with integer coordinates in the plane, it is possible to color some of these points red and the remaining ones white in such a way that for any straight line L parallel to one of the coordinate axes, the number of red colored points and the number of white colored points on L diﬀer...

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL A3

IMO 2003 SL A3 Origin: GEO | Category: Algebra Problem Consider pairs of sequences of positive real numbers a1 \geq a2 \geqa3 \geq\cdot \cdot \cdot , b1 \geqb2 \geqb3 \geq\cdot \cdot \cdot and the sums An = a1 + \cdot \cdot \cdot + an, Bn = b1 + \cdot \cdot \cdot + bn, n = 1, 2, . . . . For any pair deﬁne ci = min{ai, bi} and...

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 16

IMO 1990 SL 16 Origin: NET Problem Is there a 1990-gon with the following properties (i) and (ii)? (i) All angles are equal; (ii) The lengths of the 1990 sides are a permutation of the numbers 12, 22, . . . , 19892, 19902. Solution Let A0A1 . . . A1989 be the desired 1990-gon. We also deﬁne A1990 = A0. Let O be an arbitrary point. For 1 \leqi...

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 2

IMO 1998 SL 2 Origin: POL Problem Let ABCD be a cyclic quadrilateral. Let E and F be variable points on the sides AB and CD, respectively, such that AE : EB = CF : FD. Let P be the point on the segment EF such that PE : PF = AB : CD. Prove that the ratio between the areas of triangles APD and BPC does not depend on...

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 13

IMO 1998 SL 13 Origin: BUL Problem Determine the least possible value of f(1998), where f is a function from the set N of positive integers into itself such that for all m, n \inN, f(n2f(m)) = m[f(n)]2. Solution Denote by F the set of functions considered. Let f \inF, and let f(1) = a. Putting n = 1 and m = 1 we obtain f(f(z)) = a2z and f(az2)...

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL A6

IMO 1995 SL A6 Origin: JAP | Category: Algebra Problem Let n be an integer, n \geq3. Let x1, x2, . . . , xn be real numbers such that xi < xi+1 for 1 \leqi \leqn −1. Prove that n(n −1) i n−1 i=1 (n −i)xi ⎛ ⎝ n j=2 (j −1)xj ⎞ ⎠. Solution Let yi = xi+1 + \cdot \cdot \cdot + xn, Y...

Practice › Mathematics › IMO Shortlist › IMO 1973 Shortlist ›

IMO 1973 SL 17

IMO 1973 SL 17 Origin: POL Problem Let F be a nonempty set of functions f : R \toR of the form f(x) = ax + b, where a and b are real numbers and a ̸= 0. Suppose that F satisﬁes the following conditions: (1) If f, g \inF, then g ◦f \inF, where (g ◦f)(x) = g[f(x)]. (2) If f \inF and f(x) = ax + b, then...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 25

IMO 1979 SL 25 Origin: USA Problem Given a point P in a given plane \pi and also a given point Q not in \pi, show how to determine a point R in \pi such that QP+PR QR is a maximum. Solution Let us ﬁrst look for such a point R on a line l in \pi going through P. Let \angleQPR = 2\theta. Consider a point Q′ on l...

Practice › Mathematics › IMO Shortlist › IMO 1981 Shortlist ›

IMO 1981 SL 8

IMO 1981 SL 8 Origin: FRG Problem Let f(n, r) be the arithmetic mean of the minima of all r- subsets of the set {1, 2, . . ., n}. Prove that f(n, r) = n+1 r+1 . Solution Since the number k, k = 1, 2, . . ., n −r + 1, is the minimum in exactly n−k r−1 r-element subsets of {1, 2, . . .,...

Practice › Mathematics › IMO Shortlist › IMO 1987 Shortlist ›

IMO 1987 SL 4

IMO 1987 SL 4 Origin: FRA Problem Let ABCDEFGH be a parallelepiped with AE\parallelBF\parallelCG\parallelDH. Prove the inequality AF + AH + AC \leqAB + AD + AE + AG. In what cases does equality hold? Solution Setting x1 = −−\to AB, x2 = −−\to AD, x3 = −\to AE, we have to prove that \parallelx1 + x2\parallel+ \parallelx2 + x3\parallel+ \parallelx3 + x1\parallel\leq\parallelx1\parallel+ \parallelx2\parallel+ \parallelx3\parallel+ \parallelx1 + x2 + x3\parallel....

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL G7

IMO 2004 SL G7 Origin: RUS | Category: Geometry Problem For a given triangle ABC, let X be a variable point on the line BC such that C lies between B and X and the incircles of the triangles ABX and ACX intersect at two distinct points P and Q. Prove that the line PQ passes through a point independent of X. Solution The proof uses the following observation: Lemma....

Practice › Mathematics › IMO Shortlist › IMO 1975 Shortlist ›

IMO 1975 SL 5

IMO 1975 SL 5 Origin: SWE Problem Let M be the set of all positive integers that do not contain the digit 9 (base 10). If x1, . . . , xn are arbitrary but distinct elements in M, prove that n j=1 xj < 80. Solution There are exactly 8 \cdot 9k−1 k-digit numbers in M (the ﬁrst digit can be chosen in 8 ways, while any other...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 2

IMO 1989 SL 2 Origin: AUS Problem Ali Barber, the carpet merchant, has a rectangular piece of carpet whose dimensions are unknown. Unfortunately, his tape measure is broken and he has no other measuring instruments. However, he ﬁnds that if he lays it ﬂat on the ﬂoor of either of his storerooms, then each corner of the carpet touches a diﬀerent wall of that room. If the two rooms have...

Practice › Mathematics › IMO Shortlist › IMO 1992 Shortlist ›

IMO 1992 SL 21

IMO 1992 SL 21 Origin: GBR Problem For each positive integer n, denote by s(n) the greatest integer such that for all positive integers k \leqs(n), n2 can be expressed as a sum of squares of k positive integers. (a) Prove that s(n) \leqn2 −14 for all n \geq4. (b) Find a number n such that s(n) = n2 −14. (c) Prove that there exist inﬁnitely many positive integers n...

Practice › Mathematics › IMO Shortlist › IMO 1972 Shortlist ›

IMO 1972 SL 11

IMO 1972 SL 11 Origin: NET Problem Consider a sequence of circles K1, K2, K3, K4, . . . of radii r1, r2, r3, r4, . . . , respectively, situated inside a triangle ABC. The circle K1 is tangent to AB and AC; K2 is tangent to K1, BA, and BC; K3 is tangent to K2, CA, and CB; K4 is tangent to K3, AB, and AC; etc. (a)...

Practice › Mathematics › IMO Shortlist › IMO 1978 Shortlist ›

IMO 1978 SL 6

IMO 1978 SL 6 Origin: FRA Problem Let ϕ : {1, 2, 3, . . .} \to{1, 2, 3, . . .} be injective. Prove that for all n, n k=1 ϕ(k) k2 \geq n k=1 k . Solution For ﬁxed n and the set {ϕ(1), . . . , ϕ(n)}, there are ﬁnitely many possi- bilities for mapping ϕ to {1, . . . , n}. Suppose...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL C7

IMO 2004 SL C7 Origin: EST | Category: Combinatorics Problem Determine all m \times n rectangles that can be covered with hooks made up of 6 unit squares, as in the ﬁgure: Rotations and reﬂections of hooks are allowed. The rectangle must be covered without gaps and overlaps. No part of a hook may cover area outside the rectangle. Solution Suppose that an m\timesn rectangle can be covered by “hooks”....

Practice › Mathematics › IMO Shortlist › IMO 1976 Shortlist ›

IMO 1976 SL 10

IMO 1976 SL 10 Origin: USA Problem Find the largest number obtainable as the product of pos- itive integers whose sum is 1976. Solution Let a1 < a2 < \cdot \cdot \cdot < an be positive integers whose sum is 1976. Let M denote the maximal value of a1a2 \cdot \cdot \cdot an. We make the following observa- tions: (1) a1 = 1 does not yield the maximum, since replacing...

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL A5

IMO 1995 SL A5 Origin: UKR | Category: Algebra Problem Let R be the set of real numbers. Does there exist a function f : R \toR that simultaneously satisﬁes the following three conditions? (a) There is a positive number M such that −M \leqf(x) \leqM for all x. (b) f(1) = 1. (c) If x ̸= 0, then f x + 1 x2 = f(x) + f...

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL A4

IMO 1999 SL A4 Origin: BLR | Category: Algebra Problem Prove that the set of positive integers cannot be partitioned into three nonempty subsets such that for any two integers x, y taken from two diﬀerent subsets, the number x2 −xy +y2 belongs to the third subset. Solution Deﬁne f(x, y) = x2 −xy + y2. Let us assume that three such sets A, B, and C do exist and...

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL C8

IMO 2001 SL C8 Origin: GER | Category: Combinatorics Problem Twenty-one girls and twenty-one boys took part in a mathematical competition. It turned out that (i) each contestant solved at most six problems, and (ii) for each pair of a girl and a boy, there was at least one problem that was solved by both the girl and the boy. Show that there is a problem that was solved by...

Practice › Mathematics › IMO Shortlist › IMO 1977 Shortlist ›

IMO 1977 SL 3

IMO 1977 SL 3 Origin: FRG Problem Let a and b be natural numbers and let q and r be the quotient and remainder respectively when a2 + b2 is divided by a + b. Determine the numbers a and b if q2 + r = 1977.

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 19

IMO 1979 SL 19 Origin: ROM Problem Consider the sequences (an), (bn) deﬁned by a1 = 3, b1 = 100, an+1 = 3an, bn+1 = 100bn. Find the smallest integer m for which bm > a100. Solution Let us deﬁne ij for two positive integers i and j in the following way: i1 = i and ij+1 = iij for all positive integers j. Thus we must ﬁnd the smallest...

Practice › Mathematics › IMO Shortlist › IMO 1992 Shortlist ›

IMO 1992 SL 19

IMO 1992 SL 19 Origin: IRE Problem Let f(x) = x8 + 4x6 + 2x4 + 28x2 + 1. Let p > 3 be a prime and suppose there exists an integer z such that p divides f(z). Prove that there exist integers z1, z2, . . . , z8 such that if g(x) = (x −z1)(x −z2) \cdot \cdot \cdot (x −z8), then all coeﬃcients of f(x) −g(x) are...

Practice › Mathematics › IMO Shortlist › IMO 1985 Shortlist ›

IMO 1985 SL 17

IMO 1985 SL 17 Problem 6a.(SWE 3)IMO6 The sequence f1, f2, . . . , fn, . . . of functions is deﬁned for x > 0 recursively by f1(x) = x, fn+1(x) = fn(x) fn(x) + 1 n . Prove that there exists one and only one positive number a such that 0 < fn(a) < fn+1(a) < 1 for all integers n \geq1. Solution The statement...

Practice › Mathematics › IMO Shortlist › IMO 1994 Shortlist ›

IMO 1994 SL A4

IMO 1994 SL A4 Origin: MON | Category: Algebra Problem Let R denote the set of all real numbers and R+ the subset of all positive ones. Let \alpha and \beta be given elements in R, not necessarily distinct. Find all functions f : R+ \toR such that f(x)f(y) = y\alphaf x x\betaf y for all x and y in R+. Solution Suppose that \alpha = \beta. The given functional...

Practice › Mathematics › IMO Shortlist › IMO 1993 Shortlist ›

IMO 1993 SL 8

IMO 1993 SL 8 Origin: IND Problem Deﬁne a sequence ⟨f(n)⟩\infty n=1 of positive integers by f(1) = 1 and f(n) = . f(n −1) −n, if f(n −1) > n; f(n −1) + n, if f(n −1) \leqn, for n \geq2. Let S = {n \inN | f(n) = 1993}. (a) Prove that S is an inﬁnite set. (b) Find the least positive integer in S. (c) If all...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 8

IMO 1988 SL 8 Origin: FRA Problem Let u1, u2, . . . , um be m vectors in the plane, each of length less than or equal to 1, which add up to zero. Show that one can rear- range u1, u2, . . . , um as a sequence v1, v2, . . . , vm such that each partial sum v1, v1 + v2, v1 + v2...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL A8

IMO 1996 SL A8 Origin: ROM | Category: Algebra Problem Let $\mathbb{N}_0$ denote the set of nonnegative integers. Find all functions $f:\mathbb{N}_0 \to \mathbb{N}_0$ such that $$f(m + f(n)) = f(f(m)) + f(n), \quad \forall m,n \in \mathbb{N}_0.$$ Solution Putting $m=n=0$ we obtain $f(0)=0$ and consequently $$f(f(n)) = f(n)$$ for all $n$. Thus the given functional equation is equivalent to $$f(m + f(n)) = f(m) + f(n), \quad f(0)=0.$$ Clearly...

Practice › Mathematics › IMO Shortlist › IMO 1968 Shortlist ›

IMO 1968 SL 2

IMO 1968 SL 2 Origin: ROM Problem Prove that there exists a unique triangle whose side lengths are consecutive natural numbers and one of whose angles is twice the measure of one of the others. Solution The sides a, b, c of a triangle ABC with \angleABC = 2\angleBAC satisfy b2 = a(a + c) (this statement is the lemma in (SL98-7)). Taking into account the remaining condition that a,...

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL N3

IMO 1999 SL N3 Origin: RUS | Category: Number Theory Problem Prove that there exist two strictly increasing sequences (an) and (bn) such that an(an + 1) divides b2 n + 1 for every natural number n. Solution We ﬁrst prove the following lemma. Lemma. For d, c \inN and d2 | c2 + 1 there exists b \inN such that d2(d2 + 1) | b2 + 1. Proof. It...

Practice › Mathematics › IMO Shortlist › IMO 1974 Shortlist ›

IMO 1974 SL 7

IMO 1974 SL 7 Problem II 1 (POL 2) Let ai, bi be coprime positive integers for i = 1, 2, . . . , k, and m the least common multiple of b1, . . . , bk. Prove that the greatest common divisor of a1 m b1 , . . . , ak m bk equals the greatest common divisor of a1, . . . , ak. Solution...

Practice › Mathematics › IMO Shortlist › IMO 1982 Shortlist ›

IMO 1982 SL 3

IMO 1982 SL 3 Problem A3 (USS 4)IMO3 Consider the inﬁnite sequences {xn} of positive real numbers with the following properties: x0 = 1 and for all i \geq0, xi+1 \leqxi. (a) Prove that for every such sequence there is an n \geq1 such that x2 x1 + x2 x2 + \cdot \cdot \cdot + x2 n−1 xn \geq3.999. (b) Find such a sequence for which x2 x1 + x2...

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL G2

IMO 2001 SL G2 Origin: KOR | Category: Geometry Problem In acute triangle ABC with circumcenter O and altitude AP, ∡C \geq∡B + 30◦. Prove that ∡A + ∡COP < 90◦. Solution Since \angleOCP = 90◦−\angleA, we are led to showing that \angleOCP > \angleCOP, i.e., OP > CP. By the triangle inequality it suﬃces to prove CP < 1 2CO. Let CO = R. The law of sines yields...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 22

IMO 1989 SL 22 Origin: PHI Problem Prove that the set {1, 2, . . ., 1989} can be expressed as the disjoint union of 17 subsets A1, A2, . . . , A17 such that: (i) each Ai contains the same number of elements; (ii) the sum of all elements of each Ai is the same for i = 1, 2, . . . , 17. Solution The statement...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 1

IMO 1991 SL 1 Origin: PHI Problem Let ABC be any triangle and P any point in its interior. Let P1, P2 be the feet of the perpendiculars from P to the two sides AC and BC. Draw AP and BP, and from C drop perpendiculars to AP and BP. Let Q1 and Q2 be the feet of these perpendiculars. Prove that the lines Q1P2, Q2P1, and AB are concurrent....

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL C5

IMO 1996 SL C5 Origin: FRA | Category: Combinatorics Problem Let $p, q, n$ be three positive integers with $p + q < n$. Let $(x_0, x_1, \dots, x_n)$ be an $(n+1)$-tuple of integers satisfying the following conditions: $x_0 = x_n = 0$. For each $i$ with $1 \leq i \leq n$, either $x_i - x_{i-1} = p$ or $x_i - x_{i-1} = -q$. Show that there exists a pair...

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL A5

IMO 1999 SL A5 Origin: JAP | Category: Algebra Problem Find all the functions f : R \toR that satisfy f(x −f(y)) = f(f(y)) + xf(y) + f(x) −1 for all x, y \inR. Solution Let A = {f(x) | x \inR} and f(0) = c. Plugging in x = y = 0 we get f(−c) = f(c) + c −1, hence c ̸= 0. If x \inA, then taking...

Practice › Mathematics › IMO Shortlist › IMO 1992 Shortlist ›

IMO 1992 SL 3

IMO 1992 SL 3 Origin: CHN Problem The diagonals of a quadrilateral ABCD are perpendicular: AC\perpBD. Four squares, ABEF, BCGH, CDIJ, DAKL, are erected ex- ternally on its sides. The intersection points of the pairs of straight lines CL, DF; DF, AH; AH, BJ; BJ, CL are denoted by P1, Q1, R1, S1, respec- tively, and the intersection points of the pairs of straight lines AI, BK; BK, CE; CE,...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 2

IMO 1991 SL 2 Origin: JAP Problem For an acute triangle ABC, M is the midpoint of the segment BC, P is a point on the segment AM such that PM = BM, H is the foot of the perpendicular line from P to BC, Q is the point of intersection of segment AB and the line passing through H that is perpendicular to PB, and ﬁnally, R is the...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 29

IMO 1988 SL 29 Origin: USA Problem A number of signal lights are equally spaced along a one-way railroad track, labeled in order 1, 2, . . ., N (N \geq2). As a safety rule, a train is not allowed to pass a signal if any other train is in motion on the length of track between it and the following signal. However, there is no limit to the number...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 29

IMO 1989 SL 29 Origin: ROM Problem A ﬂock of 155 birds sit down on a circle C. Two birds Pi, Pj are mutually visible if m(PiPj) \leq10◦. Find the smallest number of mutually visible pairs of birds. (One assumes that a position (point) on C can be occupied simultaneously by several birds.) Solution Let Pi, sitting at the place A, and Pj sitting at B, be two birds that...

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL G3

IMO 2002 SL G3 Origin: KOR | Category: Geometry Problem The circle S has center O, and BC is a diameter of S. Let A be a point of S such that ∡AOB < 120◦. Let D be the midpoint of the arc AB that does not contain C. The line through O parallel to DA meets the line AC at I. The perpendicular bisector of OA meets S at...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 3

IMO 1989 SL 3 Origin: AUS Problem Ali Barber, the carpet merchant, has a rectangular piece of carpet whose dimensions are unknown. Unfortunately, his tape measure is broken and he has no other measuring instruments. However, he ﬁnds that if he lays it ﬂat on the ﬂoor of either of his storerooms, then each corner of the carpet touches a diﬀerent wall of that room. He knows that the sides...

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 25

IMO 1998 SL 25 Origin: NZL Problem Let U = {1, 2, . . ., n}, where n \geq3. A subset S of U is said to be split by an arrangement of the elements of U if an element not in S occurs in the arrangement somewhere between two elements of S. For example, 13542 splits {1, 2, 3} but not {3, 4, 5}. Prove that for any n...

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 8

IMO 1998 SL 8 Origin: IND Problem Let ABC be a triangle such that \angleA = 90◦and \angleB < \angleC. The tangent at A to its circumcircle \omega meets the line BC at D. Let E be the reﬂection of A across BC, X the foot of the perpendicular from A to BE, and Y the midpoint of AX. Let the line BY meet \omega again at Z. Prove that...

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL G3

IMO 1995 SL G3 Origin: TUR | Category: Geometry Problem The incircle of ABC touches BC, CA, and AB at D, E, and F respectively. X is a point inside ABC such that the incircle of XBC touches BC at D also, and touches CX and XB at Y and Z, respectively. Prove that EFZY is a cyclic quadrilateral. Solution If EF is parallel to BC, \triangleABC must be isosceles...

Practice › Mathematics › IMO Shortlist › IMO 1978 Shortlist ›

IMO 1978 SL 7

IMO 1978 SL 7 Origin: FRA Problem We consider three distinct half-lines Ox, Oy, Oz in a plane. Prove the existence and uniqueness of three points A \inOx, B \inOy, C \inOz such that the perimeters of the triangles OAB, OBC, OCA are all equal to a given number 2p > 0. Solution Let x = OA, y = OB, z = OC, \alpha = \angleBOC, \beta = \angleCOA, \gamma...

Practice › Mathematics › IMO Shortlist › IMO 1974 Shortlist ›

IMO 1974 SL 9

IMO 1974 SL 9 Problem II 3 (CUB 3) Let x, y, z be real numbers each of whose absolute value is diﬀerent from 1/ \sqrt 3 such that x + y + z = xyz. Prove that 3x −x3 1 −3x2 + 3y −y3 1 −3y2 + 3z −z3 1 −3z2 = 3x −x3 1 −3x2 \cdot 3y −y3 1 −3y2 \cdot 3z −z3 1 −3z2 . Solution There...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 17

IMO 1989 SL 17 Origin: MON Problem Given seven points in the plane, some of them are connected by segments so that: (i) among any three of the given points, two are connected by a segment; (i) the number of segments is minimal. How many segments does a ﬁgure satisfying (i) and (ii) contain? Give an example of such a ﬁgure. Solution A ﬁgure consisting of 9 lines is shown...

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL C2

IMO 2000 SL C2 Origin: ITA | Category: Combinatorics Problem A brick staircase with three steps of width 2 is made of twelve unit cubes. Determine all integers n for which it is possible to build a cube of side n using such bricks. Solution Since the volume of each brick is 12, the side of any such cube must be divisible by 6. Suppose that a cube of side...

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL 26

IMO 1995 SL 26 Problem S4 (NZL) Suppose that x1, x2, x3, . . . are positive real numbers for which xn n = n−1 j=0 xj n for n = 1, 2, 3, . . .. Prove that for all n, 2 − 2n−1 \leqxn < 2 −1 2n . Solution For n = 1 the result is trivial, since x1 = 1. Suppose now that n \geq2...

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 9

IMO 1990 SL 9 Origin: HUN Problem The incenter of the triangle ABC is K. The midpoint of AB is C1 and that of AC is B1. The lines C1K and AC meet at B2, the lines B1K and AB at C2. If the areas of the triangles AB2C2 and ABC are equal, what is the measure of angle \angleCAB? Solution Let a, b, c be the lengths of the...

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 4

IMO 1990 SL 4 Origin: CZS Problem Assume that the set of all positive integers is decomposed into r (disjoint) subsets A1 \cupA2 \cup\cdot \cdot \cdot Ar = N. Prove that one of them, say Ai, has the following property: There exists a positive m such that for any k one can ﬁnd numbers a1, a2, . . . , ak in Ai with 0 < aj+1 −aj \leqm (1...

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 19

IMO 1990 SL 19 Origin: POL Problem Let P be a point inside a regular tetrahedron T of unit volume. The four planes passing through P and parallel to the faces of T partition T into 14 pieces. Let f(P) be the joint volume of those pieces that are neither a tetrahedron nor a parallelepiped (i.e., pieces adjacent to an edge but not to a vertex). Find the exact bounds...

Practice › Mathematics › IMO Shortlist › IMO 1987 Shortlist ›

IMO 1987 SL 10

IMO 1987 SL 10 Origin: ICE Problem Let S1 and S2 be two spheres with distinct radii that touch externally. The spheres lie inside a cone C, and each sphere touches the cone in a full circle. Inside the cone there are n additional solid spheres arranged in a ring in such a way that each solid sphere touches the cone C, both of the spheres S1 and S2 externally,...

Practice › Mathematics › IMO Shortlist › IMO 1994 Shortlist ›

IMO 1994 SL G1

IMO 1994 SL G1 Origin: FRA | Category: Geometry Problem A semicircle \Gamma is drawn on one side of a straight line l. C and D are points on \Gamma. The tangents to \Gamma at C and D meet l at B and A respectively, with the center of the semicircle between them. Let E be the point of intersection of AC and BD, and F the point on l...

Practice › Mathematics › IMO Shortlist › IMO 1982 Shortlist ›

IMO 1982 SL 6

IMO 1982 SL 6 Problem A6 (VIE 1)IMO6 Let S be a square with sides of length 100 and let L be a path within S that does not meet itself and that is composed of linear segments A0A1, A1A2, . . . , An−1An with A0 ̸= An. Suppose that for every point P of the boundary of S there is a point of L at a distance from...

Practice › Mathematics › IMO Shortlist › IMO 1994 Shortlist ›

IMO 1994 SL C7

IMO 1994 SL C7 Origin: BRA | Category: Combinatorics Problem Prove that for any integer n \geq2, there exists a set of 2n−1 points in the plane such that no 3 lie on a line and no 2n are the vertices of a convex 2n-gon. Solution Deﬁne Sn recursively as follows: Let S2 = {(0, 0), (1, 1)} and Sn+1 = Sn \cupTn, where Tn = {(x + 2n−1, y...

Practice › Mathematics › IMO Shortlist › IMO 1984 Shortlist ›

IMO 1984 SL 19

IMO 1984 SL 19 Origin: CAN Problem The triangular array (an,k) of numbers is given by an,1 = 1/n, for n = 1, 2, . . ., an,k+1 = an−1,k −an,k, for 1 \leqk \leqn −1. Find the harmonic mean of the 1985th row. Solution First, we shall prove that the numbers in the nth row are exactly the numbers n n−1 , n n−1 , n n−1...

Practice › Mathematics › IMO Shortlist › IMO 1997 Shortlist ›

IMO 1997 SL 9

IMO 1997 SL 9 Origin: USA Problem Let A1A2A3 be a nonisosceles triangle with incenter I. Let Ci, i = 1, 2, 3, be the smaller circle through I tangent to AiAi+1 and AiAi+2 (the addition of indices being mod 3). Let Bi, i = 1, 2, 3, be the second point of intersection of Ci+1 and Ci+2. Prove that the circumcenters of the triangles A1B1I, A2B2I, A3B3I are collinear....

Practice › Mathematics › IMO Shortlist › IMO 1987 Shortlist ›

IMO 1987 SL 21

IMO 1987 SL 21 Origin: USS Problem The prolongation of the bisector AL (L \inBC) in the acute- angled triangle ABC intersects the circumscribed circle at point N. From point L to the sides AB and AC are drawn the perpendiculars LK and LM respectively. Prove that the area of the triangle ABC is equal to the area of the quadrilateral AKNM. Solution Let P be the second point of...

Practice › Mathematics › IMO Shortlist › IMO 1982 Shortlist ›

IMO 1982 SL 12

IMO 1982 SL 12 Problem B6 (FIN 3) Four distinct circles C, C1, C2, C3 and a line L are given in the plane such that C and L are disjoint and each of the circles C1, C2, C3 touches the other two, as well as C and L. Assuming the radius of C to be 1, determine the distance between its center and L. Solution Let y be the...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 32

IMO 1989 SL 32 Origin: USA Problem The vertex A of the acute triangle ABC is equidistant from the circumcenter O and the orthocenter H. Determine all possible values for the measure of angle A. Solution Let CC′ be an altitude, and R the circumradius. Then, since AH = R, we have AC′ = |R sin B| and hence (1) CC′ = |R sin B tan A|. On the other...

Practice › Mathematics › IMO Shortlist › IMO 1992 Shortlist ›

IMO 1992 SL 17

IMO 1992 SL 17 Origin: SWE Problem Let \alpha(n) be the number of digits equal to one in the binary representation of a positive integer n. Prove that: (a) the inequality \alpha(n2) \leq1 2\alpha(n)(\alpha(n) + 1) holds; (b) the above inequality is an equality for inﬁnitely many positive integers; (c) there exists a sequence (ni)\infty 1 such that \alpha(n2 i )/\alpha(ni) \to0 as i \to\infty. Alternative parts: Prove that there...

Practice › Mathematics › IMO Shortlist › IMO 1994 Shortlist ›

IMO 1994 SL G3

IMO 1994 SL G3 Origin: RUS | Category: Geometry Problem A circle \omega is tangent to two parallel lines l1 and l2. A second circle \omega1 is tangent to l1 at A and to \omega externally at C. A third circle \omega2 is tangent to l2 at B, to \omega externally at D, and to \omega1 externally at E. AD intersects BC at Q. Prove that Q is the circumcenter...

Practice › Mathematics › IMO Shortlist › IMO 1983 Shortlist ›

IMO 1983 SL 1

IMO 1983 SL 1 Origin: AUS Problem The localities P1, P2, . . . , P1983 are served by ten international airlines A1, A2, . . . , A10. It is noticed that there is direct service (without stops) between any two of these localities and that all airline schedules oﬀer round-trip ﬂights. Prove that at least one of the airlines can oﬀer a round trip with an odd number...

Practice › Mathematics › IMO Shortlist › IMO 1994 Shortlist ›

IMO 1994 SL G4

IMO 1994 SL G4 Origin: AUS-ARM | Category: Geometry Problem N is an arbitrary point on the bisector of \angleBAC. P and O are points on the lines AB and AN, respectively, such that ∡ANP = 90◦= ∡APO. Q is an arbitrary point on NP, and an arbitrary line through Q meets the lines AB and AC at E and F respectively. Prove that ∡OQE = 90◦if and only if...

Practice › Mathematics › IMO Shortlist › IMO 1974 Shortlist ›

IMO 1974 SL 6

IMO 1974 SL 6 Problem I 6 (ROM 4)IMO3 Does there exist a natural number n for which the number n k=0 2n + 1 2k + 1 23k is divisible by 5? Solution We set x = n k=0 2n + 1 2k + 1 23k = \sqrt n k=0 2n + 1 2k + 1 \sqrt 2k+1, y = n k=0 2n...

Practice › Mathematics › IMO Shortlist › IMO 1986 Shortlist ›

IMO 1986 SL 19

IMO 1986 SL 19 Origin: BUL Problem A tetrahedron ABCD is given such that AD = BC = a; AC = BD = b; AB \cdot CD = c2. Let f(P) = AP + BP + CP + DP, where P is an arbitrary point in space. Compute the least value of f(P). Solution Let M and N be the midpoints of segments AB and CD, respectively. The given conditions...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 9

IMO 1989 SL 9 Origin: FRA Problem For all integers n, n \geq0, there exist uniquely determined integers an, bn, cn such that 1 + 4 3\sqrt 2 −4 3\sqrt n = an + bn 3\sqrt 2 + cn 3\sqrt 4. Prove that cn = 0 implies n = 0. Solution From an+1 + bn+1 3\sqrt 2 + cn+1 3\sqrt 4 = (an + bn 3\sqrt 2 + cn...

Practice › Mathematics › IMO Shortlist › IMO 1974 Shortlist ›

IMO 1974 SL 10

IMO 1974 SL 10 Problem II 4 (FIN 3)IMO2 Let \triangleABC be a triangle. Prove that there exists a point D on the side AB such that CD is the geometric mean of AD and BD if and only if \sqrt sin A sin B \leqsin C 2 . Solution If we set \angleACD = \gamma1 and \angleBCD = \gamma2 for a point D on the segment AB, then by...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 9

IMO 1988 SL 9 Origin: FRG Problem Let a and b be two positive integers such that ab+1 divides a2 + b2. Show that a2+b2 ab+1 is a perfect square. Solution Let us assume a2+b2 ab+1 = k \inN. We then have a2 −kab + b2 = k. Let us assume that k is not an integer square, which implies k \geq2. Now we observe the minimal pair (a, b)...

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL G2

IMO 2002 SL G2 Origin: KOR | Category: Geometry Problem Let ABC be a triangle for which there exists an interior point F such that \angleAFB = \angleBFC = \angleCFA. Let the lines BF and CF meet the sides AC and AB at D and E respectively. Prove that AB + AC \geq4DE. Solution Construct equilateral triangles ACP and ABQ outside the triangle ABC. Since \angleAPC + \angleAFC = 60◦+...

Practice › Mathematics › IMO Shortlist › IMO 1987 Shortlist ›

IMO 1987 SL 23

IMO 1987 SL 23 Origin: YUG Problem Prove that for every natural number k (k \geq2) there exists an irrational number r such that for every natural number m, [rm] \equiv−1 (mod k). Remark. An easier variant: Find r as a root of a polynomial of second degree with integer coeﬃcients. Solution If we prove the existence of p, q \inN such that the roots r, s of f(x) =...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 13

IMO 1991 SL 13 Origin: POL Problem Given any integer n \geq2, assume that the integers a1, a2, . . . , an are not divisible by n and, moreover, that n does not divide a1 + a2 + \cdot \cdot \cdot+an. Prove that there exist at least n diﬀerent sequences (e1, e2, \cdot \cdot \cdot , en) consisting of zeros or ones such that e1a1 + e2a2 + \cdot...

Practice › Mathematics › IMO Shortlist › IMO 1994 Shortlist ›

IMO 1994 SL C3

IMO 1994 SL C3 Origin: MCD | Category: Combinatorics Problem Peter has three accounts in a bank, each with an integral number of dollars. He is only allowed to transfer money from one account to another so that the amount of money in the latter is doubled. (a) Prove that Peter can always transfer all his money into two accounts. (b) Can Peter always transfer all his money into one...

Practice › Mathematics › IMO Shortlist › IMO 1987 Shortlist ›

IMO 1987 SL 12

IMO 1987 SL 12 Origin: POL Problem Given a nonequilateral triangle ABC, the vertices listed coun- terclockwise, ﬁnd the locus of the centroids of the equilateral triangles A′B′C′ (the vertices listed counterclockwise) for which the triples of points A, B′, C′; A′, B, C′; and A′, B′, C are collinear. Solution Here all angles will be oriented and measured counterclockwise. Note that ∡CA′B = ∡AB′C = ∡BC′A = \pi/3. Let...

Practice › Mathematics › IMO Shortlist › IMO 1978 Shortlist ›

IMO 1978 SL 15

IMO 1978 SL 15 Origin: YUG Problem Let p be a prime and A = {a1, . . . , ap−1} an arbitrary subset of the set of natural numbers such that none of its elements is divisible by p. Let us deﬁne a mapping f from P(A) (the set of all subsets of A) to the set P = {0, 1, . . ., p −1} in the following...

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 5

IMO 1998 SL 5 Origin: FRA Problem Let ABC be a triangle, H its orthocenter, O its circumcenter, and R its circumradius. Let D be the reﬂection of A across BC, E that of B across CA, and F that of C across AB. Prove that D, E, and F are collinear if and only if OH = 2R. Solution Let G be the centroid of \triangleABC and H the...

Practice › Mathematics › IMO Shortlist › IMO 1977 Shortlist ›

IMO 1977 SL 11

IMO 1977 SL 11 Origin: NET Problem Let n be an integer greater than 1. Deﬁne x1 = n, y1 = 1, xi+1 = xi + yi , yi+1 = n xi+1 for i = 1, 2, . . ., where [z] denotes the largest integer less than or equal to z. Prove that min{x1, x2, . . . xn} = [\sqrtn].

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 16

IMO 1998 SL 16 Origin: UKR Problem Determine the smallest integer n \geq4 for which one can choose four diﬀerent numbers a, b, c, and d from any n distinct integers such that a + b −c −d is divisible by 20. Solution Let S be a set of integers such that for no four distinct elements a, b, c, d \in S, it holds that 20 | a +...

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 23

IMO 1990 SL 23 Origin: ROM Problem Find all positive integers n having the property that 2n+1 n2 is an integer. Solution Let us assume n > 1. Obviously n is odd. Let p \geq3 be the smallest prime divisor of n. In this case (p −1, n) = 1. Since 2n + 1 | 22n −1, we have that p | 22n −1. Thus it follows from Fermat’s little...

Practice › Mathematics › IMO Shortlist › IMO 1981 Shortlist ›

IMO 1981 SL 11

IMO 1981 SL 11 Origin: NET Problem On a semicircle with unit radius four consecutive chords AB, BC, CD, DE with lengths a, b, c, d, respectively, are given. Prove that a2 + b2 + c2 + d2 + abc + bcd < 4. Solution Let us denote the center of the semicircle by O, and \angleAOB = 2\alpha, \angleBOC = 2\beta, AC = m, CE = n. We claim...

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL G6

IMO 2000 SL G6 Origin: ARG | Category: Geometry Problem Let ABCD be a convex quadrilateral with AB not parallel to CD, let X be a point inside ABCD such that ∡ADX = ∡BCX < 90◦ and ∡DAX = ∡CBX < 90◦. If Y is the point of intersection of the perpendicular bisectors of AB and CD, prove that ∡AY B = 2∡ADX. Solution By the condition of the problem,...

Practice › Mathematics › IMO Shortlist › IMO 1992 Shortlist ›

IMO 1992 SL 12

IMO 1992 SL 12 Origin: NET Problem Let f, g, and a be polynomials with real coeﬃcients, f and g in one variable and a in two variables. Suppose f(x) −f(y) = a(x, y)(g(x) −g(y)) for all x, y \inR. Prove that there exists a polynomial h with f(x) = h(g(x)) for all x \inR. Solution Let us set deg f = n and deg g = m. We shall...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL G2

IMO 2004 SL G2 Origin: KAZ | Category: Geometry Problem The circle \Gamma and the line ℓdo not intersect. Let AB be the diameter of \Gamma perpendicular to ℓ, with B closer to ℓthan A. An arbitrary point C ̸= A, B is chosen on \Gamma. The line AC intersects ℓat D. The line DE is tangent to \Gamma at E, with B and E on the same side of...

Practice › Mathematics › IMO Shortlist › IMO 1972 Shortlist ›

IMO 1972 SL 5

IMO 1972 SL 5 Origin: GDR Problem Prove the following assertion: The four altitudes of a tetrahe- dron ABCD intersect in a point if and only if AB2 + CD2 = BC2 + AD2 = CA2 + BD2. Solution Lemma. If X, Y, Z, T are points in space, then the lines XZ and Y T are perpendicular if and only if XY 2 + ZT 2 = Y Z2...

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL N5

IMO 1999 SL N5 Origin: ARM | Category: Number Theory Problem Let n, k be positive integers such that n is not divisible by 3 and k \geqn. Prove that there exists a positive integer m that is divisible by n and the sum of whose digits in decimal representation is k. Solution Since one can arbitrarily add zeros at the end of m, which increases divis- ibility by 2...

Practice › Mathematics › IMO Shortlist › IMO 1984 Shortlist ›

IMO 1984 SL 14

IMO 1984 SL 14 Origin: ROM Problem Let ABCD be a convex quadrilateral for which the circle with diameter AB is tangent to the line CD. Show that the circle with diameter CD is tangent to the line AB if and only if the lines BC and AD are parallel. Solution Let M and N be the midpoints of AB and CD, and let M ′, N ′ be their...

Practice › Mathematics › IMO Shortlist › IMO 1982 Shortlist ›

IMO 1982 SL 19

IMO 1982 SL 19 Problem C7 (CZS 3) Let M be the set of real numbers of the form m+n \sqrt m2+n2 , where m and n are positive integers. Prove that for every pair x \inM, y \inM with x < y, there exists an element z \inM such that x < z < y. Solution Let us set x = m/n. Since f(x) = (m + n)/ \sqrt...

Practice › Mathematics › IMO Shortlist › IMO 1982 Shortlist ›

IMO 1982 SL 7

IMO 1982 SL 7 Problem B1 (CAN 2) Let p(x) be a cubic polynomial with integer coeﬃcients with leading coeﬃcient 1 and with one of its roots equal to the product of the other two. Show that 2p(−1) is a multiple of p(1)+p(−1)−2(1+p(0)). Solution Let a, b, ab be the roots of the cubic polynomial P(x) = (x−a)(x−b)(x− ab). Observe that 2p(−1) = −2(1 + a)(1 + b)(1 + ab);...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 9

IMO 1979 SL 9 Origin: FRG Problem Let S and F be two opposite vertices of a regular octagon. A counter starts at S and each second is moved to one of the two neigh- boring vertices of the octagon. The direction is determined by the toss of a coin. The process ends when the counter reaches F. We deﬁne an to be the number of distinct paths of duration...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 19

IMO 1988 SL 19 Origin: MEX Problem Let f(n) be a function deﬁned on the set of all positive integers and having its values in the same set. Suppose that f(f(m)+f(n)) = m+n for all positive integers n, m. Find all possible values for f(1988). Solution We will show that f(n) = n for every n (thus also f(1988) = 1988). Let f(1) = r and f(2) = s. We...

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL G4

IMO 2001 SL G4 Origin: FRA | Category: Geometry Problem Let M be a point in the interior of triangle ABC. Let A′ lie on BC with MA′ perpendicular to BC. Deﬁne B′ on CA and C′ on AB similarly. Deﬁne p(M) = MA′ \cdot MB′ \cdot MC′ MA \cdot MB \cdot MC . Determine, with proof, the location of M such that p(M) is maximal. Let µ(ABC) denote the...

Practice › Mathematics › IMO Shortlist › IMO 1968 Shortlist ›

IMO 1968 SL 25

IMO 1968 SL 25 Origin: MON Problem Given k parallel lines and a few points on each of them, ﬁnd the number of all possible triangles with vertices at these given points.4 Solution The answer is 1\leqp<q<r\leqk npnqnr + 1\leqp<q\leqk np nq nq np .

Practice › Mathematics › IMO Shortlist › IMO 1992 Shortlist ›

IMO 1992 SL 7

IMO 1992 SL 7 Origin: IND Problem Circles G, G1, G2 are three circles related to each other as follows: Circles G1 and G2 are externally tangent to one another at a point W and both these circles are internally tangent to the circle G. Points A, B, C are located on the circle G as follows: Line BC is a direct common tangent to the pair of circles G1...

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL G1

IMO 2003 SL G1 Origin: FIN | Category: Geometry Problem Let ABCD be a cyclic quadrilateral. Let P, Q, R be the feet of the perpendiculars from D to the lines BC, CA, AB, respectively. Show that PQ = QR if and only if the bisectors of \angleABC and \angleADC are concurrent with AC. Solution Denote by K and L the intersec- tions of the bisectors of \angleABC and \angleADC...

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL G4

IMO 1995 SL G4 Origin: UKR | Category: Geometry Problem An acute triangle ABC is given. Points A1 and A2 are taken on the side BC (with A2 between A1 and C), B1 and B2 on the side AC (with B2 between B1 and A), and C1 and C2 on the side AB (with C2 between C1 and B) such that \angleAA1A2 = \angleAA2A1 = \angleBB1B2 = \angleBB2B1 = \angleCC1C2...

Practice › Mathematics › IMO Shortlist › IMO 1973 Shortlist ›

IMO 1973 SL 4

IMO 1973 SL 4 Origin: GBR Problem Let P be a set of 7 diﬀerent prime numbers and C a set of 28 diﬀerent composite numbers each of which is a product of two (not necessarily diﬀerent) numbers from P. The set C is divided into 7 disjoint four-element subsets such that each of the numbers in one set has a com- mon prime divisor with at least two other...

Practice › Mathematics › IMO Shortlist › IMO 1981 Shortlist ›

IMO 1981 SL 5

IMO 1981 SL 5 Origin: COL Problem A cube is assembled with 27 white cubes. The larger cube is then painted black on the outside and disassembled. A blind man reassembles it. What is the probability that the cube is now completely black on the outside? Give an approximation of the size of your answer. Solution There are four types of small cubes upon disassembling: (1) 8 cubes with three...

Practice › Mathematics › IMO Shortlist › IMO 1993 Shortlist ›

IMO 1993 SL 23

IMO 1993 SL 23 Origin: GBR Problem A ﬁnite set of (distinct) positive integers is called a “DS-set” if each of the integers divides the sum of them all. Prove that every ﬁnite set of positive integers is a subset of some DS-set. Solution Let the given numbers be a1, . . . , an. Put s = a1 + \cdot \cdot \cdot + an and m = lcm(a1, ....

Practice › Mathematics › IMO Shortlist › IMO 1968 Shortlist ›

IMO 1968 SL 17

IMO 1968 SL 17 Origin: GBR Problem Given a point $O$ and lengths $x, y, z$, prove that there exists an equilateral triangle $ABC$ for which $OA = x$, $OB = y$, $OC = z$, if and only if $x+y \geq z$, $y+z \geq x$, $z+x \geq y$ (the points $O, A, B, C$ are coplanar).

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL C2

IMO 2004 SL C2 Origin: GER | Category: Combinatorics Problem Let n and k be positive integers. There are given n circles in the plane. Every two of them intersect at two distinct points, and all points of intersection they determine are distinct. Each intersection point must be colored with one of n distinct colors so that each color is used at least once, and exactly k distinct colors occur...

Practice › Mathematics › IMO Shortlist › IMO 1971 Shortlist ›

IMO 1971 SL 6

IMO 1971 SL 6 Origin: HUN Problem Let n \geq2 be a natural number. Find a way to assign nat- ural numbers to the vertices of a regular 2n-gon such that the following conditions are satisﬁed: (1) only digits 1 and 2 are used; (2) each number consists of exactly n digits; (3) diﬀerent numbers are assigned to diﬀerent vertices; (4) the numbers assigned to two neighboring vertices diﬀer at...

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL N1

IMO 2003 SL N1 Origin: POL | Category: Number Theory Problem Let m be a ﬁxed integer greater than 1. The sequence x0, x1, x2, . . . is deﬁned as follows: xi = . 2i, if 0 \leqi \leqm −1; m j=1 xi−j, if i \geqm. Find the greatest k for which the sequence contains k consecutive terms divisible by m. Solution Let ri be the remainder when xi...

Practice › Mathematics › IMO Shortlist › IMO 1983 Shortlist ›

IMO 1983 SL 6

IMO 1983 SL 6 Origin: CAN Problem Suppose that {x1, x2, . . . , xn} are positive integers for which x1 + x2 + \cdot \cdot \cdot + xn = 2(n + 1). Show that there exists an integer r with 0 \leqr \leqn −1 for which the following n −1 inequalities hold: xr+1 + \cdot \cdot \cdot + xr+i \leq2i + 1 \foralli, 1 \leqi \leqn −r; xr+1...

Practice › Mathematics › IMO Shortlist › IMO 1971 Shortlist ›

IMO 1971 SL 12

IMO 1971 SL 12 Origin: POL Problem Two congruent equilateral triangles ABC and A′B′C′ in the plane are given. Show that the midpoints of the segments AA′, BB′, CC′ either are collinear or form an equilateral triangle. Solution Let us start with the case A = A′. If the triangles ABC and A′B′C′ are oppositely oriented, then they are symmetric with respect to some axis, and the statement is true....

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL G3

IMO 1999 SL G3 Origin: EST | Category: Geometry Problem A set S of points in space will be called completely sym- metric if it has at least three elements and satisﬁes the following condition: For every two distinct points A, B from S the perpendicular bisector of the segment AB is an axis of symmetry for S. Prove that if a completely symmetric set is ﬁnite, then it consists...

Practice › Mathematics › IMO Shortlist › IMO 1981 Shortlist ›

IMO 1981 SL 6

IMO 1981 SL 6 Origin: CUB Problem Let P(z) and Q(z) be complex-variable polynomials, with degree not less than 1. Let Pk = {z \inC | P(z) = k}, Qk = {z \inC | Q(z) = k}. Let also P0 = Q0 and P1 = Q1. Prove that P(z) \equivQ(z). Solution Assume w.l.o.g. that n = degP \geqdegQ, and let P0 = {z1, z2, . . . , zk}, P1...

Practice › Mathematics › IMO Shortlist › IMO 1968 Shortlist ›

IMO 1968 SL 23

IMO 1968 SL 23 Origin: CZS Problem Find all complex numbers m such that polynomial x3 + y3 + z3 + mxyz can be represented as the product of three linear trinomials. Solution We may assume w.l.o.g. that in all the factors the coeﬃcient of x is 1. Suppose that x + ay + bz is one of the linear factors of p(x, y, z) = x3 + y3 +...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 18

IMO 1988 SL 18 Origin: LUX Problem Consider two concentric circles of radii R and r (R > r) with center O. Fix P on the small circle and consider the variable chord PA of the small circle. Points B and C lie on the large circle; B, P, C are collinear and BC is perpendicular to AP. (a) For what value(s) of \angleOPA is the sum BC2 + CA2...

Practice › Mathematics › IMO Shortlist › IMO 1993 Shortlist ›

IMO 1993 SL 11

IMO 1993 SL 11 Origin: IRE Problem Let n > 1 be an integer and let f(x) = xn + 5xn−1 + 3. Prove that there do not exist polynomials g(x), h(x), each having integer coeﬃcients and degree at least one, such that f(x) = g(x)h(x). Solution Due to the extended Eisenstein criterion, f must have an irreducible factor of degree not less than n −1. Since f has no...

Practice › Mathematics › IMO Shortlist › IMO 1973 Shortlist ›

IMO 1973 SL 1

IMO 1973 SL 1 Origin: BUL Problem Let a tetrahedron ABCD be inscribed in a sphere S. Find the locus of points P inside the sphere S for which the equality AP PA1 BP PB1 CP PC1 DP PD1 = 4 holds, where A1, B1, C1, and D1 are the intersection points of S with the lines AP, BP, CP, and DP, respectively. Solution The condition of the point P...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 19

IMO 1991 SL 19 Origin: IRE Problem Let a be a rational number with 0 < a < 1 and suppose that cos 3\pia + 2 cos2\pia = 0. (Angle measurements are in radians.) Prove that a = 2/3. Solution Set x = cos(\pia). The given equation is equivalent to 4x3+4x2−3x−2 = 0, which factorizes as (2x + 1)(2x2 + x −2) = 0. The case 2x + 1 =...

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL N6

IMO 2003 SL N6 Origin: FRA | Category: Number Theory Problem Let p be a prime number. Prove that there exists a prime number q such that for every integer n, the number np −p is not divisible by q. Solution Suppose that for every prime q, there exists an n for which np \equivp (mod q). Assume that q = kp + 1. By Fermat’s theorem we deduce that...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 5

IMO 1989 SL 5 Origin: COL Problem Consider the polynomial p(x) = xn+nxn−1+a2xn−2 +\cdot \cdot \cdot+an having all real roots. If r16 1 + r16 2 + \cdot \cdot \cdot + r16 n = n, where the rj are the roots of p(x), ﬁnd all such roots. Solution According to the Cauchy–Schwarz inequality, n i=1 ai \leq n i=1 a2 i n i=1 = n n ...

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL A6

IMO 2002 SL A6 Origin: IRN | Category: Algebra Problem Let A be a nonempty set of positive integers. Suppose that there are positive integers b1, . . . , bn and c1, . . . , cn such that (i) for each i the set biA + ci = {bia + ci | a \inA} is a subset of A, and (ii) the sets biA + ci and bjA...

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL G1

IMO 1999 SL G1 Origin: ARM | Category: Geometry Problem Let ABC be a triangle and M an interior point. Prove that min{MA, MB, MC} + MA + MB + MC < AB + AC + BC. Solution We use the following simple lemma. Lemma. Suppose that M is the interior point of a convex quadrilateral ABCD. Then it follows that MA + MB < AD + DC + CB....

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL N4

IMO 1995 SL N4 Origin: BUL | Category: Number Theory Problem Find all positive integers x and y such that x+y2+z3 = xyz, where z is the greatest common divisor of x and y. Solution Let x = za and y = zb, where a and b are relatively prime. The given Diophantine equation becomes a + zb2 + z2 = z2ab, so a = zc for some c \inZ....

Practice › Mathematics › IMO Shortlist › IMO 1994 Shortlist ›

IMO 1994 SL C6

IMO 1994 SL C6 Origin: FIN | Category: Combinatorics Problem On an inﬁnite square grid, two players alternately mark sym- bols on empty cells. The ﬁrst player always marks X’s, the second O’s. One symbol is marked per turn. The ﬁrst player wins if there are 11 con- secutive X’s in a row, column, or diagonal. Prove that the second player can prevent the ﬁrst from winning. Solution Tile the...

Practice › Mathematics › IMO Shortlist › IMO 1972 Shortlist ›

IMO 1972 SL 12

IMO 1972 SL 12 Origin: USS Problem A set of 10 positive integers is given such that the decimal expansion of each of them has two digits. Prove that there are two disjoint subsets of the set with equal sums of their elements. Solution First we observe that it is not essential to require the subsets to be disjoint (if they aren’t, one simply excludes their intersection). There are 210−1...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 11

IMO 1988 SL 11 Origin: GDR Problem The lock on a safe consists of three wheels, each of which may be set in eight diﬀerent positions. Due to a defect in the safe mechanism the door will open if any two of the three wheels are in the correct position. What is the smallest number of combinations that must be tried if one is to guarantee being able to open...

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 8

IMO 1990 SL 8 Origin: HUN Problem For a given positive integer k denote the square of the sum of its digits by f1(k) and let fn+1(k) = f1(fn(k)). Determine the value of f1991(21990). Solution Since 21990 < 8700 < 10700, we have f1(21990) < (9\cdot700)2 < 4\cdot107. We then have f2(21990) < (3+9\cdot7)2 < 4900 and ﬁnally f3(21990) < (3+9\cdot3)2 = 302. It is easily shown that fk(n) \equivfk−1(n)2...

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL C3

IMO 2001 SL C3 Origin: RUS | Category: Combinatorics Problem Deﬁne a k-clique to be a set of k people such that every pair of them are acquainted with each other. At a certain party, every pair of 3-cliques has at least one person in common, and there are no 5-cliques. Prove that there are two or fewer people at the party whose departure leaves no 3-clique remaining. Solution Consider...

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 23

IMO 1998 SL 23 Origin: BLR Problem Let n be an integer greater than 2. A positive integer is said to be attainable if it is 1 or can be obtained from 1 by a sequence of operations with the following properties: (i) The ﬁrst operation is either addition or multiplication. (ii) Thereafter, additions and multiplications are used alternately. (iii) In each addition one can choose independently whether to add...

Practice › Mathematics › IMO Shortlist › IMO 1986 Shortlist ›

IMO 1986 SL 8

IMO 1986 SL 8 Origin: USA Problem From a collection of n persons q distinct two-member teams are selected and ranked 1, . . . , q (no ties). Let m be the least integer larger than or equal to 2q/n. Show that there are m distinct teams that may be listed so that (i) each pair of consecutive teams on the list have one member in common and (ii)...

Practice › Mathematics › IMO Shortlist › IMO 1971 Shortlist ›

IMO 1971 SL 13

IMO 1971 SL 13 Origin: SWE Problem Consider the n \times n array of nonnegative integers ⎛ ⎜ ⎜ ⎜ ⎝ a11 a12 . . . a1n a21 a22 \cdot \cdot \cdot a2n ... ... ... an1 an2 . . . ann ⎞ ⎟ ⎟ ⎟ ⎠, with the following property: If an element aij is zero, then the sum of the elements of the ith row and the jth...

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 22

IMO 1990 SL 22 Origin: ROM Problem Ten localities are served by two international airlines such that there exists a direct service (without stops) between any two of these localities and all airline schedules oﬀer round-trip service between the cities they serve. Prove that at least one of the airlines can oﬀer two disjoint round trips each containing an odd number of landings. Solution We can assume without loss of...

Practice › Mathematics › IMO Shortlist › IMO 1987 Shortlist ›

IMO 1987 SL 5

IMO 1987 SL 5 Origin: GBR Problem Find, with proof, the point P in the interior of an acute-angled triangle ABC for which BL2 +CM2 +AN2 is a minimum, where L, M, N are the feet of the perpendiculars from P to BC, CA, AB respectively. Solution Assuming the notation a = BC, b = AC, c = AB; x = BL, y = CM, z = AN, from the...

Practice › Mathematics › IMO Shortlist › IMO 1970 Shortlist ›

IMO 1970 SL 10

IMO 1970 SL 10 Origin: SWE Problem Let 1 = a0 \leqa1 \leqa2 \leq\cdot \cdot \cdot \leqan \leq\cdot \cdot \cdot be a sequence of real numbers. Consider the sequence b1, b2, . . . deﬁned by: bn = n k=1 1 −ak−1 ak \sqrtak . Prove that: (a) For all natural numbers n, 0 \leqbn < 2. (b) Given an arbitrary 0 \leqb < 2, there is...

Practice › Mathematics › IMO Shortlist › IMO 1968 Shortlist ›

IMO 1968 SL 18

IMO 1968 SL 18 Origin: ITA Problem If an acute-angled triangle ABC is given, construct an equilat- eral triangle A′B′C′ in space such that lines AA′, BB′, CC′ pass through a given point. Solution The required construction is not feasible. In fact, let us consider the special case \angleBOC = 135◦, \angleAOC = 120◦, \angleAOB = 90◦, where AA′ \capBB′ \cap CC′ = {O}. Denoting OA′, OB′, OC′ by a,...

Practice › Mathematics › IMO Shortlist › IMO 1981 Shortlist ›

IMO 1981 SL 3

IMO 1981 SL 3 Origin: CAN Problem Find the minimum value of max(a + b + c, b + c + d, c + d + e, d + e + f, e + f + g) subject to the constraints (i) a, b, c, d, e, f, g \geq0, (ii) a + b + c + d + e + f + g = 1. Solution Denote max(a +...

Practice › Mathematics › IMO Shortlist › IMO 1978 Shortlist ›

IMO 1978 SL 8

IMO 1978 SL 8 Origin: GBR Problem Let S be the set of all the odd positive integers that are not multiples of 5 and that are less than 30m, m being an arbitrary positive integer. What is the smallest integer k such that in any subset of k integers from S there must be two diﬀerent integers, one of which divides the other? Solution Take the subset {ai} =...

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL G8

IMO 2001 SL G8 Origin: ISR | Category: Geometry Problem Let ABC be a triangle with ∡BAC = 60◦. Let AP bisect \angleBAC and let BQ bisect \angleABC, with P on BC and Q on AC. If AB + BP = AQ + QB, what are the angles of the triangle? Solution Let S and T respectively be the points on the extensions of AB and AQ over B and...

Practice › Mathematics › IMO Shortlist › IMO 1984 Shortlist ›

IMO 1984 SL 13

IMO 1984 SL 13 Origin: BUL Problem Prove that the volume of a tetrahedron inscribed in a right circular cylinder of volume 1 does not exceed 3\pi. Solution Let Z be the given cylinder of radius r, altitude h, and volume \pir2h = 1, k1 and k2 the circles surrounding its bases, and V the volume of an inscribed tetrahedron ABCD. We claim that there is no loss of generality...

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 4

IMO 1998 SL 4 Origin: ARM Problem Let M and N be points inside triangle ABC such that \angleMAB = \angleNAC and \angleMBA = \angleNBC. Prove that AM \cdot AN AB \cdot AC + BM \cdot BN BA \cdot BC + CM \cdot CN CA \cdot CB = 1. Solution Let K be the point on the ray BN with \angleBCK = \angleBMA. Since \angleKBC = \angleABM, we get \triangleBCK...

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL A3

IMO 2002 SL A3 Origin: POL | Category: Algebra Problem Let P be a cubic polynomial given by P(x) = ax3+bx2+cx+ d, where a, b, c, d are integers and a ̸= 0. Suppose that xP(x) = yP(y) for inﬁnitely many pairs x, y of integers with x ̸= y. Prove that the equation P(x) = 0 has an integer root. Solution Let x, y be distinct integers satisfying xP(x)...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 27

IMO 1988 SL 27 Origin: GBR Problem The triangle ABC is acute-angled. Let L be any line in the plane of the triangle and let u, v, w be the lengths of the perpendiculars from A, B, C respectively to L. Prove that u2 tan A + v2 tan B + w2 tan C \geq2∆, where ∆is the area of the triangle, and determine the lines L for which equality...

Practice › Mathematics › IMO Shortlist › IMO 1968 Shortlist ›

IMO 1968 SL 9

IMO 1968 SL 9 Origin: ROM Problem Let $ABC$ be an arbitrary triangle and $M$ a point inside it. Let $d_a, d_b, d_c$ be the distances from $M$ to sides $BC, CA, AB$; $a, b, c$ the lengths of the sides respectively, and $S$ the area of the triangle $ABC$. Prove the inequality $$abd_ad_b + bcd_bd_c + cad_cd_a \leq \frac{4S^2}{3}.$$ Prove that the left-hand side attains its maximum when $M$...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 6

IMO 1979 SL 6 Origin: CZS Problem Find the real values of p for which the equation 2p + 1 −x2 + 3x + p + 4 = x2 + 9x + 3p + 9 in x has exactly two real distinct roots ( \sqrt t means the positive square root of t). Solution Setting q = x2 + x −p, the given equation becomes (x + 1)2 −2q +...

Practice › Mathematics › IMO Shortlist › IMO 1975 Shortlist ›

IMO 1975 SL 13

IMO 1975 SL 13 Origin: ROM Problem Let A0, A1, . . . , An be points in a plane such that (i) A0A1 \leq1 2A1A2 \leq\cdot \cdot \cdot \leq 2n−1 An−1An and (ii) 0 < ∡A0A1A2 < ∡A1A2A3 < \cdot \cdot \cdot < ∡An−2An−1An < 180◦, where all these angles have the same orientation. Prove that the segments AkAk+1, AmAm+1 do not intersect for each k and n such...

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 27

IMO 1990 SL 27 Origin: USS Problem Find all natural numbers n for which every natural number whose decimal representation has n −1 digits 1 and one digit 7 is prime. Solution Let us denote by An(k) the n-digit number which consists of n −1 ones and one digit seven in the k + 1-th rightmost position (0 \leqk < n). Then An(k) = (10n + 54 \cdot 10k −1)/9....

Practice › Mathematics › IMO Shortlist › IMO 1973 Shortlist ›

IMO 1973 SL 13

IMO 1973 SL 13 Origin: YUG Problem Find the sphere of maximal radius that can be placed inside every tetrahedron that has all altitudes of length greater than or equal to 1. Solution Let S1, S2, S3, S4 denote the areas of the faces of the tetrahedron, V its volume, h1, h2, h3, h4 its altitudes, and r the radius of its inscribed sphere. Since 3V = S1h1 = S2h2...

Practice › Mathematics › IMO Shortlist › IMO 1968 Shortlist ›

IMO 1968 SL 20

IMO 1968 SL 20 Origin: CZS Problem Given $n$ ($n \geq 3$) points in space such that every three of them form a triangle with one angle greater than or equal to $120^\circ$, prove that these points can be denoted by $A_1, A_2, \ldots, A_n$ in such a way that for each $i, j, k$, $1 \leq i < j < k \leq n$, angle $A_iA_jA_k$ is greater than or...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 4

IMO 1988 SL 4 Origin: CZS Problem An n \times n chessboard (n \geq2) is numbered by the numbers 1, 2, . . . , n2 (every number occurs once). Prove that there exist two neigh- boring (which share a common edge) squares such that their numbers diﬀer by at least n. Solution Suppose that the numbers of any two neighboring squares diﬀer by at most n −1. For k...

Practice › Mathematics › IMO Shortlist › IMO 1968 Shortlist ›

IMO 1968 SL 4

IMO 1968 SL 4 Origin: BUL Problem Let $a$, $b$, $c$ be real numbers. Prove that the system of equations $$\begin{cases} ax_1^2 + bx_1 + c = x_2, \ ax_2^2 + bx_2 + c = x_3, \ \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \ ax_{n-1}^2 + bx_{n-1} + c = x_n, \ ax_n^2 + bx_n + c = x_1, \end{cases}$$ has a unique real...

Practice › Mathematics › IMO Shortlist › IMO 1974 Shortlist ›

IMO 1974 SL 11

IMO 1974 SL 11 Problem II 5 (BUL 1)IMO4 Consider a partition of an 8 \times 8 chessboard into p rectangles whose interiors are disjoint such that each of them has an equal number of white and black cells. Assume that a1 < a2 < \cdot \cdot \cdot < ap, where ai denotes the number of white cells in the ith rectangle. Find the maximal p for which such a...

Practice › Mathematics › IMO Shortlist › IMO 1971 Shortlist ›

IMO 1971 SL 17

IMO 1971 SL 17 Origin: YUG Problem Prove the inequality a1 + a3 a1 + a2 a2 + a4 a2 + a3 a3 + a1 a3 + a4 a4 + a2 a4 + a1 \geq4, where ai > 0, i = 1, 2, 3, 4. Solution We use the following obvious consequences of (a + b)2 \geq4ab: (a1 + a2)(a3 + a4) \geq (a1 + a2 + a3 + a4)2...

Practice › Mathematics › IMO Shortlist › IMO 1984 Shortlist ›

IMO 1984 SL 20

IMO 1984 SL 20 Origin: USA Problem Determine all pairs (a, b) of positive real numbers with a ̸= 1 such that loga b < loga+1(b + 1). Solution Deﬁne the set S = R+ ∖{1}. The given inequality is equivalent to ln b/ln a < ln (b + 1)/ln (a + 1). If b = 1, it is obvious that each a \inS satisﬁes this inequality. Suppose now that...

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 3

IMO 1998 SL 3 Origin: UKR Problem Let I be the incenter of triangle ABC. Let K, L, and M be the points of tangency of the incircle of ABC with AB, BC, and CA, respectively. The line t passes through B and is parallel to KL. The lines MK and ML intersect t at the points R and S. Prove that \angleRIS is acute. Solution Lemma. If U, W,...

Practice › Mathematics › IMO Shortlist › IMO 1994 Shortlist ›

IMO 1994 SL N7

IMO 1994 SL N7 Origin: GBR | Category: Number Theory Problem A wobbly number is a positive integer whose digits in base 10 are alternately nonzero and zero, the units digit being nonzero. Determine all positive integers that do not divide any wobbly number. Solution A multiple of 10 does not divide any wobbly number. Also, if 25 | n, then every multiple of n ends with 25, 50, 75,...

Practice › Mathematics › IMO Shortlist › IMO 1971 Shortlist ›

IMO 1971 SL 15

IMO 1971 SL 15 Origin: USS Problem Natural numbers from 1 to 99 (not necessarily distinct) are written on 99 cards. It is given that the sum of the numbers on any subset of cards (including the set of all cards) is not divisible by 100. Show that all the cards contain the same number. Solution Assume the opposite. Then one can numerate the cards 1 to 99, with a...

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 10

IMO 1998 SL 10 Origin: AUS Problem Let r1, r2, . . . , rn be real numbers greater than or equal to 1. Prove that r1 + 1 + r2 + 1 + \cdot \cdot \cdot + rn + 1 \geq n n\sqrtr1r2 \cdot \cdot \cdot rn + 1. Solution We shall ﬁrst prove the inequality for n of the form 2k, k = 0, 1, 2, . ....

Practice › Mathematics › IMO Shortlist › IMO 1997 Shortlist ›

IMO 1997 SL 20

IMO 1997 SL 20 Origin: IRE Problem Let D be an internal point on the side BC of a triangle ABC. The line AD meets the circumcircle of ABC again at X. Let P and Q be the feet of the perpendiculars from X to AB and AC, respectively, and let \gamma be the circle with diameter XD. Prove that the line PQ is tangent to \gamma if and only...

Practice › Mathematics › IMO Shortlist › IMO 1985 Shortlist ›

IMO 1985 SL 16

IMO 1985 SL 16 Problem 5b.(BEL 2) If possible, construct an equilateral triangle whose three vertices are on three given circles. Solution Let the three circles be \alpha(A, a), \beta(B, b), and \gamma(C, c), and assume c \leqa, b. We denote by RX,ϕ the rotation around X through an angle ϕ. Let PQR be an equilateral triangle, say of positive orientation (the case of negatively oriented \trianglePQR is analogous), with...

Practice › Mathematics › IMO Shortlist › IMO 1982 Shortlist ›

IMO 1982 SL 16

IMO 1982 SL 16 Problem C4 (GBR 2)IMO4 Prove that if n is a positive integer such that the equation x3 −3xy2 + y3 = n has a solution in integers (x, y), then it has at least three such solutions. Show that the equation has no solution in integers when n = 2891. Solution It is easy to verify that whenever (x, y) is a solution of the equation...

Practice › Mathematics › IMO Shortlist › IMO 1983 Shortlist ›

IMO 1983 SL 18

IMO 1983 SL 18 Origin: FRG Problem Let a, b, c be positive integers satisfying (a, b) = (b, c) = (c, a) = 1. Show that 2abc −ab −bc −ca is the largest integer not repre- sentable as xbc + yca + zab with nonnegative integers x, y, z. Solution Let (x0, y0, z0) be one solution of bcx + cay + abz = n (not necessarily nonnegative). By...

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL G3

IMO 2000 SL G3 Origin: IND | Category: Geometry Problem Let O be the circumcenter and H the orthocenter of an acute triangle ABC. Show that there exist points D, E, and F on sides BC, CA, and AB respectively such that OD + DH = OE + EH = OF + FH and the lines AD, BE, and CF are concurrent. Solution Let L be the point symmetric to...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 20

IMO 1991 SL 20 Origin: IRE Problem Let \alpha be the positive root of the equation x2 = 1991x + 1. For natural numbers m, n deﬁne m ∗n = mn + [\alpham][\alphan], where [x] is the greatest integer not exceeding x. Prove that for all natural numbers p, q, r, (p ∗q) ∗r = p ∗(q ∗r). Solution We prove the result with 1991 replaced by any positive integer...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 11

IMO 1979 SL 11 Origin: GDR Problem Given real numbers x1, x2, . . . , xn (n \geq2), with xi \geq1/n (i = 1, 2, . . ., n) and with x2 1 +x2 2 +\cdot \cdot \cdot+x2 n = 1, ﬁnd whether the product P = x1x2x3 \cdot \cdot \cdot xn has a greatest and/or least value and if so, give these values. Solution Let us deﬁne yi...

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL A3

IMO 2000 SL A3 Origin: BLR | Category: Algebra Problem Find all pairs of functions f : R \toR, g : R \toR such that f(x + g(y)) = xf(y) −yf(x) + g(x) for all x, y \inR. Solution Let us ﬁrst solve the problem under the assumption that g(\alpha) = 0 for some \alpha. Setting y = \alpha in the given equation yields g(x) = (\alpha+1)f(x)−xf(\alpha). Then the given...

Practice › Mathematics › IMO Shortlist › IMO 1972 Shortlist ›

IMO 1972 SL 9

IMO 1972 SL 9 Origin: NET Problem Find all solutions in positive real numbers xi (i = 1, 2, 3, 4, 5) of the following system of inequalities: (x2 1 −x3x5)(x2 2 −x3x5) \leq0, (i) (x2 2 −x4x1)(x2 3 −x4x1) \leq0, (ii) (x2 3 −x5x2)(x2 4 −x5x2) \leq0, (iii) (x2 4 −x1x3)(x2 5 −x1x3) \leq0, (iv) (x2 5 −x2x4)(x2 1 −x2x4) \leq0. (v) Solution Clearly x1 = x2 = x3...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 17

IMO 1991 SL 17 Origin: HKG Problem Find all positive integer solutions x, y, z of the equation 3x + 4y = 5z. Solution Taking the equation 3x + 4y = 5z (x, y, z > 0) modulo 3, we get that 5z \equiv1 (mod 3), hence z is even, say z = 2z1. The equation then becomes 3x = 52z1 −4y = (5z1 −2y)(5z1 + 2y). Each factor 5z1...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 14

IMO 1989 SL 14 Origin: IND Problem A bicentric quadrilateral is one that is both inscribable in and circumscribable about a circle. Show that for such a quadrilateral, the centers of the two associated circles are collinear with the point of intersection of the diagonals. Solution Lemma 1. In a quadrilateral ABCD circumscribed about a circle, with points of tangency P, Q, R, S on DA, AB, BC, CD respectively,...

Practice › Mathematics › IMO Shortlist › IMO 1977 Shortlist ›

IMO 1977 SL 1

IMO 1977 SL 1 Origin: BUL Problem Let f : N \toN be a function that satisﬁes the inequality f(n + 1) > f(f(n)) for all n \inN. Prove that f(n) = n for all natural numbers n.

Practice › Mathematics › IMO Shortlist › IMO 1984 Shortlist ›

IMO 1984 SL 7

IMO 1984 SL 7 Origin: FRG Problem (a) Decide whether the ﬁelds of the 8 \times 8 chessboard can be numbered by the numbers 1, 2, . . ., 64 in such a way that the sum of the four numbers in each of its parts of one of the forms is divisible by four. (b) Solve the analogous problem for Solution It clearly suﬃces to solve the problem for...

Practice › Mathematics › IMO Shortlist › IMO 1993 Shortlist ›

IMO 1993 SL 17

IMO 1993 SL 17 Origin: NET Problem Let n be an integer greater than 1. In a circular arrange- ment of n lamps L0, . . . , Ln−1, each one of that can be either ON or OFF, we start with the situation where all lamps are ON, and then carry out a sequence of steps, Step0,Step1,. . . . If Lj−1 (j is taken mod n) is ON,...

Practice › Mathematics › IMO Shortlist › IMO 1978 Shortlist ›

IMO 1978 SL 2

IMO 1978 SL 2 Origin: BUL Problem Two identically oriented equilateral triangles, ABC with center S and A′B′C, are given in the plane. We also have A′ ̸= S and B′ ̸= S. If M is the midpoint of A′B and N the midpoint of AB′, prove that the triangles SB′M and SA′N are similar. Solution Consider the transformation \varphi of the plane deﬁned as the homothety H with center...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL A7

IMO 2004 SL A7 Origin: IRE | Category: Algebra Problem Let a1, a2, . . . , an be positive real numbers, n > 1. Denote by gn their geometric mean, and by A1, A2, . . . , An the sequence of arithmetic means deﬁned by Ak = a1+a2+\cdot\cdot\cdot+ak k , k = 1, 2, . . ., n. Let Gn be the geometric mean of A1, A2, ....

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 4

IMO 1989 SL 4 Origin: BUL Problem Prove that for every integer n > 1 the equation xn n! + xn−1 (n −1)! + \cdot \cdot \cdot + x2 2! + x 1! + 1 = 0 has no rational roots. Solution First we note that for every integer k > 0 and prime number p, pk doesn’t divide k!. This follows from the fact that the highest exponent r...

Practice › Mathematics › IMO Shortlist › IMO 1968 Shortlist ›

IMO 1968 SL 7

IMO 1968 SL 7 Origin: HUN Problem Prove that the product of the radii of three circles exscribed to a given triangle does not exceed $3\sqrt{3}$ times the product of the side lengths of the triangle. When does equality hold? Solution Let $r_a$, $r_b$, $r_c$ denote the radii of the exscribed circles corresponding to the sides of lengths $a$, $b$, $c$ respectively, and $R$, $p$ and $S$ denote the circumradius,...

Practice › Mathematics › IMO Shortlist › IMO 1994 Shortlist ›

IMO 1994 SL N4

IMO 1994 SL N4 Origin: FRA | Category: Number Theory Problem For any positive integer x0, three sequences {xn}, {yn}, and {zn} are deﬁned as follows: (i) y0 = 4 and z0 = 1; (ii) if xn is even for n \geq0, xn+1 = xn 2 , yn+1 = 2yn, and zn+1 = zn; (iii) if xn is odd for n \geq0, xn+1 = xn −yn 2 −zn, yn+1 =...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL N1

IMO 1996 SL N1 Origin: UKR | Category: Number Theory Problem Four integers are marked on a circle. At each step we simultaneously replace each number by the difference between this number and the next number on the circle, in a given direction (that is, the numbers $a,b,c,d$ are replaced by $a-b$, $b-c$, $c-d$, $d-a$). Is it possible after $1996$ such steps to have numbers $a,b,c,d$ such that the numbers...

Practice › Mathematics › IMO Shortlist › IMO 1978 Shortlist ›

IMO 1978 SL 10

IMO 1978 SL 10 Origin: NET Problem An international society has its members in 6 diﬀerent countries. The list of members contains 1978 names, numbered 1, 2, . . . , Solution Assume the opposite. One of the countries, say A, contains at least 330 members a1, a2, . . . , a330 of the society ( 6 \cdot 329 = 1974). Consider the diﬀerences a330 −ai, = 1, 2,...

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 12

IMO 1998 SL 12 Origin: POL Problem Let n \geqk \geq0 be integers. The numbers c(n, k) are deﬁned as follows: c(n, 0) = c(n, n) = 1 for all n \geq0; c(n + 1, k) = 2kc(n, k) + c(n, k −1) for n \geqk \geq1. Prove that c(n, k) = c(n, n −k) for all n \geqk \geq0. Solution The assertion is clear for n = 0. We...

Practice › Mathematics › IMO Shortlist › IMO 1982 Shortlist ›

IMO 1982 SL 11

IMO 1982 SL 11 Problem B5 (CAN 3) (a) Find the rearrangement {a1, . . . , an} of {1, 2, . . ., n} that maximizes a1a2 + a2a3 + \cdot \cdot \cdot + ana1 = Q. (b) Find the rearrangement that minimizes Q. Solution (a) Suppose {a1, a2, . . . , an} is the arrangement that yields the maxi- mal value Qmax of Q. Note that the...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 30

IMO 1989 SL 30 Origin: SWE Problem For which positive integers n does there exist a positive integer N such that none of the integers 1 + N, 2 + N, . . . , n + N is the power of a prime number? Solution For all n such N exists. For a given n choose N = (n + 1)!2 + 1. Then 1 + j is a...

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL 27

IMO 1995 SL 27 Problem S5 (FIN) For positive integers n, the numbers f(n) are deﬁned induc- tively as follows: f(1) = 1, and for every positive integer n, f(n+1) is the greatest integer m such that there is an arithmetic progression of positive integers a1 < a2 < \cdot \cdot \cdot < am = n for which f(a1) = f(a2) = \cdot \cdot \cdot = f(am). Prove that there...

Practice › Mathematics › IMO Shortlist › IMO 1976 Shortlist ›

IMO 1976 SL 12

IMO 1976 SL 12 Origin: VIE Problem The polynomial 1976(x+x2+\cdot \cdot \cdot+xn) is decomposed into a sum of polynomials of the form a1x + a2x2 + . . . + anxn, where a1, a2, \cdot \cdot \cdot , an are distinct positive integers not greater than n. Find all values of n for which such a decomposition is possible. Solution Suppose the decomposition into k polynomials is possible. The sum...

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL C1

IMO 2000 SL C1 Origin: HUN | Category: Combinatorics Problem A magician has one hundred cards numbered 1 to 100. He puts them into three boxes, a red one, a white one, and a blue one, so that each box contains at least one card. A member of the audience draws two cards from two diﬀerent boxes and announces the sum of numbers on those cards. Given this information, the...

Practice › Mathematics › IMO Shortlist › IMO 1985 Shortlist ›

IMO 1985 SL 8

IMO 1985 SL 8 Problem 1b.(TUR 5) Find the smallest positive integer n such that (i) n has exactly 144 distinct positive divisors, and (ii) there are ten consecutive integers among the positive divisors of n. Solution Among ten consecutive integers that divide n, there must exist numbers divisible by 23, 32, 5, and 7. Thus the desired number has the form n = 2\alpha13\alpha25\alpha37\alpha411\alpha5 \cdot \cdot \cdot , where...

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL A5

IMO 2003 SL A5 Origin: KOR | Category: Algebra Problem Let R+ be the set of all positive real numbers. Find all functions f : R+ \toR+ that satisfy the following conditions: (i) f(xyz) + f(x) + f(y) + f(z) = f(\sqrtxy)f(\sqrtyz)f(\sqrtzx) for all x, y, z \in R+. (ii) f(x) < f(y) for all 1 \leqx < y. Solution Placing x = y = z = 1 in (i)...

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 18

IMO 1990 SL 18 Origin: NOR Problem Let a, b be natural numbers with 1 \leqa \leqb, and M = / a+b . Deﬁne the function f : Z \toZ by f(n) = . n + a, if n < M, n −b, if n \geqM. Let f 1(n) = f(n), f i+1(n) = f(f i(n)), i = 1, 2, . . . . Find the smallest natural number k...

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL C6

IMO 2001 SL C6 Origin: CAN | Category: Combinatorics Problem For a positive integer n deﬁne a sequence of zeros and ones to be balanced if it contains n zeros and n ones. Two balanced sequences a and b are neighbors if you can move one of the 2n symbols of a to another position to form b. For instance, when n = 4, the balanced sequences 01101001 and 00110101...

Practice › Mathematics › IMO Shortlist › IMO 1992 Shortlist ›

IMO 1992 SL 15

IMO 1992 SL 15 Origin: PRK Problem Does there exist a set M with the following properties? (i) The set M consists of 1992 natural numbers. (ii) Every element in M and the sum of any number of elements have the form mk (m, k \inN, k \geq2). Solution The result follows from the following lemma by taking n = 1992\cdot1993 and M = {d, 2d, . . . ,...

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL N1

IMO 2001 SL N1 Origin: AUS | Category: Number Theory Problem Prove that there is no positive integer n such that for k = 1, 2, . . . , 9, the leftmost digit (in decimal notation) of (n + k)! equals k. Solution For each positive integer x, deﬁne \alpha(x) = x/10r if r is the positive integer satisfying 10r \leqx < 10r+1. Observe that if \alpha(x)\alpha(y) < 10...

Practice › Mathematics › IMO Shortlist › IMO 1985 Shortlist ›

IMO 1985 SL 20

IMO 1985 SL 20 Origin: GBR Problem A circle whose center is on the side ED of the cyclic quadrilateral BCDE touches the other three sides. Prove that EB+CD = ED. Solution Let O be the center of the circle touching the three sides of BCDE and let F \in(ED) be the point such that EF = EB. Then \angleEFB = 90◦− \angleE/2 = \angleC/2 = \angleOCB, which implies that...

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL G3

IMO 2001 SL G3 Origin: GBR | Category: Geometry Problem Let ABC be a triangle with centroid G. Determine, with proof, the position of the point P in the plane of ABC such that AP \cdot AG + BP \cdot BG + CP \cdot CG is a minimum, and express this minimum value in terms of the side lengths of ABC. Solution Let us investigate a more general problem, in...

Practice › Mathematics › IMO Shortlist › IMO 1986 Shortlist ›

IMO 1986 SL 3

IMO 1986 SL 3 Origin: USA Problem Let A, B, and C be three points on the edge of a circular chord such that B is due west of C and ABC is an equilateral triangle whose side is 86 meters long. A boy swam from A directly toward B. After covering a distance of x meters, he turned and swam westward, reaching the shore after covering a distance of...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 2

IMO 1979 SL 2 Origin: BEL Problem From a bag containing 5 pairs of socks, each pair a diﬀerent color, a random sample of 4 single socks is drawn. Any complete pairs in the sample are discarded and replaced by a new pair draw from the bag. The process continues until the bag is empty or there are 4 socks of diﬀerent colors held outside the bag. What is the...

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL N1

IMO 2000 SL N1 Origin: JAP | Category: Number Theory Problem Determine all positive integers n \geq2 that satisfy the following condition: For all integers a, b relatively prime to n, a \equivb (mod n) if and only if ab \equiv1 (mod n). Solution The given condition is obviously equivalent to a2 \equiv1 (mod n) for all inte- gers a coprime to n. Let n = p\alpha1 1 p\alpha2 2...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL A3

IMO 2004 SL A3 Origin: CAN | Category: Algebra Problem Does there exist a function s: Q \to{−1, 1} such that if x and y are distinct rational numbers satisfying xy = 1 or x + y \in{0, 1}, then s(x)s(y) = −1? Justify your answer. Solution The answer is yes. Every rational number x > 0 can be uniquely expressed as a continued fraction of the form a0 +...

Practice › Mathematics › IMO Shortlist › IMO 1994 Shortlist ›

IMO 1994 SL G2

IMO 1994 SL G2 Origin: UKR | Category: Geometry Problem ABCD is a quadrilateral with BC parallel to AD. M is the midpoint of CD, P that of MA and Q that of MB. The lines DP and CQ meet at N. Prove that N is not outside triangle ABM.8 Solution Although it does not seem to have been noticed at the jury, the state- ment of the problem is...

Practice › Mathematics › IMO Shortlist › IMO 1982 Shortlist ›

IMO 1982 SL 20

IMO 1982 SL 20 Problem C8 (TUN 3) Let ABCD be a convex quadrilateral and draw regular tri- angles ABM, CDP, BCN, ADQ, the ﬁrst two outward and the other two inward. Prove that MN = AC. What can be said about the quadrilateral MNPQ? Solution Since MN is the image of AC under rotation about B for 60◦, we have MN = AC. Similarly, PQ is the image of...

Practice › Mathematics › IMO Shortlist › IMO 1977 Shortlist ›

IMO 1977 SL 8

IMO 1977 SL 8 Origin: GBR Problem Let S be a convex quadrilateral ABCD and O a point inside it. The feet of the perpendiculars from O to AB, BC, CD, DA are A1, B1, C1, D1 respectively. The feet of the perpendiculars from O to the sides of Si, the quadrilateral AiBiCiDi, are Ai+1Bi+1Ci+1Di+1, where i = 1, 2, 3. Prove that S4 is similar to S.

Practice › Mathematics › IMO Shortlist › IMO 1997 Shortlist ›

IMO 1997 SL 11

IMO 1997 SL 11 Origin: NET Problem Let P(x) be a polynomial with real coeﬃcients such that P(x) > 0 for all x \geq0. Prove that there exists a positive integer n such that (1 + x)nP(x) is a polynomial with nonnegative coeﬃcients. Solution All real roots of P(x) (if any) are negative: say −a1, −a2, . . . , −ak. Then P(x) can be factored as P(x) = C(x...

Practice › Mathematics › IMO Shortlist › IMO 1973 Shortlist ›

IMO 1973 SL 14

IMO 1973 SL 14 Origin: YUG Problem A soldier has to investigate whether there are mines in an area that has the form of an equilateral triangle. The radius of his detector is equal to one-half of an altitude of the triangle. The soldier starts from one vertex of the triangle. Determine the shortest path that the soldier has to traverse in order to check the whole region. Solution Suppose...

Practice › Mathematics › IMO Shortlist › IMO 1985 Shortlist ›

IMO 1985 SL 4

IMO 1985 SL 4 Origin: AUS Problem Each of the numbers in the set N = {1, 2, 3, . . ., n −1}, where n \geq3, is colored with one of two colors, say red or black, so that: (i) i and n −i always receive the same color, and (ii) for some j \inN, relatively prime to n, i and |j −i| receive the same color for all...

Practice › Mathematics › IMO Shortlist › IMO 1978 Shortlist ›

IMO 1978 SL 13

IMO 1978 SL 13 Origin: USA Problem Given any point P in the interior of a sphere with ra- dius R, three mutually perpendicular segments PA, PB, PC are drawn terminating on the sphere and having one common vertex in P. Con- sider the rectangular parallelepiped of which PA, PB, PC are coterminal edges. Find the locus of the point Q that is diagonally opposite P in the parallelepiped when...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 7

IMO 1988 SL 7 Origin: FRA Problem Let a be the greatest positive root of the equation x3−3x2+1 = 0. Show that [a1788] and [a1988] are both divisible by 17. ([x] denotes the integer part of x.) Solution The algebraic equation x3 −3x2 + 1 = 0 admits three real roots \beta, \gamma, a, with −0.6 < \beta < −0.5, 0.6 < \gamma < 0.7, \sqrt 8 < a <...

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 28

IMO 1998 SL 28 Origin: IRN Problem A solitaire game is played on an m \times n rectangular board, using mn markers that are white on one side and black on the other. Initially, each square of the board contains a marker with its white side up, except for one corner square, which contains a marker with its black side up. In each move, one can take away one marker...

Practice › Mathematics › IMO Shortlist › IMO 1993 Shortlist ›

IMO 1993 SL 14

IMO 1993 SL 14 Origin: ISR Problem The vertices D, E, F of an equilateral triangle lie on the sides BC, CA, AB respectively of a triangle ABC. If a, b, c are the respective lengths of these sides, and S the area of ABC, prove that DE \geq \sqrt 2S a2 + b2 + c2 + 4 \sqrt 3S . Solution Consider any point T inside the triangle ABC...

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL N5

IMO 1995 SL N5 Origin: IRE | Category: Number Theory Problem At a meeting of 12k people, each person exchanges greetings with exactly 3k + 6 others. For any two people, the number who exchange greetings with both is the same. How many people are at the meeting? Solution For each two people let n be the number of people exchanging greetings with both of them. To determine n in...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 17

IMO 1979 SL 17 Origin: NET Problem Inside an equilateral triangle ABC one constructs points P, Q and R such that \angleQAB = \anglePBA = 15◦, \angleRBC = \angleQCB = 20◦, \anglePCA = \angleRAC = 25◦. Determine the angles of triangle PQR. Solution Let K, L, and M be intersections of CQ and BR, AR and CP, and AQ and BP, respectively. Let \angleX denote the angle of the hexagon...

Practice › Mathematics › IMO Shortlist › IMO 1982 Shortlist ›

IMO 1982 SL 4

IMO 1982 SL 4 Problem A4 (BUL 2) Determine all real values of the parameter a for which the equation 16x4 −ax3 + (2a + 17)x2 −ax + 16 = 0 has exactly four distinct real roots that form a geometric progression. Solution Suppose that a satisﬁes the requirements of the problem and that x, qx, q2x, q3x are the roots of the given equation. Then x ̸= 0 and...

Practice › Mathematics › IMO Shortlist › IMO 1975 Shortlist ›

IMO 1975 SL 11

IMO 1975 SL 11 Origin: GBR Problem Let a1, a2, a3, . . . be any inﬁnite increasing sequence of pos- itive integers. (For every integer i > 0, ai+1 > ai.) Prove that there are inﬁnitely many m for which positive integers x, y, h, k can be found such that 0 < h < k < m and am = xah + yak. Solution Let (aki) be the...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 10

IMO 1979 SL 10 Origin: FIN Problem Show that for any vectors a, b in Euclidean space, |a \times b|3 \leq3 \sqrt 8 |a|2|b|2|a −b|2. Remark. Here \times denotes the vector product. Solution In the cases a = −\to0 , b = −\to0 , and a \parallelb the inequality is trivial. Otherwise, let us consider a triangle ABC such that −−\to CB = a and −\to CA = b. From...

Practice › Mathematics › IMO Shortlist › IMO 1978 Shortlist ›

IMO 1978 SL 3

IMO 1978 SL 3 Origin: CUB Problem Let n > m \geq1 be natural numbers such that the groups of the last three digits in the decimal representation of 1978m, 1978n coincide. Find the ordered pair (m, n) of such m, n for which m + n is minimal. Solution What we need are m, n for which 1978m(1978n−m −1) is divisible by 1000 = 8 \cdot125. Since 1978n−m −1...

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL G6

IMO 1999 SL G6 Origin: RUS | Category: Geometry Problem Two circles Ω1 and Ω2 touch internally the circle Ωin M and N, and the center of Ω2 is on Ω1. The common chord of the circles Ω1 and Ω2 intersects Ωin A and B. MA and MB intersect Ω1 in C and D. Prove that Ω2 is tangent to CD. Solution We ﬁrst introduce the following lemmas. Lemma 1....

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 1

IMO 1990 SL 1 Origin: AUS Problem The integer 9 can be written as a sum of two consecutive integers: 9 = 4+5. Moreover, it can be written as a sum of (more than one) consecutive positive integers in exactly two ways: 9 = 4 + 5 = 2 + 3 + 4. Is there an integer that can be written as a sum of 1990 consecutive integers and that...

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL 23

IMO 1995 SL 23 Problem S1 (UKR) Does there exist a sequence F(1), F(2), F(3), . . . of nonneg- ative integers that simultaneously satisﬁes the following three conditions? (a) Each of the integers 0, 1, 2, . . . occurs in the sequence. (b) Each positive integer occurs in the sequence inﬁnitely often. (c) For any n \geq2, F F n163 = F(F(n)) + F(F(361)). Solution By...

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 12

IMO 1990 SL 12 Origin: IRE Problem Let ABC be a triangle and L the line through C parallel to the side AB. Let the internal bisector of the angle at A meet the side BC at D and the line L at E and let the internal bisector of the angle at B meet the side AC at F and the line L at G. If GF = DE,...

Practice › Mathematics › IMO Shortlist › IMO 1982 Shortlist ›

IMO 1982 SL 5

IMO 1982 SL 5 Problem A5 (NET 2)IMO5 Let A1A2A3A4A5A6 be a regular hexagon. Each of its diagonals Ai−1Ai+1 is divided into the same ratio \lambda 1−\lambda, where 0 < \lambda < 1, by a point Bi in such a way that Ai, Bi, and Bi+2 are collinear (i \equiv 1, . . . , 6 (mod 6)). Compute \lambda. Solution We ﬁrst observe that \triangleA5B4A4 ∼= \triangleA3B2A2. Since \angleA5A3A2...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 14

IMO 1988 SL 14 Origin: HUN Problem For what values of n does there exist an n \times n array of entries −1, 0, or 1 such that the 2n sums obtained by summing the elements of the rows and the columns are all diﬀerent? Solution Consider an array [aij] of the given property and denote the sums of the rows and the columns by ri and cj respectively. Among...

Practice › Mathematics › IMO Shortlist › IMO 1983 Shortlist ›

IMO 1983 SL 22

IMO 1983 SL 22 Origin: SWE Problem Let n be a positive integer having at least two diﬀerent prime factors. Show that there exists a permutation a1, a2, . . . , an of the integers 1, 2, . . . , n such that n k=1 k \cdot cos 2\piak n = 0. Solution Decompose n into n = st, where the greatest common divisor of s and...

Practice › Mathematics › IMO Shortlist › IMO 1982 Shortlist ›

IMO 1982 SL 15

IMO 1982 SL 15 Problem C3 (CAN 5) Show that 1 −sa 1 −s \leq(1 + s)a−1 holds for every 1 ̸= s > 0 real and 0 < a \leq1 rational. Solution Let a = k/n, where n, k \inN, n \geqk. Putting tn = s, the given inequality becomes 1−tk 1−tn \leq(1 + tn)k/n−1, or equivalently (1 + t + \cdot \cdot \cdot + tk−1)n(1 + tn)n−k \leq(1...

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL G8

IMO 1999 SL G8 Origin: RUS | Category: Geometry Problem Points A, B, C divide the circumcircle Ωof the triangle ABC into three arcs. Let X be a variable point on the arc AB, and let O1, O2 be the incenters of the triangles CAX and CBX. Prove that the circumcircle of the triangle XO1O2 intersects Ωin a ﬁxed point. Solution We ﬁrst introduce the same lemma as in problem...

Practice › Mathematics › IMO Shortlist › IMO 1983 Shortlist ›

IMO 1983 SL 11

IMO 1983 SL 11 Problem (FIN 2′) Let f : [0, 1] \toR be continuous and satisfy: bf(2x) = f(x), 0 \leqx \leq1/2; f(x) = b + (1 −b)f(2x −1), 1/2 \leqx \leq1, where b = 1+c 2+c, c > 0. Show that 0 < f(x)−x < c for every x, 0 < x < 1. Solution First suppose that the binary representation of x is ﬁnite: x = 0,...

Practice › Mathematics › IMO Shortlist › IMO 1986 Shortlist ›

IMO 1986 SL 5

IMO 1986 SL 5 Origin: FRG Problem The set S = {2, 5, 13} has the property that for every a, b \inS, a ̸= b, the number ab −1 is a perfect square. Show that for every positive integer d not in S, the set S \cup{d} does not have the above property. Solution Suppose that for every a, b \in{2, 5, 13, d}, a ̸= b, the number...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 22

IMO 1991 SL 22 Origin: USA Problem Real constants a, b, c are such that there is exactly one square all of whose vertices lie on the cubic curve y = x3 + ax2 + bx + c. Prove that the square has sides of length 4\sqrt 72. Solution Suppose w.l.o.g. that the center of the square is at the origin O(0, 0). We denote the curve y = f(x)...

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 7

IMO 1998 SL 7 Origin: GBR Problem Let ABC be a triangle such that \angleACB = 2\angleABC. Let D be the point on the side BC such that CD = 2BD. The segment AD is extended to E so that AD = DE. Prove that \angleECB + 180◦= 2\angleEBC. Solution We shall use the following result. Lemma. In a triangle ABC with BC = a, CA = b, and AB...

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL N6

IMO 1999 SL N6 Origin: BLR | Category: Number Theory Problem Prove that for every real number M there exists an inﬁnite arithmetic progression such that: (i) each term is a positive integer and the common diﬀerence is not di- visible by 10; (ii) the sum of the digits of each term (in decimal representation) exceeds M. Solution Let N be the smallest integer greater than M. We take the...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 5

IMO 1988 SL 5 Origin: CZS Problem Let n be an even positive integer. Let A1, A2, . . . , An+1 be sets having n elements each such that any two of them have exactly one element in common while every element of their union belongs to at least two of the given sets. For which n can one assign to every element of the union one of the...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL G1

IMO 1996 SL G1 Origin: GBR | Category: Geometry Problem Let $\triangle ABC$ have orthocenter $H$, and let $P$ be a point on its circumcircle, distinct from $A$, $B$, $C$. Let $E$ be the foot of the altitude from $B$, let $PAQB$ and $PARC$ be parallelograms, and let $AQ$ meet $HR$ in $X$. Prove that $EX\parallel AP$. Solution We first show that $H$ is the common orthocenter of the triangles...

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL G5

IMO 2000 SL G5 Origin: GBR | Category: Geometry Problem The tangents at B and A to the circumcircle of an acute- angled triangle ABC meet the tangent at C at T and U respectively. AT meets BC at P, and Q is the midpoint of AP; BU meets CA at R, and S is the midpoint of BR. Prove that \angleABQ = \angleBAS. Determine, in terms of ratios of...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 23

IMO 1979 SL 23 Origin: USA Problem Find all natural numbers n for which 28 + 211 + 2n is a perfect square. Solution It is easily veriﬁed that no solutions exist for n \leq8. Let us now assume that n > 8. We note that 28 + 211 + 2n = 28 \cdot (9 + 2n−8). Hence 9 + 2n−8 must also be a square, say 9 + 2n−8...

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL A6

IMO 1999 SL A6 Origin: SWE | Category: Algebra Problem For n \geq3 and a1 \leqa2 \leq\cdot \cdot \cdot \leqan given real numbers we have the following instructions: (1) place the numbers in some order in a circle; (2) delete one of the numbers from the circle; (3) if just two numbers are remaining in the circle, let S be the sum of these two numbers. Otherwise, if there are...

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 6

IMO 1998 SL 6 Origin: POL Problem Let ABCDEF be a convex hexagon such that \angleB +\angleD +\angleF = 360◦and AB BC \cdot CD DE \cdot EF FA = 1. Prove that BC CA \cdot AE EF \cdot FD DB = 1. Solution Let P be the point such that \triangleCDP and \triangleCBA are similar and equally oriented. Since then \angleDCP = \angleBCA and BC CA = DC CP ,...

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL A2

IMO 2003 SL A2 Origin: AUS | Category: Algebra Problem Find all nondecreasing functions f : R \toR such that (i) f(0) = 0, f(1) = 1; (ii) f(a) + f(b) = f(a)f(b) + f(a + b −ab) for all real numbers a, b such that a < 1 < b. Solution We can rewrite (ii) as −(f(a) −1)(f(b) −1) = f(−(a −1)(b −1) + 1) −1. So putting g(x)...

Practice › Mathematics › IMO Shortlist › IMO 1984 Shortlist ›

IMO 1984 SL 6

IMO 1984 SL 6 Origin: CAN Problem Let c be a positive integer. The sequence {fn} is deﬁned as follows: f1 = 1, f2 = c, fn+1 = 2fn −fn−1 + 2 (n \geq2). Show that for each k \inN there exists r \inN such that fkfk+1 = fr. Solution From the given recurrence we infer fn+1−fn = fn−fn−1+2. Consequently, fn+1 −fn = (f2 −f1) + 2(n −1) = c...

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL G8

IMO 1995 SL G8 Origin: COL | Category: Geometry Problem Let ABC be a triangle. A circle passing through B and C in- tersects the sides AB and AC again at C′ and B′, respectively. Prove that BB′, CC′, and HH′ are concurrent, where H and H′ are the orthocenters of triangles ABC and AB′C′ respectively. Solution Let BB′ cut CC′ at P. Since \angleB′BC′ = \angleB′CC′, it follows that...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL G8

IMO 2004 SL G8 Origin: SMN | Category: Geometry Problem A cyclic quadrilateral ABCD is given. The lines AD and BC intersect at E, with C between B and E; the diagonals AC and BD intersect at F. Let M be the midpoint of the side CD, and let N ̸= M be a point on the circumcircle of the triangle ABM such that AN/BN = AM/BM. Prove that the...

Practice › Mathematics › IMO Shortlist › IMO 1983 Shortlist ›

IMO 1983 SL 7

IMO 1983 SL 7 Origin: CAN Problem Let a be a positive integer and let {an} be deﬁned by a0 = 0 and an+1 = (an + 1)a + (a + 1)an + 2 a(a + 1)an(an + 1) (n = 1, 2 . . .). Show that for each positive integer n, an is a positive integer. Solution Clearly, each an is positive and \sqrtan+1 = \sqrtan \sqrta +...

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL N8

IMO 2003 SL N8 Origin: IRN | Category: Number Theory Problem Let p be a prime number and let A be a set of positive integers that satisﬁes the following conditions: (i) the set of prime divisors of the elements in A consists of p−1 elements; (ii) for any nonempty subset of A, the product of its elements is not a perfect pth power. What is the largest possible number...

Practice › Mathematics › IMO Shortlist › IMO 1984 Shortlist ›

IMO 1984 SL 12

IMO 1984 SL 12 Origin: NET Problem Find two positive integers a, b such that none of the num- bers a, b, a + b is divisible by 7 and (a + b)7 −a7 −b7 is divisible by 77. Solution By the binomial formula we have (a + b)7 −a7 −b7 = 7ab[(a5 + b5) + 3ab(a3 + b3) + 5a2b2(a + b)] = 7ab(a + b)(a2 + ab +...

Practice › Mathematics › IMO Shortlist › IMO 1981 Shortlist ›

IMO 1981 SL 1

IMO 1981 SL 1 Origin: BEL Problem (a) For which values of n > 2 is there a set of n consecutive positive integers such that the largest number in the set is a divisor of the least common multiple of the remaining n −1 numbers? (b) For which values of n > 2 is there a unique set having the stated property? Solution Assume that the set {a−n+ 1,...

Practice › Mathematics › IMO Shortlist › IMO 1997 Shortlist ›

IMO 1997 SL 8

IMO 1997 SL 8 Origin: GBR Problem Four diﬀerent points A, B, C, D are chosen on a circle \Gamma such that the triangle BCD is not right-angled. Prove that: (a) The perpendicular bisectors of AB and AC meet the line AD at cer- tain points W and V , respectively, and that the lines CV and BW meet at a certain point T . (b) The length of one...

Practice › Mathematics › IMO Shortlist › IMO 1981 Shortlist ›

IMO 1981 SL 4

IMO 1981 SL 4 Origin: CAN Problem Let {fn} be the Fibonacci sequence {1, 1, 2, 3, 5, . . .}. (a) Find all pairs (a, b) of real numbers such that for each n, afn + bfn+1 is a member of the sequence. (b) Find all pairs (u, v) of positive real numbers such that for each n, uf 2 n + vf 2 n+1 is a member of...

Practice › Mathematics › IMO Shortlist › IMO 1994 Shortlist ›

IMO 1994 SL C2

IMO 1994 SL C2 Origin: COL | Category: Combinatorics Problem In a certain city, age is reckoned in terms of real numbers rather than integers. Every two citizens x and x′ either know each other or do not know each other. Moreover, if they do not, then there exists a chain of citizens x = x0, x1, . . . , xn = x′ for some integer n \geq2 such...

Practice › Mathematics › IMO Shortlist › IMO 1976 Shortlist ›

IMO 1976 SL 6

IMO 1976 SL 6 Origin: NET Problem A rectangular box can be ﬁlled completely with unit cubes. If one places cubes with volume 2 in the box such that their edges are parallel to the edges of the box, one can ﬁll exactly 40% of the box. Determine all possible (interior) sizes of the box. Solution Suppose a1 \leqa2 \leqa3 are the dimensions of the box. If we set bi...

Practice › Mathematics › IMO Shortlist › IMO 1997 Shortlist ›

IMO 1997 SL 1

IMO 1997 SL 1 Origin: BLR Problem An inﬁnite square grid is colored in the chessboard pattern. For any pair of positive integers m, n consider a right-angled triangle whose vertices are grid points and whose legs, of lengths m and n, run along the lines of the grid. Let Sb be the total area of the black part of the triangle and Sw the total area of its white...

Practice › Mathematics › IMO Shortlist › IMO 1974 Shortlist ›

IMO 1974 SL 2

IMO 1974 SL 2 Problem I 2 (POL 1) Prove that the squares with sides 1/1, 1/2, 1/3, . . . may be put into the square with side 3/2 in such a way that no two of them have any interior point in common. Solution We denote by qi the square with side 1 i . Let us divide the big square into rectangles ri by parallel lines, where...

Practice › Mathematics › IMO Shortlist › IMO 1997 Shortlist ›

IMO 1997 SL 10

IMO 1997 SL 10 Origin: CZE Problem Find all positive integers k for which the following statement is true: If F(x) is a polynomial with integer coeﬃcients satisfying the condition 0 \leqF(c) \leqk for each c \in{0, 1, . . ., k + 1}, then F(0) = F(1) = \cdot \cdot \cdot = F(k + 1). Solution Suppose that k \geq4. Consider any polynomial F(x) with integer coeﬃ- cients such...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 19

IMO 1989 SL 19 Origin: MON Problem A positive integer is written in each square of an m\timesn board. The allowed move is to add an integer k to each of two adjacent numbers in such a way that no negative numbers are obtained. (Two squares are adjacent if they have a common side.) Find a necessary and suﬃcient condition for it to be possible for all the numbers to...

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL A2

IMO 2001 SL A2 Origin: POL | Category: Algebra Problem Let a0, a1, a2, . . . be an arbitrary inﬁnite sequence of positive numbers. Show that the inequality 1 + an > an−1 n\sqrt 2 holds for inﬁnitely many positive integers n. Solution It follows from Bernoulli’s inequality that for each n \inN, 1 + 1 n n \geq2, or n\sqrt 2 \leq1 + 1 n. Consequently, it...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 7

IMO 1991 SL 7 Origin: CHN Problem Let O be the center of the circumsphere of a tetrahedron ABCD. Let L, M, N be the midpoints of BC, CA, AB respectively, and assume that AB+BC = AD+CD, BC+CA = BD+AD, and CA+AB = CD + BD. Prove that \angleLOM = \angleMON = \angleNOL. Solution The given equations imply AB = CD, AC = BD, AD = BC. Let L1, M1,...

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL C3

IMO 2002 SL C3 Origin: COL | Category: Combinatorics Problem Let n be a positive integer. A sequence of n positive integers (not necessarily distinct) is called full if it satisﬁes the following condition: For each positive integer k \geq2, if the number k appears in the sequence, then so does the number k −1, and moreover, the ﬁrst occurrence of k −1 comes before the last occurrence of k....

Practice › Mathematics › IMO Shortlist › IMO 1992 Shortlist ›

IMO 1992 SL 11

IMO 1992 SL 11 Origin: JAP Problem In a triangle ABC, let D and E be the intersections of the bisec- tors of \angleABC and \angleACB with the sides AC, AB, respectively. Determine the angles \angleA, \angleB, \angleC if ∡BDE = 24◦, ∡CED = 18◦. Solution Let I be the incenter of \triangleABC. Since 90◦+ \alpha/2 = \angleBIC = \angleDIE = 138◦, we obtain that \angleA = 96◦. A B...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL A7

IMO 1996 SL A7 Origin: ARM | Category: Algebra Problem Let $f$ be a function from the set of real numbers $\mathbb{R}$ into itself such that for all $x\in\mathbb{R}$, we have $|f(x)|\leq 1$ and $$f\left(x+\frac13\right)+f(x)=f\left(x+\frac16\right)+f\left(x+\frac17\right).$$ Prove that $f$ is a periodic function (that is, there exists a nonzero real number $c$ such that $f(x+c)=f(x)$ for all $x\in\mathbb{R}$). Solution We are given that $$f(x+a+b)-f(x+a)=f(x+b)-f(x),$$ where $$a=\frac16,\qquad b=\frac17.$$ Summing up these equations...

Practice › Mathematics › IMO Shortlist › IMO 1987 Shortlist ›

IMO 1987 SL 20

IMO 1987 SL 20 Origin: USS Problem Let f(x) = x2 + x + p, p \inN. Prove that if the numbers f(0), f(1), . . . , f([ p/3 ]) are primes, then all the numbers f(0), f(1), . . . , f(p −2) are primes. Solution Let y be the smallest nonnegative integer with y \leqp −2 for which f(y) is a composite number. Denote by q the...

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL C6

IMO 2000 SL C6 Origin: FRA | Category: Combinatorics Problem Let p and q be relatively prime positive integers. A subset S of {0, 1, 2, . . .} is called ideal if 0 \inS and for each element n \inS, the integers n + p and n + q belong to S. Determine the number of ideal subsets of {0, 1, 2 . . .}. Solution Every integer z...

Practice › Mathematics › IMO Shortlist › IMO 1994 Shortlist ›

IMO 1994 SL C1

IMO 1994 SL C1 Origin: UKR | Category: Combinatorics Problem On a 5 \times 5 board, two players alternately mark numbers on empty cells. The ﬁrst player always marks 1’s, the second 0’s. One number is marked per turn, until the board is ﬁlled. For each of the nine 3 \times 3 squares the sum of the nine numbers on its cells is computed. Denote by A the maximum of...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 9

IMO 1991 SL 9 Origin: FRA Problem In the plane we are given a set E of 1991 points, and certain pairs of these points are joined with a path. We suppose that for every point of E, there exist at least 1593 other points of E to which it is joined by a path. Show that there exist six points of E every pair of which are joined by...

Practice › Mathematics › IMO Shortlist › IMO 1981 Shortlist ›

IMO 1981 SL 15

IMO 1981 SL 15 Origin: GBR Problem Find the point P inside the triangle ABC for which BC PD + CA PE + AB PF is minimal, where PD, PE, PF are the perpendiculars from P to BC, CA, AB respectively. Solution Set BC = a, CA = b, AB = c, and denote the area of \triangleABC by P, and a/PD + b/PE + c/PF by S. Since a...

Practice › Mathematics › IMO Shortlist › IMO 1986 Shortlist ›

IMO 1986 SL 10

IMO 1986 SL 10 Origin: HUN Problem Three persons A, B, C, are playing the following game: A k- element subset of the set {1, . . . , 1986} is randomly chosen, with an equal probability of each choice, where k is a ﬁxed positive integer less than or equal to 1986. The winner is A, B or C, respectively, if the sum of the chosen numbers leaves a...

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL C4

IMO 2000 SL C4 Origin: CZE | Category: Combinatorics Problem Let n and k be positive integers such that n/2 < k \leq2n/3. Find the least number m for which it is possible to place m pawns on m squares of an n \times n chessboard so that no column or row contains a block of k adjacent unoccupied squares. Solution By a good placement of pawns we mean the...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 15

IMO 1979 SL 15 Origin: ISR Problem The nonnegative real numbers x1, x2, x3, x4, x5, a satisfy the following relations: i=1 ixi = a, i=1 i3xi = a2, i=1 i5xi = a3. What are the possible values of a? Solution We note that i=1 i(a−i2)2xi = a2 i=1 ixi−2a i=1 i3xi+ i=1 i5xi = a2\cdota−2a\cdota2+a3 = 0. Since the terms in the...

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL N4

IMO 2001 SL N4 Origin: VIE | Category: Number Theory Problem Let p \geq5 be a prime number. Prove that there exists an integer a with 1 \leqa \leqp −2 such that neither ap−1 −1 nor (a + 1)p−1 −1 is divisible by p2. Solution Let C be the set of those a \in{1, 2, . . ., p−1} for which ap−1 \equiv1 (mod p2). At ﬁrst, we observe that...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 13

IMO 1989 SL 13 Origin: ICE Problem The quadrilateral ABCD has the following properties: (i) AB = AD + BC; (ii) there is a point P inside it at a distance x from the side CD such that AP = x + AD and BP = x + BC. Show that \sqrtx \geq \sqrt AD + \sqrt BC . Solution Let us construct the circles \sigma1 with center A and...

Practice › Mathematics › IMO Shortlist › IMO 1971 Shortlist ›

IMO 1971 SL 9

IMO 1971 SL 9 Origin: POL Problem Let Tk = k −1 for k = 1, 2, 3, 4 and T2k−1 = T2k−2 + 2k−2, T2k = T2k−5 + 2k (k \geq3). Show that for all k, 1 + T2n−1 = 7 2n−1 and 1 + T2n = 7 2n−1 , where [x] denotes the greatest integer not exceeding x. Solution We use induction. Since T1 = 0, T2 =...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL A4

IMO 2004 SL A4 Origin: KOR | Category: Algebra Problem Find all polynomials P(x) with real coeﬃcients that satisfy the equality P(a −b) + P(b −c) + P(c −a) = 2P(a + b + c) for all triples a, b, c of real numbers such that ab + bc + ca = 0. Solution Let P(x) = a0 + a1x + \cdot \cdot \cdot + anxn. For every x \inR...

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL G6

IMO 2003 SL G6 Origin: POL | Category: Geometry Problem Each pair of opposite sides of a convex hexagon has the following property: The distance between their midpoints is equal to \sqrt 3/2 times the sum of their lengths. Prove that all the angles of the hexagon are equal. Solution Let ABCDEF be the given hexagon. We shall use the following lemma. Lemma. If \angleXZY \geq60◦and if M is the...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 24

IMO 1988 SL 24 Origin: SWE Problem Let {ak}\infty be a sequence of nonnegative real numbers such that ak −2ak+1 + ak+2 \geq0 and k j=1 aj \leq1 for all k = 1, 2, . . . . Prove that 0 \leq(ak −ak+1) < k2 for all k = 1, 2, . . . . Solution The ﬁrst condition means in fact that ak−ak+1 is decreasing. In particular, if ak...

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL N2

IMO 1999 SL N2 Origin: ARM | Category: Number Theory Problem Prove that every positive rational number can be repre- sented in the form a3 + b3 c3 + d3 , where a, b, c, d are positive integers. Solution We ﬁrst prove that every rational number in the interval (1, 2) can be represented in the form a3+b3 a3+d3 . Taking b, d such that b ̸= d and...

Practice › Mathematics › IMO Shortlist › IMO 1985 Shortlist ›

IMO 1985 SL 13

IMO 1985 SL 13 Problem 4a.(BUL 1) Let m boxes be given, with some balls in each box. Let n < m be a given integer. The following operation is performed: choose n of the boxes and put 1 ball in each of them. Prove: (a) If m and n are relatively prime, then it is possible, by performing the operation a ﬁnite number of times, to arrive at the...

Practice › Mathematics › IMO Shortlist › IMO 1986 Shortlist ›

IMO 1986 SL 20

IMO 1986 SL 20 Origin: CAN Problem Prove that the sum of the face angles at each vertex of a tetra- hedron is a straight angle if and only if the faces are congruent triangles. Solution If the faces of the tetrahedron ABCD are congruent triangles, we must have AB = CD, AC = BD, and AD = BC. Then the sum of angles at A is \angleBAC + \angleCAD...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 22

IMO 1979 SL 22 Origin: USS Problem There are two circles in the plane. Let a point A be one of the points of intersection of these circles. Two points begin moving simultaneously with constant speeds from the point A, each point along its own circle. The two points return to the point A at the same time. Prove that there is a point P in the plane such that...

Practice › Mathematics › IMO Shortlist › IMO 1993 Shortlist ›

IMO 1993 SL 12

IMO 1993 SL 12 Origin: IRE Problem Let n, k be positive integers with k \leqn and let S be a set containing n distinct real numbers. Let T be the set of all real numbers of the form x1 + x2 + \cdot \cdot \cdot + xk, where x1, x2, . . . , xk are distinct elements of S. Prove that T contains at least k(n −k) +...

Practice › Mathematics › IMO Shortlist › IMO 1987 Shortlist ›

IMO 1987 SL 14

IMO 1987 SL 14 Origin: FRG Problem How many words with n digits can be formed from the alphabet {0, 1, 2, 3, 4}, if neighboring digits must diﬀer by exactly one? Solution Let xn be the total number of counted words of length n, and yn, zn, un, zn, yn the numbers of counted words of length n starting with 0, 1, 2, 3, 4, respectively (indeed, by symmetry,...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL G9

IMO 1996 SL G9 Origin: UKR | Category: Geometry Problem In the plane are given a point $O$ and a polygon $F$ (not necessarily convex). Let $P$ denote the perimeter of $F$, $D$ the sum of the distances from $O$ to the vertices of $F$, and $H$ the sum of the distances from $O$ to the lines containing the sides of $F$. Prove that $$D^2 - H^2 \geq \frac{P^2}{4}.$$ Solution...

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 5

IMO 1990 SL 5 Origin: FRA Problem Given \triangleABC with no side equal to another side, let G, K, and H be its centroid, incenter, and orthocenter, respectively. Prove that \angleGKH > 90◦. Solution Let O be the circumcenter of ABC, E the midpoint of OH, and R and r the radii of the circumcircle and incircle respectively. We use the following facts from elementary geometry: −−\to OH = 3−−\to...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 12

IMO 1979 SL 12 Origin: GDR Problem Let R be a set of exactly 6 elements. A set F of subsets of R is called an S-family over R if and only if it satisﬁes the following three conditions: (i) For no two sets X, Y in F is X \subseteqY ; (ii) For any three sets X, Y, Z in F, X \cupY \cupZ ̸= R, (iii) % X\inF...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 21

IMO 1989 SL 21 Origin: NET Problem Prove that the intersection of a plane and a regular tetrahedron can be an obtuse-angled triangle and that the obtuse angle in any such triangle is always smaller than 120◦. Solution In order to obtain a triangle as the intersection we must have three points P, Q, R on three sides of the tetrahedron passing through one vertex, say T . It is...

Practice › Mathematics › IMO Shortlist › IMO 1983 Shortlist ›

IMO 1983 SL 16

IMO 1983 SL 16 Origin: GDR Problem Let F(n) be the set of polynomials P(x) = a0+a1x+\cdot \cdot \cdot+anxn, with a0, a1, . . . , an \inR and 0 \leqa0 = an \leqa1 = an−1 \leq\cdot \cdot \cdot \leqa[n/2] = a[(n+1)/2]. Prove that if f \inF(m) and g \inF(n), then fg \inF(m + n). Solution Set hn,i(x) = xi + \cdot \cdot \cdot + xn−i, 2i \leqn. The set...

Practice › Mathematics › IMO Shortlist › IMO 1975 Shortlist ›

IMO 1975 SL 7

IMO 1975 SL 7 Origin: GDR Problem Prove that from x + y = 1 (x, y \inR) it follows that xm+1 n j=0 m + j j yj + yn+1 m i=0 n + i i xi = 1 (m, n = 0, 1, 2, . . .). Solution We use induction on m. Denote by Sm the left-hand side of the equality to be...

Practice › Mathematics › IMO Shortlist › IMO 1983 Shortlist ›

IMO 1983 SL 19

IMO 1983 SL 19 Origin: ROM Problem Let (Fn)n\geq1 be the Fibonacci sequence F1 = F2 = 1, Fn+2 = Fn+1 + Fn (n \geq1), and P(x) the polynomial of degree 990 satisfying P(k) = Fk, for k = 992, . . ., 1982. Prove that P(1983) = F1983 −1. Solution For all n, there exists a unique polynomial Pn of degree n such that Pn(k) = Fk for n...

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL G7

IMO 2003 SL G7 Origin: SAF | Category: Geometry Problem Let ABC be a triangle with semiperimeter s and inradius r. The semicircles with diameters BC, CA, AB are drawn outside of the triangle ABC. The circle tangent to all three semicircles has radius t. Prove that s 2 < t \leqs 2 + 1 − \sqrt r. Solution Let D, E, F be the midpoints of BC, CA, AB,...

Practice › Mathematics › IMO Shortlist › IMO 1986 Shortlist ›

IMO 1986 SL 21

IMO 1986 SL 21 Origin: TUR Problem Let ABCD be a tetrahedron having each sum of opposite sides equal to 1. Prove that rA + rB + rC + rD \leq \sqrt 3 , where rA, rB, rC, rD are the inradii of the faces, equality holding only if ABCD is regular. Solution Since the sum of all edges of ABCD is 3, the statement of the problem is an...

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 25

IMO 1990 SL 25 Origin: TUR Problem Let Q+ be the set of positive rational numbers. Construct a function f : Q+ \toQ+ such that f(xf(y)) = f(x) y , for all x, y in Q+. Solution Plugging in x = 1 we get f(f(y)) = f(1)/y and hence f(y1) = f(y2) implies y1 = y2 i.e. that the function is bijective. Plugging in y = 1 gives us f(xf(1))...

Practice › Mathematics › IMO Shortlist › IMO 1975 Shortlist ›

IMO 1975 SL 9

IMO 1975 SL 9 Origin: NET Problem Let f(x) be a continuous function deﬁned on the closed interval 0 \leqx \leq1. Let G(f) denote the graph of f(x): G(f) = {(x, y) \inR2 | 0 \leq x \leq1, y = f(x)}. Let Ga(f) denote the graph of the translated function f(x −a) (translated over a distance a), deﬁned by Ga(f) = {(x, y) \in R2 | a \leqx \leqa +...

Practice › Mathematics › IMO Shortlist › IMO 1997 Shortlist ›

IMO 1997 SL 22

IMO 1997 SL 22 Origin: UKR Problem (a) Do there exist functions f : R \toR and g : R \toR such that f(g(x)) = x2 and g(f(x)) = x3 for all x \inR? (b) Do there exist functions f : R \toR and g : R \toR such that f(g(x)) = x2 and g(f(x)) = x4 for all x \inR? Solution (a) Suppose that f and g are such...

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL N6

IMO 1995 SL N6 Origin: POL | Category: Number Theory Problem Let p be an odd prime. Find the number of p-element subsets A of {1, 2, . . ., 2p} such that the sum of all elements of A is divisible by p. Solution We shall consider the set M = {0, 1, . . ., 2p −1} instead. Let M1 = {0, 1, . . ., p −1}...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL G8

IMO 1996 SL G8 Origin: RUS | Category: Geometry Problem Let $\triangle ABCD$ be a convex quadrilateral, and let $R_A$, $R_B$, $R_C$, and $R_D$ denote the circumradii of the triangles $\triangle DAB$, $\triangle ABC$, $\triangle BCD$, and $\triangle CDA$ respectively. Prove that $$R_A+R_C>R_B+R_D$$ if and only if $$\angle A+\angle C>\angle B+\angle D.$$ Solution Let the diagonals $AC$ and $BD$ meet at $X$. Either $\angle AXB$ or $\angle AXD$ is greater...

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL G5

IMO 2001 SL G5 Origin: GRE | Category: Geometry Problem Let ABC be an acute triangle. Let DAC, EAB, and FBC be isosceles triangles exterior to ABC, with DA = DC, EA = EB, and FB = FC such that \angleADC = 2\angleBAC, \angleBEA = 2\angleABC, \angleCFB = 2\angleACB. Let D′ be the intersection of lines DB and EF, let E′ be the intersection of EC and DF, and let...

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL 25

IMO 1995 SL 25 Problem S3 (POL) For an integer x \geq1, let p(x) be the least prime that does not divide x, and deﬁne q(x) to be the product of all primes less than p(x). In particular, p(1) = 2. For x such that p(x) = 2, deﬁne q(x) = 1. Consider the sequence x0, x1, x2, . . . deﬁned by x0 = 1 and xn+1 = xnp(xn)...

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL 24

IMO 1995 SL 24 Problem S2 (POL)IMO4 The positive real numbers x0, x1, . . . , x1995 satisfy x0 = x1995 and xi−1 + xi−1 = 2xi + 1 xi for i = 1, 2, . . ., 1995. Find the maximum value that x0 can have. Solution The given condition is equivalent to (2xi −xi−1)(xixi−1 −1) = 0, so either xi = 1 2xi−1 or xi = xi−1...

Practice › Mathematics › IMO Shortlist › IMO 1971 Shortlist ›

IMO 1971 SL 3

IMO 1971 SL 3 Origin: GDR Problem Knowing that the system x + y + z = 3, x3 + y3 + z3 = 15, x4 + y4 + z4 = 35, has a real solution x, y, z for which x2 + y2 + z2 < 10, ﬁnd the value of x5 + y5 + z5 for that solution. Solution Let x, y, z be a solution of the...

Practice › Mathematics › IMO Shortlist › IMO 1970 Shortlist ›

IMO 1970 SL 1

IMO 1970 SL 1 Origin: BEL Problem Consider a regular 2n-gon and the n diagonals of it that pass through its center. Let P be a point of the inscribed circle and let a1, a2, . . . , an be the angles in which the diagonals mentioned are visible from the point P. Prove that n i=1 tan2 ai = 2ncos2 \pi 2n sin4 \pi 2n . Solution...

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL A4

IMO 1995 SL A4 Origin: USA | Category: Algebra Problem Let a, b, and c be given positive real numbers. Determine all positive real numbers x, y, and z such that x + y + z = a + b + c and 4xyz −(a2x + b2y + c2z) = abc. Solution The second equation is equivalent to a2 yz + b2 zx + c2 xy + abc xyz =...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL C1

IMO 1996 SL C1 Origin: FIN | Category: Combinatorics Problem We are given a positive integer $r$ and a rectangular board $ABCD$ with dimensions $|AB| = 20$, $|BC| = 12$. The rectangle is divided into a grid of $20 \times 12$ unit squares. The following moves are permitted on the board: one can move from one square to another only if the distance between the centers of the two squares...

Practice › Mathematics › IMO Shortlist › IMO 1982 Shortlist ›

IMO 1982 SL 9

IMO 1982 SL 9 Problem B3 (GBR 1) Let ABC be a triangle, and let P be a point inside it such that ∡PAC = ∡PBC. The perpendiculars from P to BC and CA meet these lines at L and M, respectively, and D is the midpoint of AB. Prove that DL = DM. Solution Let X and Y be the midpoints of the segments AP and BP. Then DY...

Practice › Mathematics › IMO Shortlist › IMO 1997 Shortlist ›

IMO 1997 SL 21

IMO 1997 SL 21 Origin: RUS Problem Let x1, x2, . . . , xn be real numbers satisfying the conditions |x1 + x2 + \cdot \cdot \cdot + xn| = 1 and |xi| \leqn + 1 for i = 1, 2, . . ., n. Show that there exists a permutation y1, . . . , yn of the sequence x1, . . . , xn such that |y1...

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL N6

IMO 2002 SL N6 Origin: ROM | Category: Number Theory Problem Find all pairs of positive integers m, n \geq3 for which there exist inﬁnitely many positive integers a such that am + a −1 an + a2 −1 is itself an integer. Solution Suppose that (m, n) is such a pair. Assume that division of the polynomial F(x) = xm + x −1 by G(x) = xn + x2...

Practice › Mathematics › IMO Shortlist › IMO 1974 Shortlist ›

IMO 1974 SL 3

IMO 1974 SL 3 Problem I 3 (SWE 3)IMO6 Let P(x) be a polynomial with integer coeﬃcients. If n(P) is the number of (distinct) integers k such that P 2(k) = 1, prove that n(P) −deg(P) \leq2, where deg(P) denotes the degree of the polynomial P. Solution For deg(P) \leq2 the statement is obvious, since n(P) \leqdeg(P 2) = 2 deg(P) \leqdeg(P) + 2. Suppose now that deg(P) \geq3 and...

Practice › Mathematics › IMO Shortlist › IMO 1992 Shortlist ›

IMO 1992 SL 4

IMO 1992 SL 4 Origin: CHN Problem Given nine points in space, no four of which are coplanar, ﬁnd the minimal natural number n such that for any coloring with red or blue of n edges drawn between these nine points there always exists a triangle having all edges of the same color. Solution There are 36 possible edges in total. If not more than 3 edges are left undrawn,...

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL A1

IMO 1995 SL A1 Origin: RUS | Category: Algebra Problem Let a, b, and c be positive real numbers such that abc = 1. Prove that a3(b + c) + b3(a + c) + c3(a + b) \geq3 2. Solution Let x = 1 a, y = 1 b, z = 1 c. Then xyz = 1 and S = a3(b + c) + b3(c + a) + c3(a +...

Practice › Mathematics › IMO Shortlist › IMO 1975 Shortlist ›

IMO 1975 SL 3

IMO 1975 SL 3 Origin: USA Problem Find the integer represented by 109 n=1 n−2/3 . Here [x] denotes the greatest integer less than or equal to x (e.g. [ \sqrt 2] = 1). Solution From (k + 1)2/3 + (k + 1)1/3k1/3 + k2/3 (k + 1)1/3 −k1/3 = 1 and 3k2/3 < (k + 1)2/3+ (k + 1)1/3k1/3 + k2/3 < 3(k + 1)2/3 we obtain...

Practice › Mathematics › IMO Shortlist › IMO 1987 Shortlist ›

IMO 1987 SL 18

IMO 1987 SL 18 Origin: ROM Problem For any integer r \geq1, determine the smallest integer h(r) \geq1 such that for any partition of the set {1, 2, . . ., h(r)} into r classes, there are integers a \geq0, 1 \leqx \leqy, such that a + x, a + y, a + x + y belong to the same class. Solution Note ﬁrst that the statement that some a...

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL 28

IMO 1995 SL 28 Problem S6 (IND) Let N denote the set of all positive integers. Prove that there exists a unique function f : N \toN satisfying f(m + f(n)) = n + f(m + 95) for all m and n in N. What is the value of 19 k=1 f(k)? Solution Let F(x) = f(x) −95 for x \geq1. Writing k for m + 95, the given condition...

Practice › Mathematics › IMO Shortlist › IMO 1982 Shortlist ›

IMO 1982 SL 18

IMO 1982 SL 18 Problem C6 (FRA 2) Let O be a point of three-dimensional space and let l1, l2, l3 be mutually perpendicular straight lines passing through O. Let S denote the sphere with center O and radius R, and for every point M of S, let SM denote the sphere with center M and radius R. We denote by P1, P2, P3 the intersection of SM with the...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 18

IMO 1989 SL 18 Origin: MON Problem Given a convex polygon A1A2 . . . An with area S, and a point M in the same plane, determine the area of polygon M1M2 . . . Mn, where Mi is the image of M under rotation R\alpha Ai around Ai by \alpha, i = 1, 2, . . . , n. Solution Consider the triangle MAiMi. Obviously, the point Mi...

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL C6

IMO 1999 SL C6 Origin: GBR | Category: Combinatorics Problem Suppose that every integer has been given one of the colors red, blue, green, yellow. Let x and y be odd integers such that |x| ̸= |y|. Show that there are two integers of the same color whose diﬀerence has one of the following values: x, y, x + y, x −y. Solution We denote colors by capital initial letters....

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL G6

IMO 1995 SL G6 Origin: USA | Category: Geometry Problem Let A1A2A3A4 be a tetrahedron, G its centroid, and A′ 1, A′ 2, A′ 3, and A′ 4 the points where the circumsphere of A1A2A3A4 in- tersects GA1, GA2, GA3, and GA4, respectively. Prove that GA1 \cdot GA2 \cdot GA3 \cdot GA4 \leqGA′ 1 \cdot GA′ 2 \cdot GA′ 3 \cdot GA′ and GA′ + GA′ + GA′ + GA′...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 23

IMO 1991 SL 23 Origin: IND Problem Let f and g be two integer-valued functions deﬁned on the set of all integers such that (a) f(m + f(f(n))) = −f(f(m + 1) −n for all integers m and n; (b) g is a polynomial function with integer coeﬃcients and g(n) = g(f(n)) for all integers n. Determine f(1991) and the most general form of g. Solution From (i), replacing m...

Practice › Mathematics › IMO Shortlist › IMO 1971 Shortlist ›

IMO 1971 SL 7

IMO 1971 SL 7 Origin: NET Problem Given a tetrahedron ABCD whose all faces are acute- angled triangles, set \sigma = ∡DAB + ∡BCD −∡ABC −∡CDA. Consider all closed broken lines XY ZTX whose vertices X, Y, Z, T lie in the interior of segments AB, BC, CD, DA respectively. Prove that: (a) if \sigma ̸= 0, then there is no broken line XY ZT of minimal length; (b) if...

Practice › Mathematics › IMO Shortlist › IMO 1983 Shortlist ›

IMO 1983 SL 25

IMO 1983 SL 25 Origin: USS Problem Prove that every partition of 3-dimensional space into three disjoint subsets has the following property: One of these subsets contains all possible distances; i.e., for every a \inR+, there are points M and N inside that subset such that distance between M and N is exactly a. Solution Suppose the contrary, that R3 = P1 \cupP2 \cupP3 is a partition such that a1...

Practice › Mathematics › IMO Shortlist › IMO 1975 Shortlist ›

IMO 1975 SL 14

IMO 1975 SL 14 Origin: YUG Problem Let x0 = 5 and xn+1 = xn + xn (n = 0, 1, 2, . . .). Prove that 45 < x1000 < 45, 1. Solution We shall prove that for every n \inN, \sqrt2n + 25 \leqxn \leq\sqrt2n + 25 + 0.1. Note that for n = 1000 this gives us exactly the desired inequalities. First, notice that the recurrent relation...

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL A2

IMO 2002 SL A2 Origin: YUG | Category: Algebra Problem Let a1, a2, . . . be an inﬁnite sequence of real numbers for which there exists a real number c with 0 \leqai \leqc for all i such that |ai −aj| \geq i + j for all i, j with i ̸= j. Prove that c \geq1. Solution For n \geq2, let (k1, k2, . . . , kn)...

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL N3

Practice › Mathematics › IMO Shortlist › IMO 1971 Shortlist ›

IMO 1971 SL 10

IMO 1971 SL 10 Origin: POL Problem Prove that the sequence 2n −3 (n > 1) contains a subse- quence of numbers relatively prime in pairs. Solution We use induction. Suppose that every two of the numbers a1 = 2n1 − 3, a2 = 2n2 −3, . . . , ak = 2nk −3, where 2 = n1 < n2 < \cdot \cdot \cdot < nk, are coprime. Then one...

Practice › Mathematics › IMO Shortlist › IMO 1985 Shortlist ›

IMO 1985 SL 18

IMO 1985 SL 18 Problem 6b.(CAN 5) Let x1, x2, . . . , xn be positive numbers. Prove that x2 x2 1 + x2x3 + x2 x2 2 + x3x4 \cdot \cdot \cdot + x2 n−1 x2 n−1 + xnx1 x2 n x2n + x1x2 \leqn −1. Supplementary Problems Solution Set yi = x2 i xi+1xi+2 , where xn+i = xi. Then $n i=1 yi = 1 and the...

Practice › Mathematics › IMO Shortlist › IMO 1984 Shortlist ›

IMO 1984 SL 18

IMO 1984 SL 18 Origin: USA Problem Inside triangle ABC there are three circles k1, k2, k3 each of which is tangent to two sides of the triangle and to its incircle k. The radii of k1, k2, k3 are 1, 4, and 9. Determine the radius of k. Solution Suppose that circles k1(O1, r1), k2(O2, r2), and k3(O3, r3) touch the edges of the angles \angleBAC, \angleABC, and \angleACB,...

Practice › Mathematics › IMO Shortlist › IMO 1974 Shortlist ›

IMO 1974 SL 5

IMO 1974 SL 5 Problem I 5 (GBR 3) Let Ar, Br, Cr be points on the circumference of a given circle S. From the triangle ArBrCr, called \triangler, the triangle \triangler+1 is ob- tained by constructing the points Ar+1, Br+1, Cr+1 on S such that Ar+1Ar is parallel to BrCr, Br+1Br is parallel to CrAr, and Cr+1Cr is parallel to ArBr. Each angle of \triangle1 is an integer number...

Practice › Mathematics › IMO Shortlist › IMO 1985 Shortlist ›

IMO 1985 SL 9

IMO 1985 SL 9 Problem 2a.(USA 3) Determine the radius of a sphere S that passes through the centroids of each face of a given tetrahedron T inscribed in a unit sphere with center O. Also, determine the distance from O to the center of S as a function of the edges of T . Solution Let −\toa , −\tob , −\toc , −\tod denote the vectors −\to OA, −−\to...

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL N5

IMO 2002 SL N5 Origin: IRN | Category: Number Theory Problem Let m, n \geq2 be positive integers, and let a1, a2, . . . , an be integers, none of which is a multiple of mn−1. Show that there exist integers e1, e2, . . . , en, not all zero, with |ei| < m for all i, such that e1a1 + e2a2 + \cdot \cdot \cdot + enan...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL C5

IMO 2004 SL C5 Origin: NZL | Category: Combinatorics Problem Let N be a positive integer. Two players A and B, taking turns, write numbers from the set {1, . . . , N} on a blackboard. A begins the game by writing 1 on his ﬁrst move. Then, if a player has written n on a certain move, his adversary is allowed to write n+1 or 2n (provided the...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL C8

IMO 2004 SL C8 Origin: POL | Category: Combinatorics Problem For a ﬁnite graph G, let f(G) be the number of triangles and g(G) the number of tetrahedra formed by edges of G. Find the least constant c such that g(G)3 \leqc \cdot f(G)4 for every graph G. Solution Denote by V1, . . . , Vn the vertices of a graph G and by E the set of its...

Practice › Mathematics › IMO Shortlist › IMO 1992 Shortlist ›

IMO 1992 SL 10

IMO 1992 SL 10 Origin: ITA Problem Let V be a ﬁnite subset of Euclidean space consisting of points (x, y, z) with integer coordinates. Let S1, S2, S3 be the projections of V onto the yz, xz, xy planes, respectively. Prove that |V |2 \leq|S1||S2||S3| (|X| denotes the number of elements of X). Solution Let us set S(x) = {(y, z) | (x, y, z) \inV }, Sy(x) =...

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 17

IMO 1990 SL 17 Origin: NET Problem Unit cubes are made into beads by drilling a hole through them along a diagonal. The beads are put on a string in such a way that they can move freely in space under the restriction that the vertices of two neighboring cubes are touching. Let A be the beginning vertex and B be the end vertex. Let there be p \times q...

Practice › Mathematics › IMO Shortlist › IMO 1992 Shortlist ›

IMO 1992 SL 13

IMO 1992 SL 13 Origin: NZL Problem Find all integer triples (p, q, r) such that 1 < p < q < r and (p −1)(q −1)(r −1) is a divisor of (pqr −1). Solution Let us deﬁne F(p, q, r) = (pqr −1) (p −1)(q −1)(r −1) = 1 + p −1 + q −1 + r −1 + (p −1)(q −1) + (q −1)(r −1) + (r −1)(p...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 23

IMO 1989 SL 23 Origin: POL Problem We consider permutations (x1, . . . , x2n) of the set {1, . . . , 2n} such that |xi −xi+1| = n for at least one i \in{1, . . ., 2n−1}. For every natural number n, ﬁnd out whether permutations with this property are more or less numerous than the remaining permutations of {1, . . . , 2n}. Solution...

Practice › Mathematics › IMO Shortlist › IMO 1986 Shortlist ›

IMO 1986 SL 11

IMO 1986 SL 11 Origin: BUL Problem Let f(n) be the least number of distinct points in the plane such that for each k = 1, 2, . . ., n there exists a straight line containing exactly k of these points. Find an explicit expression for f(n). Simpliﬁed version. Show that f(n) = / n+1 0 / n+2 ([x] denoting the great- est integer not exceeding x). Solution Let...

Practice › Mathematics › IMO Shortlist › IMO 1977 Shortlist ›

IMO 1977 SL 10

IMO 1977 SL 10 Origin: NET Problem Let n be an integer greater than 2. Deﬁne V = {1 + kn | k = 1, 2, . . . }. A number p \inV is called indecomposable in V if it is not possible to ﬁnd numbers q1, q2 \inV such that q1q2 = p. Prove that there exists a number N \inV that can be factorized into indecomposable factors...

Practice › Mathematics › IMO Shortlist › IMO 1997 Shortlist ›

IMO 1997 SL 24

IMO 1997 SL 24 Origin: LIT Problem For a positive integer n, let f(n) denote the number of ways to represent n as the sum of powers of 2 with nonnegative integer exponents. Representations that diﬀer only in the ordering in their summands are not considered to be distinct. (For instance, f(4) = 4 because the number 4 can be represented in the following four ways: 4; 2+2; 2+1+1; 1+1+1+1.)...

Practice › Mathematics › IMO Shortlist › IMO 1985 Shortlist ›

IMO 1985 SL 14

IMO 1985 SL 14 Problem 4b.(IRE 4) A set of 1985 points is distributed around the circumference of a circle and each of the points is marked with 1 or −1. A point is called “good” if the partial sums that can be formed by starting at that point and proceeding around the circle for any distance in either direction are all strictly positive. Show that if the number of...

Practice › Mathematics › IMO Shortlist › IMO 1970 Shortlist ›

IMO 1970 SL 11

IMO 1970 SL 11 Origin: SWE Problem Let P, Q, R be polynomials and let S(x) = P(x3) + xQ(x3) + x2R(x3) be a polynomial of degree n whose roots x1, . . . , xn are distinct. Construct with the aid of the polynomials P, Q, R a polynomial T of degree n that has the roots x3 1, x3 2, . . . , x3 n. Solution Let...

Practice › Mathematics › IMO Shortlist › IMO 1993 Shortlist ›

IMO 1993 SL 9

IMO 1993 SL 9 Origin: IND Problem (a) Show that the set Q+ of all positive rational numbers can be par- titioned into three disjoint subsets A, B, C satisfying the following conditions: BA = B, B2 = C, BC = A, where HK stands for the set {hk | h \inH, k \inK} for any two subsets H, K of Q+ and H2 stands for HH. (b) Show that...

Practice › Mathematics › IMO Shortlist › IMO 1984 Shortlist ›

IMO 1984 SL 5

IMO 1984 SL 5 Origin: FRG Problem Let x, y, z be nonnegative real numbers with x+y +z = 1. Show that 0 \leqxy + yz + zx −2xyz \leq7 27. Solution Let f(x, y, z) = xy + yz + zx −2xyz. The ﬁrst inequality follows imme- diately by adding xy \geqxyz, yz \geqxyz, and zx \geqxyz (in fact, a stronger inequality xy + yz + zx −9xyz \geq0...

Practice › Mathematics › IMO Shortlist › IMO 1994 Shortlist ›

IMO 1994 SL N2

IMO 1994 SL N2 Origin: AUS | Category: Number Theory Problem Determine all pairs (m, n) of positive integers such that n3+1 mn−1 is an integer. Solution Since mn −1 and m3 are relatively prime, mn −1 divides n3 + 1 if and only if it divides m3(n3 + 1) = (m3n3 −1) + m3 + 1. Thus n3 + 1 mn −1 \inZ ⇔m3 + 1 mn −1 \inZ;...

Practice › Mathematics › IMO Shortlist › IMO 1977 Shortlist ›

IMO 1977 SL 12

IMO 1977 SL 12 Origin: NET Problem On the sides of a square ABCD one constructs inwardly equilateral triangles ABK, BCL, CDM, DAN. Prove that the midpoints of the four segments KL, LM, MN, NK, together with the midpoints of the eight segments AK, BK, BL, CL, CM, DM, DN, AN, are the 12 vertices of a regular dodecagon.

Practice › Mathematics › IMO Shortlist › IMO 1997 Shortlist ›

IMO 1997 SL 3

IMO 1997 SL 3 Origin: GER Problem For each ﬁnite set U of nonzero vectors in the plane we deﬁne l(U) to be the length of the vector that is the sum of all vectors in U. Given a ﬁnite set V of nonzero vectors in the plane, a subset B of V is said to be maximal if l(B) is greater than or equal to l(A) for each nonempty...

Practice › Mathematics › IMO Shortlist › IMO 1985 Shortlist ›

IMO 1985 SL 2

IMO 1985 SL 2 Origin: BRA Problem A polyhedron has 12 faces and is such that: (i) all faces are isosceles triangles, (ii) all edges have length either x or y, (iii) at each vertex either 3 or 6 edges meet, and (iv) all dihedral angles are equal. Find the ratio x/y. Solution The polyhedron has 3 \cdot 12/2 = 18 edges, and by Euler’s formula, 8 vertices. Let v1...

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL N3

IMO 2003 SL N3 Origin: BUL | Category: Number Theory Problem Determine all pairs (a, b) of positive integers such that a2 2ab2 −b3 + 1 is a positive integer. Solution Let a and b be positive integers for which a2 2ab2−b3+1 = k is a positive integer. Since k > 0, it follows that 2ab2 \geqb3, so 2a \geqb. If 2a > b, then from 2ab2 −b3 + 1...

Practice › Mathematics › IMO Shortlist › IMO 1986 Shortlist ›

IMO 1986 SL 15

IMO 1986 SL 15 Origin: NET Problem Let ABCD be a convex quadrilateral whose vertices do not lie on a circle. Let A′B′C′D′ be a quadrangle such that A′, B′, C′, D′ are the centers of the circumcircles of triangles BCD, ACD, ABD, and ABC. We write T (ABCD) = A′B′C′D′. Let us deﬁne A′′B′′C′′D′′ = T (A′B′C′D′) = T (T (ABCD)). (a) Prove that ABCD and A′′B′′C′′D′′ are similar....

Practice › Mathematics › IMO Shortlist › IMO 1981 Shortlist ›

IMO 1981 SL 12

IMO 1981 SL 12 Origin: NET Problem Determine the maximum value of m2 + n2 where m and n are integers satisfying m, n \in{1, 2, . . ., 100} and (n2 −mn −m2)2 = 1. Solution We will solve the contest problem (in which m, n \in{1, 2, . . ., 1981}). For m = 1, n can be either 1 or 2. If m > 1, then n(n...

Practice › Mathematics › IMO Shortlist › IMO 1973 Shortlist ›

IMO 1973 SL 3

IMO 1973 SL 3 Origin: CZS Problem Prove that the sum of an odd number of unit vectors passing through the same point O and lying in the same half-plane whose border passes through O has length greater than or equal to 1. Solution We use induction on odd numbers n. For n = 1 there is nothing to prove. Suppose that the result holds for n−2 vectors, and let...

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL G1

IMO 2000 SL G1 Origin: NET | Category: Geometry Problem In the plane we are given two circles intersecting at X and Y . Prove that there exist four points A, B, C, D with the following property: For every circle touching the two given circles at A and B, and meeting the line XY at C and D, each of the lines AC, AD, BC, BD passes through one...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 13

IMO 1979 SL 13 Origin: GRE Problem Show that 20 60 < sin 20◦< 21 60. Solution From elementary trigonometry we have sin 3t = 3 sin t −4 sin3 t. Hence, if we denote y = sin 20◦, we have \sqrt 3/2 = sin 60◦= 3y −4y3. Obviously 0 < y < 1/2 = sin 30◦. The function f(x) = 3x −4x3 is strictly increasing on [0, 1/2) because...

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL C4

IMO 2003 SL C4 Origin: IRN | Category: Combinatorics Problem Let x1, . . . , xn and y1, . . . , yn be real numbers. Let A = (aij)1\leqi,j\leqn be the matrix with entries aij = . 1, if xi + yj \geq0; 0, if xi + yj < 0. Suppose that B is an n \times n matrix whose entries are 0, 1 such that the sum...

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL C1

IMO 2003 SL C1 Origin: BRA | Category: Combinatorics Problem Let A be a 101-element subset of the set S = {1, 2, . . . , 1000000}. Prove that there exist numbers t1, t2, . . . , t100 in S such that the sets Aj = {x + tj | x \inA}, j = 1, 2, . . . , 100, are pairwise disjoint. Solution Consider the set...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 14

IMO 1979 SL 14 Origin: GRE Problem Find all bases of logarithms in which a real positive number can be equal to its logarithm or prove that none exist. Solution Let us assume that a \inR \ {1} is such that there exist a and x such that x = loga x, or equivalently f(x) := ln x/x = ln a. Then a is a value of the function f(x)...

Practice › Mathematics › IMO Shortlist › IMO 1975 Shortlist ›

IMO 1975 SL 1

IMO 1975 SL 1 Origin: FRA Problem There are six ports on a lake. Is it possible to organize a series of routes satisfying the following conditions: (i) Every route includes exactly three ports; (ii) No two routes contain the same three ports; (iii) The series oﬀers exactly two routes to each tourist who desires to visit two diﬀerent arbitrary ports? Solution First, we observe that there cannot exist three...

Practice › Mathematics › IMO Shortlist › IMO 1994 Shortlist ›

IMO 1994 SL N1

IMO 1994 SL N1 Origin: BUL | Category: Number Theory Problem M is a subset of {1, 2, 3, . . ., 15} such that the product of any three distinct elements of M is not a square. Determine the maximum number of elements in M. Solution Since for each of the subsets {1, 4, 9}, {2, 6, 12}, {3, 5, 15} and {7, 8, 14} the product of its...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 4

IMO 1991 SL 4 Origin: FRA Problem Let ABC be a triangle and M an interior point in ABC. Show that at least one of the angles ∡MAB, ∡MBC, and ∡MCA is less than or equal to 30◦. Solution Assume the contrary, that \angleMAB, \angleMBC, \angleMCA are all greater than 30◦. By the sine Ceva theorem, it holds that sin \angleMAC sin \angleMBA sin \angleMCB = sin \angleMAB sin \angleMBC...

Practice › Mathematics › IMO Shortlist › IMO 1983 Shortlist ›

IMO 1983 SL 17

IMO 1983 SL 17 Origin: GDR Problem Let P1, P2, . . . , Pn be distinct points of the plane, n \geq2. Prove that max 1\leqi<j\leqn PiPj > \sqrt 2 (\sqrtn −1) min 1\leqi<j\leqn PiPj. Solution Set a = min PiPj, b = max PiPj. We use the following lemma. Lemma. There exists a disk of radius less than or equal to b/ \sqrt 3 containing all the Pi’s....

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL A3

IMO 2001 SL A3 Origin: ROM | Category: Algebra Problem Let x1, x2, . . . , xn be arbitrary real numbers. Prove the inequality x1 1 + x2 + x2 1 + x2 1 + x2 \cdot \cdot \cdot + xn 1 + x2 1 + \cdot \cdot \cdot + x2n < \sqrtn. Solution By the arithmetic–quadratic mean inequality, it suﬃces to prove that x2 (1 + x2 1)2...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 12

IMO 1991 SL 12 Origin: CHN Problem Let S = {1, 2, 3, . . ., 280}. Find the minimal natural num- ber n such that in any n-element subset of S there are ﬁve numbers that are pairwise relatively prime. Solution Let Am be the set of those elements of S which are divisible by m. By the inclusion-exclusion principle, the number of elements divisible by 2, 3, 5...

Practice › Mathematics › IMO Shortlist › IMO 1985 Shortlist ›

IMO 1985 SL 1

IMO 1985 SL 1 Origin: MON Problem Given a set M of 1985 positive integers, none of which has a prime divisor larger than 26, prove that the set has four distinct elements whose geometric mean is an integer. Solution Since there are 9 primes (p1 = 2 < p2 = 3 < \cdot \cdot \cdot < p9 = 23) less than 26, each number xj \inM is of the...

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL N4

IMO 1999 SL N4 Origin: FRA | Category: Number Theory Problem Denote by S the set of all primes p such that the decimal representation of 1/p has its fundamental period divisible by 3. For every p \inS such that 1/p has its fundamental period 3r one may write 1/p = 0.a1a2 . . . a3ra1a2 . . . a3r . . . , where r = r(p); for every...

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL G4

IMO 2000 SL G4 Origin: RUS | Category: Geometry Problem Let A1A2 . . . An be a convex polygon, n \geq4. Prove that A1A2 . . . An is cyclic if and only if to each vertex Aj one can assign a pair (bj, cj) of real numbers, j = 1, 2, . . .n, such that AiAj = bjci −bicj for all i, j with 1 \leqi \leqj...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL N3

IMO 1996 SL N3 Origin: BUL | Category: Number Theory Problem A finite sequence of integers $a_0, a_1, \ldots, a_n$ is called quadratic if for each $i \in {1, 2, \ldots, n}$ we have the equality $|a_i - a_{i-1}| = i^2$. (a) Prove that for any two integers $b$ and $c$, there exist a natural number $n$ and a quadratic sequence with $a_0 = b$ and $a_n = c$. (b)...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 12

IMO 1989 SL 12 Origin: HUN Problem At n distinct points of a circular race course there are n cars ready to start. Each car moves at a constant speed and covers the circle in an hour. On hearing the initial signal, each of them selects a direction and starts moving immediately. If two cars meet, both of them change directions and go on without loss of speed. Show that...

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL A1

IMO 1999 SL A1 Origin: POL | Category: Algebra Problem Let n \geq2 be a ﬁxed integer. Find the least constant C such that the inequality i<j xixj(x2 i + x2 j) \leqC i xi holds for every x1, . . . , xn \geq0 (the sum on the left consists of n sum- mands). For this constant C, characterize the instances of equality. Solution For all...

Practice › Mathematics › IMO Shortlist › IMO 1993 Shortlist ›

IMO 1993 SL 16

IMO 1993 SL 16 Origin: MCD Problem Let n \inN, n \geq2, and A0 = (a01, a02, . . . , a0n) be any n-tuple of natural numbers such that 0 \leqa0i \leqi−1, for i = 1, . . . , n. The n-tuples A1 = (a11, a12, . . . , a1n), A2 = (a21, a22, . . . , a2n), . . . are deﬁned by ai+1,j =...

Practice › Mathematics › IMO Shortlist › IMO 1984 Shortlist ›

IMO 1984 SL 4

IMO 1984 SL 4 Origin: MON Problem Let d be the sum of the lengths of all diagonals of a convex polygon of n (n > 3) vertices and let p be its perimeter. Prove that n −3 < d p < 1 n n + 1 −2 . Solution Consider the convex n-gon A1A2 . . . An (the indices are considered modulo n). For any diagonal...

Practice › Mathematics › IMO Shortlist › IMO 1984 Shortlist ›

IMO 1984 SL 15

IMO 1984 SL 15 Origin: LUX Problem Angles of a given triangle ABC are all smaller than 120◦. Equilateral triangles AFB, BDC and CEA are constructed in the exterior of \triangleABC. (a) Prove that the lines AD, BE, and CF pass through one point S. (b) Prove that SD + SE + SF = 2(SA + SB + SC). Solution (a) Since rotation by 60◦around A transforms the triangle CAF...

Practice › Mathematics › IMO Shortlist › IMO 1984 Shortlist ›

IMO 1984 SL 9

IMO 1984 SL 9 Origin: POL Problem Let a, b, c be positive numbers with \sqrta+ \sqrt b+\sqrtc = \sqrt 2 . Prove that the system of equations \sqrty −a + \sqrtz −a = 1, \sqrt z −b + \sqrt x −b = 1, \sqrtx −c + \sqrty −c = 1, has exactly one solution (x, y, z) in real numbers. Solution Let us show ﬁrst that the system has...

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 11

IMO 1998 SL 11 Origin: RUS Problem Let x, y, and z be positive real numbers such that xyz = 1. Prove that x3 (1 + y)(1 + z) + y3 (1 + z)(1 + x) + z3 (1 + x)(1 + y) \geq3 4. Solution The given inequality is equivalent to x3(x + 1) + y3(y + 1) + z3(z + 1) \geq 4(x + 1)(y + 1)(z +...

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL A5

IMO 2002 SL A5 Origin: IND | Category: Algebra Problem Let n be a positive integer that is not a perfect cube. Deﬁne real numbers a, b, c by a = 3\sqrtn, b = a −[a], c = b −[b], where [x] denotes the integer part of x. Prove that there are inﬁnitely many such integers n with the property that there exist integers r, s, t, not all zero,...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL C6

IMO 2004 SL C6 Origin: IRN | Category: Combinatorics Problem For an n \times n matrix A, let Xi be the set of entries in row i, and Yj the set of entries in column j, 1 \leqi, j \leqn. We say that A is golden if X1, . . . , Xn, Y1, . . . , Yn are distinct sets. Find the least integer n such that there...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 15

IMO 1989 SL 15 Origin: IRE Problem Let a, b, c, d, m, n be positive integers such that a2+b2+c2+d2 = 1989, a+b+c+d = m2, and the largest of a, b, c, d is n2. Determine, with proof, the values of m and n. Solution By Cauchy’s inequality, 44 < \sqrt 1989 < a + b + c + d \leq \sqrt 2 \cdot 1989 < 90. Since m2 =...

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 6

IMO 1990 SL 6 Origin: FRG Problem Two players A and B play a game in which they choose numbers alternately according to the following rule: At the beginning, an initial natural number n0 > 1 is given. Knowing n2k, player A may choose any n2k+1 \inN such that n2k \leqn2k+1 \leqn2 2k. Then player B chooses a number n2k+2 \inN such that n2k+1 n2k+2 = pr, where p is...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 16

IMO 1991 SL 16 Origin: ROM Problem Let n > 6 and a1 < a2 < \cdot \cdot \cdot < ak be all natural numbers that are less than n and relatively prime to n. Show that if a1, a2, . . . , ak is an arithmetic progression, then n is a prime number or a natural power of two. Solution Let p be the least prime number that...

Practice › Mathematics › IMO Shortlist › IMO 1970 Shortlist ›

IMO 1970 SL 7

IMO 1970 SL 7 Origin: USS Problem For which digits a do exist integers n \geq4 such that each digit of n(n+1) equals a? Solution For a = 5 one can take n = 10, while for a = 6 one takes n = 11. Now assume a ̸\in{5, 6}. If there exists an integer n such that each digit of n(n + 1)/2 is equal to a, then there...

Practice › Mathematics › IMO Shortlist › IMO 1997 Shortlist ›

IMO 1997 SL 5

IMO 1997 SL 5 Origin: ROM Problem Let ABCD be a regular tetrahedron and M, N distinct points in the planes ABC and ADC respectively. Show that the segments MN, BN, MD are the sides of a triangle. Solution We shall prove ﬁrst the 2-dimensional analogue: Lemma. Given an equilateral triangle ABC and two points M, N on the sides AB and AC respectively, there exists a triangle with sides...

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 26

IMO 1998 SL 26 Origin: IND Problem In a contest, there are m candidates and n judges, where n \geq3 is an odd integer. Each candidate is evaluated by each judge as either pass or fail. Suppose that each pair of judges agrees on at most k candidates. Prove that k m \geqn−1 2n . Solution Put n = 2r + 1. Since each of the n pairs of...

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 18

IMO 1998 SL 18 Origin: BUL Problem Determine all positive integers n for which there exists an integer m such that 2n −1 is a divisor of m2 + 9. Solution We claim that, if 2n −1 divides m2 + 9 for some m \inN, then n must be a power of 2. Suppose otherwise that n has an odd divisor d > 1. Then 2d −1 | 2n −1...

Practice › Mathematics › IMO Shortlist › IMO 1976 Shortlist ›

IMO 1976 SL 7

IMO 1976 SL 7 Problem (POL 1b) Let I = (0, 1] be the unit interval of the real line. For a given number a \in(0, 1) we deﬁne a map T : I \toI by the formula T (x, y) = . x + (1 −a) if 0 < x \leqa, x −a if a < x \leq1. Show that for every interval J \subsetI there exists an integer...

Practice › Mathematics › IMO Shortlist › IMO 1985 Shortlist ›

IMO 1985 SL 10

IMO 1985 SL 10 Problem 2b.(VIE 1) Prove that for every point M on the surface of a regular tetrahedron there exists a point M ′ such that there are at least three diﬀerent curves on the surface joining M to M ′ with the smallest possible length among all curves on the surface joining M to M ′. Solution If M is at a vertex of the regular tetrahedron...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 1

IMO 1989 SL 1 Origin: AUS Problem Let ABC be a triangle. The bisector of angle A meets the circumcircle of triangle ABC in A1. Points B1 and C1 are deﬁned similarly. Let AA1 meet the lines that bisect the two external angles at B and C in point A0. Deﬁne B0 and C0 similarly. If SX1X2...Xn denotes the area of the polygon X1X2 . . . Xn, prove that...

Practice › Mathematics › IMO Shortlist › IMO 1971 Shortlist ›

IMO 1971 SL 1

IMO 1971 SL 1 Origin: BUL Problem Consider a sequence of polynomials P0(x), P1(x), P2(x), . . . , Pn(x), . . . , where P0(x) = 2, P1(x) = x and for every n \geq1 the following equality holds: Pn+1(x) + Pn−1(x) = xPn(x). Prove that there exist three real numbers a, b, c such that for all n \geq1, (x2 −4)[P 2 n(x) −4] = [aPn+1(x) + bPn(x)...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL A4

IMO 1996 SL A4 Origin: KOR | Category: Algebra Problem Let $a_1, a_2, \dots, a_n$ be nonnegative real numbers, not all zero. (a) Prove that $$x^n - a_1 x^{n-1} - a_2 x^{n-2} - \cdots - a_{n-1} x - a_n = 0$$ has precisely one positive real root. (b) Let $$A = \sum_{j=1}^{n} a_j, \quad B = \sum_{j=1}^{n} j a_j,$$ and let $R$ be the positive real root of the equation...

Practice › Mathematics › IMO Shortlist › IMO 1974 Shortlist ›

IMO 1974 SL 1

IMO 1974 SL 1 Problem I 1 (USA 4)IMO1 Alice, Betty, and Carol took the same series of exam- inations. There was one grade of A, one grade of B, and one grade of C for each examination, where A, B, C are diﬀerent positive integers. The ﬁnal test scores were Alice Betty Carol If Betty placed ﬁrst in the arithmetic examination, who placed second in the spelling examination? Solution...

Practice › Mathematics › IMO Shortlist › IMO 1973 Shortlist ›

IMO 1973 SL 5

IMO 1973 SL 5 Origin: FRA Problem A circle of radius 1 is located in a right-angled trihedron and touches all its faces. Find the locus of centers of such circles. Solution Let O be the vertex of the trihedron, Z the center of a circle k inscribed in the trihedron, and A, B, C points in which the plane of the circle meets the edges of the trihedron. We...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL A3

IMO 1996 SL A3 Origin: GRE | Category: Algebra Problem Let $a>2$ be given, and define recursively $$a_0 = 1, \quad a_1 = a, \quad a_{n+1} = \left(\frac{a_n^2}{a_{n-1}^2}-2\right)a_n.$$ Show that for all $k \in \mathbb{N}$, we have $$\frac{1}{a_0+1} + \frac{1}{a_1+1} + \frac{1}{a_2+1} + \cdots + \frac{1}{a_k+1} < \frac{1}{2} + a - \sqrt{a^2-4}.$$ Solution Since $a_1>2$, it can be written as $a_1 = b + b^{-1}$ for some $b>0$. Furthermore, $$a_1^2...

Practice › Mathematics › IMO Shortlist › IMO 1972 Shortlist ›

IMO 1972 SL 4

IMO 1972 SL 4 Origin: GDR Problem Let n1, n2 be positive integers. Consider in a plane E two dis- joint sets of points M1 and M2 consisting of 2n1 and 2n2 points, respec- tively, and such that no three points of the union M1 \cupM2 are collinear. Prove that there exists a straightline g with the following property: Each of the two half-planes determined by g on E (g...

Practice › Mathematics › IMO Shortlist › IMO 1983 Shortlist ›

IMO 1983 SL 12

IMO 1983 SL 12 Origin: GBR Problem Find all functions f deﬁned on the positive real numbers and taking positive real values that satisfy the following conditions: (i) f(xf(y)) = yf(x) for all positive real x, y. (ii) f(x) \to0 as x \to+\infty. Solution Putting y = x in (1) we see that there exist positive real numbers z such that f(z) = z (this is true for every z...

Practice › Mathematics › IMO Shortlist › IMO 1968 Shortlist ›

IMO 1968 SL 3

IMO 1968 SL 3 Origin: POL Problem Prove that in any tetrahedron there is a vertex such that the lengths of its sides through that vertex are sides of a triangle. Solution A triangle cannot be formed out of three lengths if and only if one of them is larger than the sum of the other two. Let us assume this is the case for all triplets of edges out...

Practice › Mathematics › IMO Shortlist › IMO 1985 Shortlist ›

IMO 1985 SL 21

IMO 1985 SL 21 Origin: IRE Problem The tangents at B and C to the circumcircle of the acute-angled triangle ABC meet at X. Let M be the midpoint of BC. Prove that (a) \angleBAM = \angleCAX, and (b) AM AX = cos \angleBAC. Solution Let B1 and C1 be the points on the rays AC and AB respectively such that XB1 = XC = XB = XC1. Then \angleXB1C...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 28

IMO 1991 SL 28 Origin: NET Problem Given a real number a > 1, construct an inﬁnite and bounded sequence x0, x1, x2, . . . such that for all natural numbers i and j, i ̸= j, the following inequality holds: |xi −xj||i −j|a \geq1. Solution Let xn = c(n \sqrt 2 −[n \sqrt 2]) for some constant c > 0. For i > j, putting p = [i...

Practice › Mathematics › IMO Shortlist › IMO 1981 Shortlist ›

IMO 1981 SL 14

IMO 1981 SL 14 Origin: ROM Problem Prove that a convex pentagon (a ﬁve-sided polygon) ABCDE with equal sides and for which the interior angles satisfy the condition \angleA \geq\angleB \geq\angleC \geq\angleD \geq\angleE is a regular pentagon. Solution We need the following lemma. Lemma. If a convex quadrilateral PQRS satisﬁes PS = QR and \angleSPQ \geq \angleRQP, then \angleQRS \geq\anglePSR. Proof. If the lines PS and QR are parallel, then...

Practice › Mathematics › IMO Shortlist › IMO 1983 Shortlist ›

IMO 1983 SL 3

IMO 1983 SL 3 Origin: BEL Problem We say that a set E of points of the Euclidian plane is “Pythagorean” if for any partition of E into two sets A and B, at least one of the sets contains the vertices of a right-angled triangle. Decide whether the following sets are Pythagorean: (a) a circle; (b) an equilateral triangle (that is, the set of three vertices and the points...

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL N2

IMO 2003 SL N2 Origin: USA | Category: Number Theory Problem Each positive integer a undergoes the following procedure in order to obtain the number d = d(a): (1) move the last digit of a to the ﬁrst position to obtain the number b; (2) square b to obtain the number c; (3) move the ﬁrst digit of c to the end to obtain the number d. (All the numbers...

Practice › Mathematics › IMO Shortlist › IMO 1981 Shortlist ›

IMO 1981 SL 7

IMO 1981 SL 7 Origin: FIN Problem Assume that f(x, y) is deﬁned for all positive integers x and y, and that the following equations are satisﬁed: f(0, y) = y + 1, f(x + 1, 0) = f(x, 1), f(x + 1, y + 1) = f(x, f(x + 1, y)). Determine f(2, 2), f(3, 3) and f(4, 4). Alternative version: Determine f(4, 1981). Solution We immediately ﬁnd that...

Practice › Mathematics › IMO Shortlist › IMO 1981 Shortlist ›

IMO 1981 SL 16

IMO 1981 SL 16 Origin: GBR Problem A sequence of real numbers u1, u2, u3, . . . is determined by u1 and the following recurrence relation for n \geq1: 4un+1 = 3\sqrt64un + 15. Describe, with proof, the behavior of un as n \to\infty. Solution The sequence {un} is bounded, whatever u1 is. Indeed, assume the opposite, and let um be the ﬁrst member of the sequence such that...

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 1

IMO 1998 SL 1 Origin: LUX Problem A convex quadrilateral ABCD has perpendicular diagonals. The perpendicular bisectors of AB and CD meet at a unique point P inside ABCD. Prove that ABCD is cyclic if and only if triangles ABP and CDP have equal areas. Solution We begin with the following observation: Suppose that P lies in \triangleAEB, where E is the intersection of AC and BD (the other cases...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 4

IMO 1979 SL 4 Origin: BUL Problem A pentagonal prism A1A2 . . . A5B1B2 . . . B5 is given. The edges, the diagonals of the lateral walls and the internal diagonals of the prism are each colored either red or green in such a way that no triangle whose vertices are vertices of the prism has its three edges of the same color. Prove that all edges of...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL A2

IMO 1996 SL A2 Origin: IRE | Category: Algebra Problem Let $a_1 \geq a_2 \geq \cdots \geq a_n$ be real numbers such that $a_1^k + a_2^k + \cdots + a_n^k \geq 0$ for all integers $k > 0$. Let $p = \max{|a_1|, \ldots, |a_n|}$. Prove that $p = a_1$ and that $(x - a_1)(x - a_2) \cdots (x - a_n) \leq x^n - a_1 x^{n-1} + \cdots + (-1)^n a_n$...

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL A4

IMO 2003 SL A4 Origin: IRE | Category: Algebra Problem Let n be a positive integer and let x1 \leqx2 \leq\cdot \cdot \cdot \leqxn be real numbers. (a) Prove that ⎛ ⎝ n i,j=1 |xi −xj| ⎞ ⎠ \leq2(n2 −1) n i,j=1 (xi −xj)2. (b) Show that equality holds if and only if x1, . . . , xn is an arithmetic progession. Solution By the Cauchy–Schwarz inequality...

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL C4

IMO 2002 SL C4 Origin: BUL | Category: Combinatorics Problem Let T be the set of ordered triples (x, y, z), where x, y, z are integers with 0 \leqx, y, z \leq9. Players A and B play the following guessing game: Player A chooses a triple (x, y, z) in T , and Player B has to discover A’s triple in as few moves as possible. A move consists...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL A5

IMO 2004 SL A5 Origin: THA | Category: Algebra Problem Let a, b, c > 0 and ab + bc + ca = 1. Prove the inequality a + 6b + b + 6c + c + 6a \leq abc. Solution By the general mean inequality (M1 \leqM3), the LHS of the inequality to be proved does not exceed E = 3\sqrt a + 1 b...

Practice › Mathematics › IMO Shortlist › IMO 1968 Shortlist ›

IMO 1968 SL 13

IMO 1968 SL 13 Origin: POL Problem Given two congruent triangles $A_1A_2A_3$ and $B_1B_2B_3$ $(A_iA_k = B_iB_k)$, prove that there exists a plane such that the orthogonal projections of these triangles onto it are congruent and equally oriented. Solution Translating one of the triangles if necessary, we may assume w.l.o.g. that $B_1 \equiv A_1$. We also assume that $B_2 \not\equiv A_2$ and $B_3 \not\equiv A_3$, since the result is obvious...

Practice › Mathematics › IMO Shortlist › IMO 1981 Shortlist ›

IMO 1981 SL 9

IMO 1981 SL 9 Origin: FRG Problem A sequence (an) is deﬁned by means of the recursion a1 = 1, an+1 = 1 + 4an + \sqrt1 + 24an . Find an explicit formula for an. Solution If we put 1 + 24an = b2 n, the given recurrent relation becomes 3b2 n+1 = 3 2 + b2 n 6 + bn = 2 3 2 + bn 2 ,...

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 7

IMO 1990 SL 7 Origin: GRE Problem Let f(0) = f(1) = 0 and f(n + 2) = 4n+2f(n + 1) −16n+1f(n) + n \cdot 2n2, n = 0, 1, 2, 3, . . .. Show that the numbers f(1989), f(1990), f(1991) are divisible by 13. Solution Let f(n) = g(n)2n2 for all n. The recursion then transforms into g(n + 2) −2g(n + 1) + g(n) = n \cdot...

Practice › Mathematics › IMO Shortlist › IMO 1983 Shortlist ›

IMO 1983 SL 9

IMO 1983 SL 9 Origin: USA Problem If a, b, and c are sides of a triangle, prove that a2b(a −b) + b2c(b −c) + c2a(c −a) \geq0. Determine when there is equality. Solution For any triangle of sides a, b, c there exist 3 nonnegative numbers x, y, z such that a = y +z, b = z +x, c = x+y (these numbers correspond to the division of...

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 14

IMO 1990 SL 14 Origin: JAP Problem In the coordinate plane a rectangle with vertices (0, 0), (m, 0), (0, n), (m, n) is given where both m and n are odd integers. The rectangle is partitioned into triangles in such a way that (i) each triangle in the partition has at least one side (to be called a “good” side) that lies on a line of the form x...

Practice › Mathematics › IMO Shortlist › IMO 1981 Shortlist ›

IMO 1981 SL 10

IMO 1981 SL 10 Origin: FRA Problem Determine the smallest natural number n having the following property: For every integer p, p \geqn, it is possible to subdivide (partition) a given square into p squares (not necessarily equal). Solution It is easy to see that partitioning into p = 2k squares is possible for k \geq2 (Fig. 1). Furthermore, whenever it is possible to partition the square into p squares,...

Practice › Mathematics › IMO Shortlist › IMO 1993 Shortlist ›

IMO 1993 SL 22

IMO 1993 SL 22 Origin: GBR Problem A, B, C, D are four points in the plane, with C, D on the same side of the line AB, such that AC \cdot BD = AD \cdot BC and ∡ADB = 90◦+ ∡ACB. Find the ratio AB \cdot CD AC \cdot BD, and prove that circles ACD, BCD are orthogonal. (Intersecting circles are said to be orthogonal if at either common...

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL N3

IMO 2000 SL N3 Origin: RUS | Category: Number Theory Problem Does there exist a positive integer n such that n has exactly 2000 prime divisors and 2n + 1 is divisible by n? Solution More generally, we will prove by induction on k that for each k \inN there exists nk \inN that has exactly k distinct prime divisors such that nk | 2nk + 1 and 3 |...

Practice › Mathematics › IMO Shortlist › IMO 1973 Shortlist ›

IMO 1973 SL 11

IMO 1973 SL 11 Origin: SWE Problem Determine the minimum of a2 + b2 if a and b are real numbers for which the equation x4 + ax3 + bx2 + ax + 1 = 0 has at least one real solution. Solution Putting x + 1 x = t we also get x2 + x2 = t2 −2, and the given equation reduces to t2 + at + b...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 8

IMO 1979 SL 8 Origin: FRG Problem For all rational x satisfying 0 \leqx < 1, f is deﬁned by f(x) = f(2x)/4, for 0 \leqx < 1/2, 3/4 + f(2x −1)/4, for 1/2 \leqx < 1. Given that x = 0.b1b2b3 . . . is the binary representation of x, ﬁnd f(x). Solution By the deﬁnition of f, it holds that f(0.b1b2 . . . ) = 3b1/4+f(0.b2b3 ....

Practice › Mathematics › IMO Shortlist › IMO 1987 Shortlist ›

IMO 1987 SL 11

IMO 1987 SL 11 Origin: POL Problem Find the number of partitions of the set {1, 2, . . ., n} into three subsets A1, A2, A3, some of which may be empty, such that the following conditions are satisﬁed: (i) After the elements of every subset have been put in ascending order, every two consecutive elements of any subset have diﬀerent parity. (ii) If A1, A2, A3 are all...

Practice › Mathematics › IMO Shortlist › IMO 1972 Shortlist ›

IMO 1972 SL 2

IMO 1972 SL 2 Origin: CZS Problem We are given 3n points A1, A2, . . . , A3n in the plane, no three of them collinear. Prove that one can construct n disjoint triangles with vertices at the points Ai. Solution We use induction. For n = 1 the assertion is obvious. Assume that it is true for a positive integer n. Let A1, A2, . . . ,...

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 19

IMO 1998 SL 19 Origin: BLR Problem For any positive integer n, let \tau(n) denote the number of its positive divisors (including 1 and itself). Determine all positive integers m for which there exists a positive integer n such that \tau(n2) \tau(n) = m. Solution For n = p\alpha1 1 p\alpha2 2 \cdot \cdot \cdot p\alphar r , where pi are distinct primes and \alphai natural num- bers, we have...

Practice › Mathematics › IMO Shortlist › IMO 1994 Shortlist ›

IMO 1994 SL A1

IMO 1994 SL A1 Origin: USA | Category: Algebra Problem Let a0 = 1994 and an+1 = a2 n an+1 for each nonnegative integer n. Prove that 1994 −n is the greatest integer less than or equal to an, 0 \leqn \leq998. Solution Obviously a0 > a1 > a2 > \cdot \cdot \cdot . Since ak −ak+1 = 1 − ak+1, we have an = a0 +(a1 −a0)+\cdot \cdot \cdot+(an...

Practice › Mathematics › IMO Shortlist › IMO 1978 Shortlist ›

IMO 1978 SL 4

IMO 1978 SL 4 Origin: CZS Problem Let T1 be a triangle having a, b, c as lengths of its sides and let T2 be another triangle having u, v, w as lengths of its sides. If P, Q are the areas of the two triangles, prove that 16PQ \leqa2(−u2 + v2 + w2) + b2(u2 −v2 + w2) + c2(u2 + v2 −w2). When does equality hold? Solution Let...

Practice › Mathematics › IMO Shortlist › IMO 1983 Shortlist ›

IMO 1983 SL 8

IMO 1983 SL 8 Origin: SPA Problem In a test, 3n students participate, who are located in three rows of n students in each. The students leave the test room one by one. If N1(t), N2(t), N3(t) denote the numbers of students in the ﬁrst, second, and third row respectively at time t, ﬁnd the probability that for each t during the test, |Ni(t) −Nj(t)| < 2, i ̸= j,...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 31

IMO 1988 SL 31 Origin: USS Problem Around a circular table an even number of persons have a discussion. After a break they sit again around the circular table in a diﬀerent order. Prove that there are at least two people such that the number of participants sitting between them before and after the break is the same. Solution Denote the number of participants by 2n, and assign to each...

Practice › Mathematics › IMO Shortlist › IMO 1997 Shortlist ›

IMO 1997 SL 26

IMO 1997 SL 26 Origin: ITA Problem For every integer n \geq2 determine the minimum value that the sum a0 + a1 + \cdot \cdot \cdot + an can take for nonnegative numbers a0, a1, . . . , an satisfying the condition a0 = 1, ai \leqai+1 + ai+2 for i = 0, . . . , n −2. Solution Let us ﬁrst examine the case that all the...

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL N6

IMO 2001 SL N6 Origin: RUS | Category: Number Theory Problem Is it possible to ﬁnd 100 positive integers not exceeding 25,000 such that all pairwise sums of them are diﬀerent? Solution Yes. The desired result is an immediate consequence of the following fact applied on p = 101. Lemma. For any odd prime number p, there exist p nonnegative integers less than 2p2 with all pairwise sums mutually distinct....

Practice › Mathematics › IMO Shortlist › IMO 1983 Shortlist ›

IMO 1983 SL 23

IMO 1983 SL 23 Origin: USS Problem Let K be one of the two intersection points of the circles W1 and W2. Let O1 and O2 be the centers of W1 and W2. The two common tangents to the circles meet W1 and W2 respectively in P1 and P2, the ﬁrst tangent, and Q1 and Q2, the second tangent. Let M1 and M2 be the midpoints of P1Q1 and P2Q2,...

Practice › Mathematics › IMO Shortlist › IMO 1987 Shortlist ›

IMO 1987 SL 19

IMO 1987 SL 19 Origin: USS Problem Let \alpha, \beta, \gamma be positive real numbers such that \alpha + \beta + \gamma < \pi, \alpha + \beta > \gamma, \beta + \gamma > \alpha, \gamma + \alpha > \beta. Prove that with the segments of lengths sin \alpha, sin \beta, sin \gamma we can construct a triangle and that its area is not greater than 8(sin 2\alpha + sin 2\beta...

Practice › Mathematics › IMO Shortlist › IMO 1976 Shortlist ›

IMO 1976 SL 11

IMO 1976 SL 11 Origin: VIE Problem Prove that there exist inﬁnitely many positive integers n such that the decimal representation of 5n contains a block of 1976 consecutive zeros. Solution We shall show by induction that 52k −1 = 2k+2qk for each k = 0, 1, . . . , where qk \inN. Indeed, the statement is true for k = 0, and if it holds for some k...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL N4

IMO 1996 SL N4 Origin: BUL | Category: Number Theory Problem Find all positive integers $a$ and $b$ for which $$\left[\frac{a^2}{b}\right] + \left[\frac{b^2}{a}\right] = \left[\frac{a^2 + b^2}{ab}\right] + ab$$ where as usual, $[t]$ refers to greatest integer that is less than or equal to $t$. Solution Let $a, b \in \mathbb{N}$ satisfy the given equation. It is not possible that $a = b$ (since it leads to $a^2 + 2...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 16

IMO 1988 SL 16 Origin: IRE Problem Show that the solution set of the inequality k=1 k x −k \geq5 is a union of disjoint intervals the sum of whose lengths is 1988. Solution Let f(x) = 70 k=1 k x−k. For all integers i = 1, . . . , 70 we have that f(x) tends to plus inﬁnity as x tends downward to i, and f(x) tends...

Practice › Mathematics › IMO Shortlist › IMO 1971 Shortlist ›

IMO 1971 SL 11

IMO 1971 SL 11 Origin: POL Problem The matrix ⎛ ⎜ ⎝ a11 . . . a1n ... \cdot \cdot \cdot ... an1 . . . ann ⎞ ⎟ ⎠ satisﬁes the inequality n j=1 |aj1x1 + \cdot \cdot \cdot + ajnxn| \leqM for each choice of numbers xi equal to \pm1. Show that |a11 + a22 + \cdot \cdot \cdot + ann| \leqM. Solution We use induction. The statement...

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL G6

IMO 2002 SL G6 Origin: UKR | Category: Geometry Problem Let n \geq3 be a positive integer. Let C1, C2, C3, . . . , Cn be unit circles in the plane, with centers O1, O2, O3, . . . , On respectively. If no line meets more than two of the circles, prove that 1\leqi<j\leqn OiOj \leq(n −1)\pi . Solution Let l(MN) denote the length of the shorter...

Practice › Mathematics › IMO Shortlist › IMO 1976 Shortlist ›

IMO 1976 SL 5

IMO 1976 SL 5 Origin: NET Problem Let a set of p equations be given, a11x1 + \cdot \cdot \cdot + a1qxq = 0, a21x1 + \cdot \cdot \cdot + a2qxq = 0, ... ap1x1 + \cdot \cdot \cdot + apqxq = 0, with coeﬃcients aij satisfying aij = −1, 0, or +1 for all i = 1, . . . , p and j = 1, . . ....

Practice › Mathematics › IMO Shortlist › IMO 1981 Shortlist ›

IMO 1981 SL 17

IMO 1981 SL 17 Origin: USS Problem Three equal circles touch the sides of a triangle and have one common point O. Show that the center of the circle inscribed in and of the circle circumscribed about the triangle ABC and the point O are collinear. Solution Let us denote by SA, SB, SC the centers of the given circles, where SA lies on the bisector of \angleA, etc. Then...

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL A1

IMO 2001 SL A1 Origin: IND | Category: Algebra Problem Let T denote the set of all ordered triples (p, q, r) of nonneg- ative integers. Find all functions f : T \toR such that f(p, q, r) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ if pqr = 0, 1 + 1 6 (f(p + 1, q −1, r) + f(p −1, q + 1, r) +f(p...

Practice › Mathematics › IMO Shortlist › IMO 1983 Shortlist ›

IMO 1983 SL 14

IMO 1983 SL 14 Origin: POL Problem Prove or disprove: From the interval [1, . . . , 30000] one can select a set of 1000 integers containing no arithmetic triple (three consecutive numbers of an arithmetic progression). Solution Let Tn be the set of all nonnegative integers whose ternary representations consist of at most n digits and do not contain a digit 2. The cardinality of Tn is 2n,...

Practice › Mathematics › IMO Shortlist › IMO 1978 Shortlist ›

IMO 1978 SL 17

IMO 1978 SL 17 Origin: FRA Problem Prove that for any positive integers x, y, z with xy−z2 = 1 one can ﬁnd nonnegative integers a, b, c, d such that x = a2 + b2, y = c2 + d2, z = ac + bd. Set z = (2q)! to deduce that for any prime number p = 4q + 1, p can be represented as the sum of...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 30

IMO 1991 SL 30 Origin: BUL Problem Two students A and B are playing the following game: Each of them writes down on a sheet of paper a positive integer and gives the sheet to the referee. The referee writes down on a blackboard two integers, one of which is the sum of the integers written by the players. After that, the referee asks student A: “Can you tell the...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL A6

IMO 2004 SL A6 Origin: RUS | Category: Algebra Problem Find all functions f : R \toR satisfying the equation f x2 + y2 + 2f(xy) = (f(x + y))2 for all x, y \inR. Solution Let us make the substitution z = x + y, t = xy. Given z, t \inR, x, y are real if and only if 4t \leqz2. Deﬁne g(x) = 2(f(x) −x)....

Practice › Mathematics › IMO Shortlist › IMO 1977 Shortlist ›

IMO 1977 SL 14

IMO 1977 SL 14 Problem (FIN 2‘) Let E be a ﬁnite set of points such that E is not contained in a plane and no three points of E are collinear. Show that at least one of the following alternatives holds: (i) E contains ﬁve points that are vertices of a convex pyramid having no other points in common with E; (ii) some plane contains exactly three points from...

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL N5

IMO 2001 SL N5 Origin: BUL | Category: Number Theory Problem Let a > b > c > d be positive integers and suppose ac + bd = (b + d + a −c)(b + d −a + c). Prove that ab + cd is not prime. Solution The given equality is equivalent to a2 −ac + c2 = b2 + bd + d2. Hence (ab + cd)(ad + bc)...

Practice › Mathematics › IMO Shortlist › IMO 1975 Shortlist ›

IMO 1975 SL 8

IMO 1975 SL 8 Origin: NET Problem On the sides of an arbitrary triangle ABC, triangles BPC, CQA, and ARB are externally erected such that ∡PBC = ∡CAQ = 45◦, ∡BCP = ∡QCA = 30◦, ∡ABR = ∡BAR = 15◦. Prove that ∡QRP = 90◦and QR = RP. Solution Let K and L be the feet of perpendiculars from P and Q to BC and AC respectively. Let M, N...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 13

IMO 1988 SL 13 Origin: GRE Problem In a right-angled triangle ABC, let AD be the altitude drawn to the hypotenuse and let the straight line joining the incenters of the triangles ABD, ACD intersect the sides AB, AC at the points K, L respectively. If E and E1 denote the areas of the triangles ABC and AKL respectively, show that E E1 \geq2. Solution Let AB = c, AC...

Practice › Mathematics › IMO Shortlist › IMO 1994 Shortlist ›

IMO 1994 SL G5

IMO 1994 SL G5 Origin: CYP | Category: Geometry Problem A line l does not meet a circle \omega with center O. E is the point on l such that OE is perpendicular to l. M is any point on l other than E. The tangents from M to \omega touch it at A and B. C is the point on MA such that EC is perpendicular to MA. D...

Practice › Mathematics › IMO Shortlist › IMO 1993 Shortlist ›

IMO 1993 SL 3

IMO 1993 SL 3 Origin: SPA Problem Consider the triangle ABC, its circumcircle k with center O and radius R, and its incircle with center I and radius r. Another circle kc is tangent to the sides CA, CB at D, E, respectively, and it is internally tangent to k. Show that the incenter I is the midpoint of DE.

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL G7

IMO 2001 SL G7 Origin: BUL | Category: Geometry Problem Let O be an interior point of acute triangle ABC. Let A1 lie on BC with OA1 perpendicular to BC. Deﬁne B1 on CA and C1 on AB similarly. Prove that O is the circumcenter of ABC if and only if the perimeter of A1B1C1 is not less than any one of the perimeters of AB1C1, BC1A1, and CA1B1. Solution...

Practice › Mathematics › IMO Shortlist › IMO 1978 Shortlist ›

IMO 1978 SL 1

IMO 1978 SL 1 Origin: BUL Problem The set M = {1, 2, . . ., 2n} is partitioned into k nonintersecting subsets M1, M2, . . . , Mk, where n \geqk3 + k. Prove that there exist even numbers 2j1, 2j2, . . . , 2jk+1 in M that are in one and the same subset Mi (1 \leqi \leqk) such that the numbers 2j1 −1, 2j2 −1,...

Practice › Mathematics › IMO Shortlist › IMO 1978 Shortlist ›

IMO 1978 SL 5

IMO 1978 SL 5 Origin: GDR Problem For every integer d \geq1, let Md be the set of all positive integers that cannot be written as a sum of an arithmetic progression with diﬀerence d, having at least two terms and consisting of positive integers. Let A = M1, B = M2 ∖{2}, C = M3. Prove that every c \inC may be written in a unique way as c...

Practice › Mathematics › IMO Shortlist › IMO 1968 Shortlist ›

IMO 1968 SL 26

IMO 1968 SL 26 Origin: GDR Problem Let $a > 0$ be a real number and $f(x)$ a real function defined on all of $\mathbb{R}$, satisfying for all $x \in \mathbb{R}$, $$f(x + a) = \frac{1}{2} + \sqrt{f(x) - f(x)^2}.$$ (a) Prove that the function $f$ is periodic; i.e., there exists $b > 0$ such that for all $x$, $f(x + b) = f(x)$. (b) Give an example of such...

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL N5

IMO 2000 SL N5 Origin: BUL | Category: Number Theory Problem Prove that there exist inﬁnitely many positive integers n such that p = nr, where p and r are respectively the semiperimeter and the inradius of a triangle with integer side lengths. Solution It is known that the area of the triangle is S = pr = p2/n and S = p(p −a)(p −b)(p −c). It follows that p3...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL G6

IMO 1996 SL G6 Origin: ARM | Category: Geometry Problem Let the sides of two rectangles be ${a, b}$ and ${c, d}$ with $a < c \leq d < b$ and $ab < cd$. Prove that the first rectangle can be placed within the second one if and only if $(b^2 - a^2)^2 \leq (bd - ac)^2 + (bc - ad)^2.$ Solution Denote by $ABCD$ and $EFGH$ the two rectangles,...

Practice › Mathematics › IMO Shortlist › IMO 1987 Shortlist ›

IMO 1987 SL 16

IMO 1987 SL 16 Origin: FRG Problem Let S be a set of n elements. We denote the number of all permutations of S that have exactly k ﬁxed points by pn(k). Prove: (a) n k=0 kpn(k) = n!; (b) n k=0(k −1)2pn(k) = n!. Solution We assume that S = {1, 2, . . ., n}, and use the obvious fact n k=0 pn(k) = n! (0) (a)...

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL N2

IMO 1995 SL N2 Origin: RUS | Category: Number Theory Problem Let Z denote the set of all integers. Prove that for any integers A and B, one can ﬁnd an integer C for which M1 = {x2 +Ax+B : x \inZ} and M2 = {2x2 + 2x + C : x \inZ} do not intersect. Solution If A is odd, then every number in M1 is of the form...

Practice › Mathematics › IMO Shortlist › IMO 1998 Shortlist ›

IMO 1998 SL 14

IMO 1998 SL 14 Origin: GBR Problem Determine all pairs (x, y) of positive integers such that x2y+ x + y is divisible by xy2 + y + 7. Solution If x2y + x + y is divisible by xy2 + y + 7, then so is the number y(x2y + x + y) −x(xy2 + y + 7) = y2 −7x. If y2 −7x \geq0, then since y2 −7x...

Practice › Mathematics › IMO Shortlist › IMO 1997 Shortlist ›

IMO 1997 SL 7

IMO 1997 SL 7 Origin: RUS Problem Let ABCDEF be a convex hexagon such that AB = BC, CD = DE, EF = FA. Prove that BC BE + DE DA + FA FC \geq3 2. When does equality occur? Solution Let us set AC = a, CE = b, EA = c. Applying Ptolemy’s inequality for the quadrilateral ACEF we get AC \cdot EF + CE \cdot AF \geqAE...

Practice › Mathematics › IMO Shortlist › IMO 1977 Shortlist ›

IMO 1977 SL 5

IMO 1977 SL 5 Origin: FRG Problem There are 2n words of length n over the alphabet {0, 1}. Prove that the following algorithm generates the sequence w0, w1, . . . , w2n−1 of all these words such that any two consecutive words diﬀer in exactly one digit. (1) w0 = 00 . . .0 (n zeros). (2) Suppose wm−1 = a1a2 . . . an, ai \in{0, 1}....

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 27

IMO 1991 SL 27 Origin: POL Problem Determine the maximum value of the sum i<j xixj(xi + xj) over all n-tuples (x1, . . . , xn), satisfying xi \geq0 and n i=1 xi = 1. Solution Write F(x1, . . . , xn) = i<j xixj(xi+xj). Choose an n-tuple (x1, . . . , xn), n i=1 xi = 1, xi \geq0 with at least three nonzero...

Practice › Mathematics › IMO Shortlist › IMO 1968 Shortlist ›

IMO 1968 SL 15

IMO 1968 SL 15 Origin: GBR Problem Let $[x]$ denote the integer part of $x$, i.e., the greatest integer not exceeding $x$. If $n$ is a positive integer, express as a simple function of $n$ the sum $$\frac{n+1}{2} + \frac{n+2}{2^2} + \cdots + \frac{n+2^i}{2^{i+1}} + \cdots .$$ Solution Set $$f(n)= \frac{n+1}{2} + \frac{n+2}{2^2} + \cdots + \frac{n+2^i}{2^{i+1}} + \cdots .$$ We prove by induction that $f(n)=n$. This obviously holds for...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL C3

IMO 1996 SL C3 Origin: USA | Category: Combinatorics Problem Let $k, m, n$ be integers such that $1 < n \leq m-1 \leq k$. Determine the maximum size of a subset $S$ of the set ${1, 2, 3, \ldots, k}$ such that no $n$ distinct elements of $S$ add up to $m$. Solution Denote the required maximum size by $M_k(m, n)$. If $m < \frac{n(n+1)}{2}$, then trivially $M_k(m, n)...

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 21

IMO 1990 SL 21 Problem (ROM 1′) Let n be a composite natural number and p a proper divisor of n. Find the binary representation of the smallest natural number N such that (1+2p+2n−p)N−1 2n is an integer. Solution We must solve the congruence (1 + 2p + 2n−p)N \equiv1 (mod 2n). Since (1 + 2p + 2n−p) and 2n are coprime, there clearly exists a unique N satisfying this...

Practice › Mathematics › IMO Shortlist › IMO 1968 Shortlist ›

IMO 1968 SL 1

IMO 1968 SL 1 Origin: SWE Problem Two ships sail on the sea with constant speeds and fixed directions. It is known that at $9{:}00$ the distance between them was $20$ miles; at $9{:}35$, $15$ miles; and at $9{:}55$, $13$ miles. At what moment were the ships the smallest distance from each other, and what was that distance? Solution Since the ships are sailing with constant speeds and directions, the...

Practice › Mathematics › IMO Shortlist › IMO 1986 Shortlist ›

IMO 1986 SL 14

IMO 1986 SL 14 Origin: IRE Problem The circle inscribed in a triangle ABC touches the sides BC, CA, AB in D, E, F, respectively, and X, Y, Z are the midpoints of EF, FD, DE, respectively. Prove that the centers of the inscribed circle and of the circles around XY Z and ABC are collinear. Solution We shall use the following simple fact. Lemma. If <k is the image...

Practice › Mathematics › IMO Shortlist › IMO 1987 Shortlist ›

IMO 1987 SL 7

IMO 1987 SL 7 Origin: NET Problem Given ﬁve real numbers u0, u1, u2, u3, u4, prove that it is always possible to ﬁnd ﬁve real numbers v0, v1, v2, v3, v4 that satisfy the following conditions: (i) ui −vi \inN. (ii) 0\leqi<j\leq4(vi −vj)2 < 4. Solution For all real numbers v the following inequality holds: 0\leqi<j\leq4 (vi −vj)2 \leq5 i=0 (vi −v)2. (1) Indeed, 0\leqi<j\leq4...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 20

IMO 1989 SL 20 Origin: NET Problem Given a set S in the plane containing n points and satis- fying the conditions: (i) no three points of S are collinear, (ii) for every point P of S there exist at least k points in S that have the same distance to P, prove that the following inequality holds: k < 1 2 + \sqrt 2n. Solution Suppose k \geq1/2 +...

Practice › Mathematics › IMO Shortlist › IMO 1993 Shortlist ›

IMO 1993 SL 7

IMO 1993 SL 7 Origin: GEO Problem Let a, b, c be given integers a > 0, ac −b2 = P = P1 \cdot \cdot \cdot Pm where P1, . . . , Pm are (distinct) prime numbers. Let M(n) denote the num- ber of pairs of integers (x, y) for which ax2 + 2bxy + cy2 = n. Prove that M(n) is ﬁnite and M(n) = M(P k \cdot...

Practice › Mathematics › IMO Shortlist › IMO 1985 Shortlist ›

IMO 1985 SL 5

IMO 1985 SL 5 Origin: ROM Problem Let D be the interior of the circle C and let A \inC. Show that the function f : D \toR, f(M) = |MA| |MM′|, where M ′ = (AM \capC, is strictly convex; i.e., f(P) < f(M1)+f(M2) , \forallM1, M2 \inD, M1 ̸= M2, where P is the midpoint of the segment M1M2. Solution Let w.l.o.g. circle C have unit radius. For...

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL G6

IMO 2001 SL G6 Origin: IND | Category: Geometry Problem Let ABC be a triangle and P an exterior point in the plane of the triangle. Suppose AP, BP, CP meet the sides BC, CA, AB (or ex- tensions thereof) in D, E, F, respectively. Suppose further that the areas of triangles PBD, PCE, PAF are all equal. Prove that each of these areas is equal to the area of...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL C6

IMO 1996 SL C6 Origin: CAN | Category: Combinatorics Problem A finite number of beans are placed on an infinite row of squares. A sequence of moves is performed as follows: at each stage a square containing more than one bean is chosen. Two beans are taken from this square; one of them is placed on the square immediately to the left, and the other is placed on the square...

Practice › Mathematics › IMO Shortlist › IMO 1975 Shortlist ›

IMO 1975 SL 10

IMO 1975 SL 10 Origin: GBR Problem The function f(x, y) is a homogeneous polynomial of the nth degree in x and y. If f(1, 0) = 1 and for all a, b, c, f(a + b, c) + f(b + c, a) + f(c + a, b) = 0, prove that f(x, y) = (x −2y)(x + y)n−1. Solution We shall prove that for all x, y with x+y...

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL A1

IMO 2002 SL A1 Origin: CZE | Category: Algebra Problem Find all functions f from the reals to the reals such that f(f(x) + y) = 2x + f(f(y) −x) for all real x, y. Solution We observe ﬁrst that f is surjective. Indeed, setting y = −f(x) gives f(f(−f(x)) −x) = f(0) −2x, where the right-hand expression can take any real value. In particular, there exists x0 for which...

Practice › Mathematics › IMO Shortlist › IMO 1994 Shortlist ›

IMO 1994 SL C5

IMO 1994 SL C5 Origin: SWE | Category: Combinatorics Problem At a round table are 1994 girls, playing a game with a deck of n cards. Initially, one girl holds all the cards. In each turn, if at least one girl holds at least two cards, one of these girls must pass a card to each of her two neighbors. The game ends when and only when each girl is...

Practice › Mathematics › IMO Shortlist › IMO 1968 Shortlist ›

IMO 1968 SL 5

IMO 1968 SL 5 Origin: BUL Problem Let $h_n$ be the apothem (distance from the center to one of the sides) of a regular $n$-gon ($n \geq 3$) inscribed in a circle of radius $r$. Prove the inequality $(n + 1)h_{n+1} - n h_n > r.$ Also prove that if $r$ on the right side is replaced with a greater number, the inequality will not remain true for all $n...

Practice › Mathematics › IMO Shortlist › IMO 1985 Shortlist ›

IMO 1985 SL 12

IMO 1985 SL 12 Problem 3b.(GBR 4) A sequence of polynomials Pm(x, y, z), m = 0, 1, 2, . . ., in x, y, and z is deﬁned by P0(x, y, z) = 1 and by Pm(x, y, z) = (x + z)(y + z)Pm−1(x, y, z + 1) −z2Pm−1(x, y, z) for m > 0. Prove that each Pm(x, y, z) is symmetric, in other words, is unaltered...

Practice › Mathematics › IMO Shortlist › IMO 1993 Shortlist ›

IMO 1993 SL 2

IMO 1993 SL 2 Origin: CAN Problem Let triangle ABC be such that its circumradius R is equal to Solution It is well known that r \leq1 2R. Therefore 1 3(1 + r)2 \leq1 1 + 1 2 = 3 4. It remains only to show that p \leq

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 24

IMO 1990 SL 24 Origin: THA Problem Let a, b, c, d be nonnegative real numbers such that ab + bc + cd + da = 1. Show that a3 b + c + d + b3 a + c + d + c3 a + b + d + d3 a + b + c \geq1 3. Solution Let us denote A = b + c + d, B...

Practice › Mathematics › IMO Shortlist › IMO 1994 Shortlist ›

IMO 1994 SL A3

IMO 1994 SL A3 Origin: GBR | Category: Algebra Problem Let S be the set of real numbers greater than −1. Find all functions f : S \toS such that f(x + f(y) + xf(y)) = y + f(x) + yf(x) for all x and y in S, and f(x)/x is strictly increasing for −1 < x < 0 and for 0 < x. Solution The last condition implies that...

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL A5

IMO 2001 SL A5 Origin: BUL | Category: Algebra Problem Find all positive integers a1, a2, . . . , an such that 100 = a0 a1 a1 a2 \cdot \cdot \cdot + an−1 an , where a0 = 1 and (ak+1 −1)ak−1 \geqa2 k(ak −1) for k = 1, 2, . . ., n −1. Solution Let a1, a2, . . . , an satisfy the conditions of the...

Practice › Mathematics › IMO Shortlist › IMO 1990 Shortlist ›

IMO 1990 SL 2

IMO 1990 SL 2 Origin: CAN Problem Given n countries with three representatives each, m commit- tees A(1), A(2), . . . A(m) are called a cycle if (i) each committee has n members, one from each country; (ii) no two committees have the same membership; (iii) for i = 1, 2, . . . , m, committee A(i) and committee A(i + 1) have no member in common, where...

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL G5

IMO 2002 SL G5 Origin: AUS | Category: Geometry Problem For any set S of ﬁve points in the plane, no three of which are collinear, let M(S) and m(S) denote the greatest and smallest areas, respectively, of triangles determined by three points from S. What is the minimum possible value of M(S)/m(S)? Solution When S is the set of vertices of a regular pentagon, then it is easily veriﬁed...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 26

IMO 1991 SL 26 Origin: CZS Problem Let n \geq2 be a natural number and let the real numbers p, a1, a2, . . . , an, b1, b2, . . . , bn satisfy 1/2 \leqp \leq1, 0 \leqai, 0 \leqbi \leqp, i = 1, . . . , n, and n i=1 ai = n i=1 bi = 1. Prove the inequality n i=1 bi n j=1...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 6

IMO 1988 SL 6 Origin: CZS Problem In a given tetrahedron ABCD let K and L be the centers of edges AB and CD respectively. Prove that every plane that contains the line KL divides the tetrahedron into two parts of equal volume. Solution Let \omega be the plane through AB, parallel to CD. Deﬁne the point trans- formation f : X "\toX′ in space as follows. If X \inKL,...

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL C6

IMO 2003 SL C6 Origin: SAF | Category: Combinatorics Problem Let f(k) be the number of integers n that satisfy the following conditions: (i) 0 \leqn < 10k, so n has exactly k digits (in decimal notation), with leading zeros allowed; (ii) the digits of n can be permuted in such a way that they yield an integer divisible by 11. Prove that f(2m) = 10f(2m −1) for every positive...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 31

IMO 1989 SL 31 Origin: SWE Problem Let a1 \geqa2 \geqa3 be given positive integers and let N(a1, a2, a3) be the number of solutions (x1, x2, x3) of the equation a1 x1 a2 x2 a3 x3 = 1, where x1, x2, and x3 are positive integers. Show that N(a1, a2, a3) \leq6a1a2(3 + ln(2a1)). Solution Let us denote by Npqr the number of solutions for which ap/xp \geqaq/xq \geq...

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL N4

IMO 2003 SL N4 Origin: ROM | Category: Number Theory Problem Let b be an integer greater than 5. For each positive integer n, consider the number xn = 11 . . .1 n−1 22 . . .2 n 5, written in base b. Prove that the following condition holds if and only if b = 10: There exists a positive integer M such...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL A2

IMO 2004 SL A2 Origin: ROM | Category: Algebra Problem An inﬁnite sequence a0, a1, a2, . . . of real numbers satisﬁes the condition an = |an+1 −an+2| for every n \geq0 with a0 and a1 positive and distinct. Can this sequence be bounded? Solution We claim that the sequence {an} must be unbounded. The condition of the sequence is equivalent to an > 0 and an+1 = an+an−1...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL G2

IMO 1996 SL G2 Origin: CAN | Category: Geometry Problem Let $P$ be a point inside $\triangle ABC$ such that $\angle APB - \angle C = \angle APC - \angle B.$ Let $D, E$ be the incenters of $\triangle APB$ and $\triangle APC$, respectively. Show that $AP$, $BD$, and $CE$ meet in a point. Solution Let $X, Y, Z$ respectively be the feet of the perpendiculars from $P$ to $BC$,...

Practice › Mathematics › IMO Shortlist › IMO 1993 Shortlist ›

IMO 1993 SL 1

IMO 1993 SL 1 Origin: BRA Problem Show that there exists a ﬁnite set A \subsetR2 such that for every X \inA there are points Y1, Y2, . . . , Y1993 in A such that the distance between X and Yi is equal to 1, for every i. Solution First we notice that for a rational point O (i.e., with rational coordinates), there exist 1993 rational points in each...

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL C2

IMO 2002 SL C2 Origin: ARM | Category: Combinatorics Problem For n an odd positive integer, the unit squares of an n \times n chessboard are colored alternately black and white, with the four corners colored black. A tromino is an L-shape formed by three connected unit squares. For which values of n is it possible to cover all the black squares with nonoverlapping trominos? When it is possible, what...

Practice › Mathematics › IMO Shortlist › IMO 1982 Shortlist ›

IMO 1982 SL 17

IMO 1982 SL 17 Problem C5 (USS 5) The right triangles ABC and AB1C1 are similar and have opposite orientation. The right angles are at C and C1, and we also have ∡CAB = ∡C1AB1. Let M be the point of intersection of the lines BC1 and B1C. Prove that if the lines AM and CC1 exist, they are perpendic- ular. Solution Let A be the origin of the Cartesian...

Practice › Mathematics › IMO Shortlist › IMO 1973 Shortlist ›

IMO 1973 SL 9

IMO 1973 SL 9 Origin: ROM Problem Let Ox, Oy, Oz be three rays, and G a point inside the trihe- dron Oxyz. Consider all planes passing through G and cutting Ox, Oy, Oz at points A, B, C, respectively. How is the plane to be placed in order to yield a tetrahedron OABC with minimal perimeter? Solution Let a, b, c be vectors going along Ox, Oy, Oz, respectively,...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 2

IMO 1988 SL 2 Origin: BUL Problem Let n be a positive integer. Find the number of odd coeﬃcients of the polynomial un(x) = (x2 + x + 1)n. Solution For polynomials f(x), g(x) with integer coeﬃcients, we use the notation f(x) ∼g(x) if all the coeﬃcients of f −g are even. Let n = 2s. It is immediately shown by induction that (x2 + x + 1)2s ∼x2s+1 +...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 8

IMO 1989 SL 8 Origin: FRA Problem Let R be a rectangle that is the union of a ﬁnite number of rectangles Ri, 1 \leqi \leqn, satisfying the following conditions: (i) The sides of every rectangle Ri are parallel to the sides of R. (ii) The interiors of any two diﬀerent Ri are disjoint. (iii) Every Ri has at least one side of integral length. Prove that R has at...

Practice › Mathematics › IMO Shortlist › IMO 1983 Shortlist ›

IMO 1983 SL 24

IMO 1983 SL 24 Origin: USS Problem Let dn be the last nonzero digit of the decimal representation of n!. Prove that dn is aperiodic; that is, there do not exist T and n0 such that for all n \geqn0, dn+T = dn. Solution See the solution of (SL91-15).

Practice › Mathematics › IMO Shortlist › IMO 1982 Shortlist ›

IMO 1982 SL 14

IMO 1982 SL 14 Problem C2 (AUS 4) Let ABCD be a convex plane quadrilateral and let A1 denote the circumcenter of \triangleBCD. Deﬁne B1, C1, D1 in a corresponding way. (a) Prove that either all of A1, B1, C1, D1 coincide in one point, or they are all distinct. Assuming the latter case, show that A1, C1 are on opposite sides of the line B1D1, and similarly, B1, D1...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 27

IMO 1989 SL 27 Origin: ROM Problem Let m be a positive odd integer, m \geq2. Find the smallest positive integer n such that 21989 divides mn −1. Solution Let n be the required exponent, and suppose n = 2kq, where q is an odd integer. Then we have mn −1 = (m2k −1)[(m2k(q−1) + \cdot \cdot \cdot + m2k + 1] = (m2k −1)A, where A is odd. Therefore...

Practice › Mathematics › IMO Shortlist › IMO 1975 Shortlist ›

IMO 1975 SL 4

IMO 1975 SL 4 Origin: SWE Problem Let a1, a2, . . . , an, . . . be a sequence of real numbers such that 0 \leqan \leq1 and an −2an+1 + an+2 \geq0 for n = 1, 2, 3, . . .. Prove that 0 \leq(n + 1)(an −an+1) \leq2 for n = 1, 2, 3, . . .. Solution Put ∆an = an −an+1. By the imposed...

Practice › Mathematics › IMO Shortlist › IMO 1972 Shortlist ›

IMO 1972 SL 6

IMO 1972 SL 6 Origin: GDR Problem Show that for any n ̸\equiv0 (mod 10) there exists a multiple of n not containing the digit 0 in its decimal expansion. Solution Let n = 2\alpha5\betam, where \alpha = 0 or \beta = 0. These two cases are analogous, and we treat only \alpha = 0, n = 5\betam. The case m = 1 is settled by the following lemma. Lemma....

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL C2

IMO 2001 SL C2 Origin: CAN | Category: Combinatorics Problem Let n be an odd integer greater than 1 and let c1, c2, . . . , cn be integers. For each permutation a = (a1, a2, . . . , an) of {1, 2, . . ., n}, deﬁne S(a) = n i=1 ciai. Prove that there exist permutations a ̸= b of {1, 2, . . ., n}...

Practice › Mathematics › IMO Shortlist › IMO 1994 Shortlist ›

IMO 1994 SL N3

IMO 1994 SL N3 Origin: FIN | Category: Number Theory Problem Find a set A of positive integers such that for any inﬁnite set P of prime numbers, there exist positive integers m \inA and n ̸\inA, both the product of the same number of distinct elements of P. Solution Let A be the set of all numbers of the form p1p2 . . . pp1, where p1 < p2...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 23

IMO 1988 SL 23 Origin: SIN Problem Let Q be the center of the inscribed circle of a triangle ABC. Prove that for any point P, a(PA)2+b(PB)2+c(PC)2 = a(QA)2+b(QB)2+c(QC)2+(a+b+c)(QP)2, where a = BC, b = CA, and c = AB. Solution Denote by R the intersection point of lines AQ and BC. We know that BR : RC = c : b and AQ : QR = (b + c)...

Practice › Mathematics › IMO Shortlist › IMO 1977 Shortlist ›

IMO 1977 SL 7

IMO 1977 SL 7 Origin: GBR Problem Let a, b, A, B be given constant real numbers and f(x) = 1 −a cos x −b sin x −A cos 2x −B sin 2x. Prove that if f(x) \geq0 for all real x, then a2 + b2 \leq2 and A2 + B2 \leq1.

Practice › Mathematics › IMO Shortlist › IMO 1993 Shortlist ›

IMO 1993 SL 13

IMO 1993 SL 13 Origin: IRE Problem Let S be the set of all pairs (m, n) of relatively prime positive integers m, n with n even and m < n. For s = (m, n) \inS write n = 2kn0, where k, n0 are positive integers with n0 odd and deﬁne f(s) = (n0, m + n −n0). Prove that f is a function from S to S and...

Practice › Mathematics › IMO Shortlist › IMO 1986 Shortlist ›

IMO 1986 SL 2

IMO 1986 SL 2 Origin: SWE Problem Let f(x) = xn where n is a ﬁxed positive integer and x = 1, 2, . . . . Is the decimal expansion a = 0.f(1)f(2)f(3) . . . rational for any value of n? The decimal expansion of a is deﬁned as follows: If f(x) = d1(x)d2(x) . . . . . . dr(x)(x) is the decimal expansion of f(x), then...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL N2

IMO 2004 SL N2 Origin: RUS | Category: Number Theory Problem The function \psi from the set N of positive integers into itself is deﬁned by the equality \psi(n) = n k=1 (k, n), n \inN, where (k, n) denotes the greatest common divisor of k and n. (a) Prove that \psi(mn) = \psi(m)\psi(n) for every two relatively prime m, n \in N. (b) Prove that for each a...

Practice › Mathematics › IMO Shortlist › IMO 1984 Shortlist ›

IMO 1984 SL 11

IMO 1984 SL 11 Origin: CAN Problem Let n be a natural number and a1, a2, . . . , a2n mutually distinct integers. Find all integers x satisfying (x −a1) \cdot (x −a2) \cdot \cdot \cdot (x −a2n) = (−1)n(n!)2. Solution Suppose that an integer x satisﬁes the equation. Then the numbers x − a1, x −a2, . . . , x −a2n are 2n distinct integers whose product...

Practice › Mathematics › IMO Shortlist › IMO 1977 Shortlist ›

IMO 1977 SL 4

IMO 1977 SL 4 Origin: FRG Problem Describe all closed bounded ﬁgures \Phi in the plane any two points of which are connectable by a semicircle lying in \Phi.

Practice › Mathematics › IMO Shortlist › IMO 1984 Shortlist ›

IMO 1984 SL 2

IMO 1984 SL 2 Origin: CAN Problem Prove: (a) There are inﬁnitely many triples of positive integers m, n, p such that 4mn −m −n = p2 −1. (b) There are no positive integers m, n, p such that 4mn −m −n = p2. Solution (a) For m = t(t −1)/2 and n = t(t + 1)/2 we have 4mn −m −n = (t2 −1)2 −1. (b) Suppose that 4mn...

Practice › Mathematics › IMO Shortlist › IMO 1987 Shortlist ›

IMO 1987 SL 9

IMO 1987 SL 9 Origin: HUN Problem Does there exist a set M in usual Euclidean space such that for every plane \lambda the intersection M \cap\lambda is ﬁnite and nonempty? Solution The answer is yes. Consider the curve C = {(x, y, z) | x = t, y = t3, z = t5, t \inR}. Any plane deﬁned by an equation of the form ax+by+cz+d = 0 intersects the...

Practice › Mathematics › IMO Shortlist › IMO 1975 Shortlist ›

IMO 1975 SL 12

IMO 1975 SL 12 Origin: GRE Problem Consider on the ﬁrst quadrant of the trigonometric circle the arcs AM1 = x1, AM2 = x2, AM3 = x3, . . . , AM\nu = x\nu, such that x1 < x2 < x3 < \cdot \cdot \cdot < x\nu. Prove that \nu−1 i=0 sin 2xi − \nu−1 i=0 sin(xi −xi+1) < \pi 2 + \nu−1 i=0 sin(xi + xi+1)....

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 30

IMO 1988 SL 30 Origin: USS Problem A point M is chosen on the side AC of the triangle ABC in such a way that the radii of the circles inscribed in the triangles ABM and BMC are equal. Prove that BM 2 = ∆cot B 2 , where ∆is the area of the triangle ABC. Solution Let ∆1, s1, r′ denote the area, semiperimeter, and inradius of triangle ABM,...

Practice › Mathematics › IMO Shortlist › IMO 1968 Shortlist ›

IMO 1968 SL 10

IMO 1968 SL 10 Origin: ROM Problem Consider two segments of length $a, b$ ($a > b$) and a segment of length $c = \sqrt{ab}$. (a) For what values of $a/b$ can these segments be sides of a triangle? (b) For what values of $a/b$ is this triangle right-angled, obtuse-angled, or acute-angled?

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL C3

IMO 1999 SL C3 Origin: GBR | Category: Combinatorics Problem A biologist watches a chameleon. The chameleon catches ﬂies and rests after each catch. The biologist notices that: (i) the ﬁrst ﬂy is caught after a resting period of one minute; (ii) the resting period before catching the 2mth ﬂy is the same as the resting period before catching the mth ﬂy and one minute shorter than the resting period...

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL G7

IMO 1999 SL G7 Origin: ARM | Category: Geometry Problem The point M inside the convex quadrilateral ABCD is such that MA = MC, \angleAMB = \angleMAD+\angleMCD, \angleCMD = \angleMCB+\angleMAB. Prove that AB \cdot CM = BC \cdot MD and BM \cdot AD = MA \cdot CD. Solution Let us construct a convex quadrilateral PQRS and an interior point T such that \trianglePTQ ∼= \triangleAMB, \triangleQTR ∼\triangleAMD, and \trianglePTS ∼\triangleCMD....

Practice › Mathematics › IMO Shortlist › IMO 1982 Shortlist ›

IMO 1982 SL 10

IMO 1982 SL 10 Problem B4 (BRA 1) A box contains p white balls and q black balls. Beside the box there is a pile of black balls. Two balls are taken out of the box. If they have the same color, a black ball from the pile is put into the box. If they have diﬀerent colors, the white ball is put back into the box. This procedure is...

Practice › Mathematics › IMO Shortlist › IMO 1992 Shortlist ›

IMO 1992 SL 2

IMO 1992 SL 2 Origin: CHN Problem Let R+ be the set of all nonnegative real numbers. Given two positive real numbers a and b, suppose that a mapping f : R+ \toR+ satisﬁes the functional equation f(f(x)) + af(x) = b(a + b)x. Prove that there exists a unique solution of this equation. Solution Let us deﬁne xn inductively as xn = f(xn−1), where x0 \geq0 is a ﬁxed...

Practice › Mathematics › IMO Shortlist › IMO 1989 Shortlist ›

IMO 1989 SL 25

IMO 1989 SL 25 Origin: KOR Problem Let a, b be integers that are not perfect squares. Prove that if x2 −ay2 −bz2 + abw2 = 0 has a nontrivial solution in integers, then so does x2 −ay2 −bz2 = 0. Solution We may assume w.l.o.g. that a > 0 (because a, b < 0 is impossible, and a, b ̸= 0 from the condition of the problem). Let (x0,...

Practice › Mathematics › IMO Shortlist › IMO 1977 Shortlist ›

IMO 1977 SL 2

IMO 1977 SL 2 Origin: CZS Problem A lattice point in the plane is a point both of whose coordinates are integers. Each lattice point has four neighboring points: upper, lower, left, and right. Let k be a circle with radius r \geq2, that does not pass through any lattice point. An interior boundary point is a lattice point lying inside the circle k that has a neighboring point lying...

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL G8

IMO 2002 SL G8 Origin: ARM | Category: Geometry Problem Let S1 and S2 be circles meeting at the points A and B. A line through A meets S1 at C and S2 at D. Points M, N, K lie on the line segments CD, BC, BD respectively, with MN parallel to BD and MK parallel to BC. Let E and F be points on those arcs BC of S1...

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL C4

IMO 1999 SL C4 Origin: GBR | Category: Combinatorics Problem Let A be a set of N residues (mod N 2). Prove that there exists a set B of N residues (mod N 2) such that the set A + B = {a + b | a \inA, b \inB} contains at least half of all residues (mod N 2). Solution Let S = {0, 1, . . ., N...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL G5

IMO 1996 SL G5 Origin: ARM | Category: Geometry Problem Let $ABCDEF$ be a convex hexagon such that $AB \parallel DE$, $BC \parallel EF$, and $CD \parallel AF$. Let $R_A$, $R_C$, $R_E$ be the circumradii of triangles $FAB$, $BCD$, $DEF$ respectively, and let $P$ denote the perimeter of the hexagon. Prove that $$R_A + R_C + R_E \geq \frac{P}{2}.$$ Solution Let $a, b, c, d, e, f$ denote the lengths...

Practice › Mathematics › IMO Shortlist › IMO 1999 Shortlist ›

IMO 1999 SL A2

IMO 1999 SL A2 Origin: RUS | Category: Algebra Problem The numbers from 1 to n2 are randomly arranged in the cells of a n \times n square (n \geq2). For any pair of numbers situated in the same row or in the same column, the ratio of the greater number to the smaller one is calculated. Let us call the characteristic of the arrangement the smallest of these n2(n−1)...

Practice › Mathematics › IMO Shortlist › IMO 2000 Shortlist ›

IMO 2000 SL A2

IMO 2000 SL A2 Origin: GBR | Category: Algebra Problem Let a, b, c be positive integers satisfying the conditions b > 2a and c > 2b. Show that there exists a real number t with the property that all the three numbers ta, tb, tc have their fractional parts lying in the interval (1/3, 2/3]. Solution We note that {ta} lies in 1 3, 2 if and only...

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