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Practice › Mathematics › IMO Shortlist › IMO 1968 Shortlist ›

IMO 1968 SL 21

IMO 1968 SL 21 Origin: CZS Problem Let a0, a1, . . . , ak (k \geq1) be positive integers. Find all positive integers y such that a0 | y; (a0 + a1) | (y + a1); . . . ; (a0 + an) | (y + an). Solution The given conditions are equivalent to y −a0 being divisible by a0, a0 + a1, a0 +a2, . . . ,...

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 1996 Shortlist ›

IMO 1996 SL C4

IMO 1996 SL C4 Origin: FIN | Category: Combinatorics Problem Determine whether or not there exist two disjoint infinite sets $A$ and $B$ of points in the plane satisfying the following conditions: No three points in $A \cup B$ are collinear, and the distance between any two points in $A \cup B$ is at least $1$. There is a point of $A$ in any triangle whose vertices are in $B$,...

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 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 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 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 1978 Shortlist ›

IMO 1978 SL 16

IMO 1978 SL 16 Origin: YUG Problem Determine all the triples (a, b, c) of positive real numbers such that the system ax + by −cz = 0, a 1 −x2 + b 1 −y2 −c 1 −z2 = 0, is compatible in the set of real numbers, and then ﬁnd all its real solutions. Solution Clearly |x| \leq1. As x runs over [−1, 1], the vector u = (ax,...

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 1985 Shortlist ›

IMO 1985 SL 15

IMO 1985 SL 15 Problem 5a.(FRA 3) Let K and K′ be two squares in the same plane, their sides of equal length. Is it possible to decompose K into a ﬁnite number of tri- angles T1, T2, . . . , Tp with mutually disjoint interiors and ﬁnd translations t1, t2, . . . , tp such that K′ = p2 i=1 ti(Ti)? Solution There is no loss of...

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 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 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 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 1988 Shortlist ›

IMO 1988 SL 1

IMO 1988 SL 1 Origin: BUL Problem An integer sequence is deﬁned by an = 2an−1 + an−2 (n > 1), a0 = 0, a1 = 1. Prove that 2k divides an if and only if 2k divides n. Solution Assume that p and q are real and b0, b1, b2, . . . is a sequence such that bn = pbn−1 + qbn−2 for all n > 1. From...

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 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 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 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 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 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 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 1996 Shortlist ›

IMO 1996 SL N5

IMO 1996 SL N5 Origin: ROM | Category: Number Theory Problem Let $N_0$ denote the set of nonnegative integers. Find a bijective function $f$ from $N_0$ into $N_0$ such that for all $m, n \in N_0$, $$f(3mn + m + n) = 4f(m)f(n) + f(m) + f(n).$$

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 2001 Shortlist ›

IMO 2001 SL N3

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 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 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 1989 Shortlist ›

IMO 1989 SL 28

IMO 1989 SL 28 Origin: ROM Problem Consider in a plane \Pi the points O, A1, A2, A3, A4 such that \sigma(OAiAj) \geq1 for all i, j = 1, 2, 3, 4, i ̸= j. Prove that there is at least one pair i0, j0 \in{1, 2, 3, 4} such that \sigma(OAi0Aj0) \geq \sqrt 2. (We have denoted by \sigma(OAiAj) the area of triangle OAiAj.) Solution Assume w.l.o.g. that the...

Practice › Mathematics › IMO Shortlist › IMO 1992 Shortlist ›

IMO 1992 SL 9

IMO 1992 SL 9 Origin: IRN Problem Let f(x) be a polynomial with rational coeﬃcients and \alpha be a real number such that \alpha3 −\alpha = f(\alpha)3 −f(\alpha) = 331992. Prove that for each n \geq1, (f (n)(\alpha))3 −f (n)(\alpha) = 331992, where f (n)(x) = f(f(. . . f(x))), and n is a positive integer. Solution Since the equation x3 −x −c = 0 has only one real root...

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 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 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 2004 Shortlist ›

IMO 2004 SL G6

IMO 2004 SL G6 Origin: GBR | Category: Geometry Problem Let P be a convex polygon. Prove that there is a convex hexagon that is contained in P and that occupies at least 75 percent of the area of P. Solution Let ABC be the triangle of maximum area S contained in P (it exists because of compactness of P). Draw parallels to BC, CA, AB through A, B, C,...

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 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 1991 Shortlist ›

IMO 1991 SL 25

IMO 1991 SL 25 Origin: USA Problem Suppose that n \geq2 and x1, x2, . . . , xn are real numbers between 0 and 1 (inclusive). Prove that for some index i between 1 and n −1 the inequality xi(1 −xi+1) \geq1 4x1(1 −xn) holds. Solution Since replacing x1 by 1 can only reduce the set of indices i for which the desired inequality holds, we may assume x1...

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 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 1997 Shortlist ›

IMO 1997 SL 19

IMO 1997 SL 19 Origin: IRE Problem Let a1 \geq\cdot \cdot \cdot \geqan \geqan+1 = 0 be a sequence of real numbers. Prove that n k=1 ak \leq n k=1 \sqrt k(\sqrtak −\sqrtak+1). Solution Using that an+1 = 0 we can transform the desired inequality into a1 + a2 + \cdot \cdot \cdot + an+1 \leq \sqrt 1\sqrta1 + ( \sqrt 2 − \sqrt 1)\sqrta2 + \cdot \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 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 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 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 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 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 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 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 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 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 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 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 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 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 2004 Shortlist ›

IMO 2004 SL N5

IMO 2004 SL N5 Origin: IRN | Category: Number Theory Problem We call a positive integer alternate if its decimal digits are alternately odd and even. Find all positive integers n such that n has an alternate multiple. Solution If n is divisible by 20, then every multiple of n has two last digits even and hence it is not alternate. We shall show that any other n has an...

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 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 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 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 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 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 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 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 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 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 12

IMO 1997 SL 12 Origin: ITA Problem Let p be a prime number and let f(x) be a polynomial of degree d with integer coeﬃcients such that: (i) f(0) = 0, f(1) = 1; (ii) for every positive integer n, the remainder of the division of f(n) by p is either 0 or 1. Prove that d \geqp −1. Solution Lemma. For any polynomial P of degree at most n,...

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 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 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 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 1988 Shortlist ›

IMO 1988 SL 15

IMO 1988 SL 15 Origin: ICE Problem Let ABC be an acute-angled triangle. Three lines LA, LB, and LC are constructed through the vertices A, B, and C respectively according to the following prescription: Let H be the foot of the altitude drawn from the vertex A to the side BC; let SA be the circle with diameter AH; let SA meet the sides AB and AC at M and...

Practice › Mathematics › IMO Shortlist › IMO 1984 Shortlist ›

IMO 1984 SL 3

IMO 1984 SL 3 Origin: USS Problem Find all positive integers n such that n = d2 6 + d2 7 −1, where 1 = d1 < d2 < \cdot \cdot \cdot < dk = n are all positive divisors of the number n. Solution From the equality n = d2 6 + d2 7 −1 we see that d6 and d7 are relatively prime and d7 | d2 6...

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 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 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 1993 Shortlist ›

IMO 1993 SL 19

IMO 1993 SL 19 Origin: ROM Problem Let a, b, n be positive integers, b > 1 and bn −1 | a. Show that the representation of the number a in the base b contains at least n digits diﬀerent from zero. Solution Let s be the minimum number of nonzero digits that can appear in the b- adic representation of any number divisible by bn −1. Among all numbers...

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 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 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 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 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 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 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 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 1995 Shortlist ›

IMO 1995 SL G7

IMO 1995 SL G7 Origin: LAT | Category: Geometry Problem O is a point inside a convex quadrilateral ABCD of area S. K, L, M, and N are interior points of the sides AB, BC, CD, and DA respectively. If OKBL and OMDN are parallelograms, prove that \sqrt S \geq\sqrtS1 + \sqrtS2, where S1 and S2 are the areas of ONAK and OLCM respectively. Solution If O lies on AC,...

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 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 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 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 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 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 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 1998 Shortlist ›

IMO 1998 SL 17

IMO 1998 SL 17 Origin: GBR Problem A sequence of integers a1, a2, a3, . . . is deﬁned as follows: a1 = 1, and for n \geq1, an+1 is the smallest integer greater than an such that ai + aj ̸= 3ak for any i, j, k in {1, 2, . . ., n + 1}, not necessarily distinct. Determine a1998. Solution Initially, we determine that the ﬁrst few...

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 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 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 1988 Shortlist ›

IMO 1988 SL 21

IMO 1988 SL 21 Origin: POL Problem Forty-nine students solve a set of three problems. The score for each problem is a whole number of points from 0 to 7. Prove that there exist two students A and B such that for each problem, A will score at least as many points as B. Solution Let X be the set of all ordered triples a = (a1, a2, a3) for...

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 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 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 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 1970 Shortlist ›

IMO 1970 SL 3

IMO 1970 SL 3 Origin: BUL Problem In the tetrahedron SABC the angle BSC is a right angle, and the projection of the vertex S to the plane ABC is the intersection of the altitudes of the triangle ABC. Let z be the radius of the inscribed circle of the triangle ABC. Prove that SA2 + SB2 + SC2 \geq18z2. Solution We shall use the following lemma Lemma. If an...

Practice › Mathematics › IMO Shortlist › IMO 1995 Shortlist ›

IMO 1995 SL G2

IMO 1995 SL G2 Origin: GER | Category: Geometry Problem Let A, B, and C be noncollinear points. Prove that there is a unique point X in the plane of ABC such that XA2 + XB2 + AB2 = XB2 + XC2 + BC2 = XC2 + XA2 + CA2. Solution Let A′, B′, C′ be the points symmetric to A, B, C with respect to the midpoints of BC,...

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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 1991 Shortlist ›

IMO 1991 SL 3

IMO 1991 SL 3 Origin: PRK Problem Let S be any point on the circumscribed circle of \trianglePQR. Then the feet of the perpendiculars from S to the three sides of the triangle lie on the same straight line. Denote this line by l(S, PQR). Suppose that the hexagon ABCDEF is inscribed in a circle. Show that the four lines l(A, BDF), l(B, ACE), l(D, ABF), and l(E, ABC) intersect...

Practice › Mathematics › IMO Shortlist › IMO 1984 Shortlist ›

IMO 1984 SL 1

IMO 1984 SL 1 Origin: FRA Problem Find all solutions of the following system of n equations in n variables: x1|x1| −(x1 −a)|x1 −a| = x2|x2|, x2|x2| −(x2 −a)|x2 −a| = x3|x3|, \cdot \cdot \cdot xn|xn| −(xn −a)|xn −a| = x1|x1|, where a is a given number. Solution This is the same problem as (SL83-20).

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 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 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 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 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 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 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 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 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 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 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 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 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 1994 Shortlist ›

IMO 1994 SL N5

IMO 1994 SL N5 Origin: ROM | Category: Number Theory Problem For any positive integer k, Ak is the subset of {k+1, k+ 2, . . . , 2k} consisting of all elements whose digits in base 2 contain exactly three 1’s. Let f(k) denote the number of elements in Ak. (a) Prove that for any positive integer m, f(k) = m has at least one Solution (a) Denote by...

Practice › Mathematics › IMO Shortlist › IMO 1983 Shortlist ›

IMO 1983 SL 10

IMO 1983 SL 10 Origin: FIN Problem Let p and q be integers. Show that there exists an interval I of length 1/q and a polynomial P with integral coeﬃcients such that P(x) −p q < 1 q2 for all x \inI. Solution Choose P(x) = p q (qx −1)2n+1 + 1 , I = [1/2q, 3/2q]. Then all the coef- ﬁcients of P are integers, and P(x)...

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 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 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 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 25

IMO 1988 SL 25 Origin: GBR Problem A positive integer is called a double number if its decimal rep- resentation consists of a block of digits, not commencing with 0, followed immediately by an identical block. For instance, 360360 is a double num- ber, but 36036 is not. Show that there are inﬁnitely many double numbers that are perfect squares. Solution Observe that 1001 = 7 \cdot 143, i.e., 103...

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 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 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 2004 Shortlist ›

IMO 2004 SL C1

IMO 2004 SL C1 Origin: PUR | Category: Combinatorics Problem There are 10001 students at a university. Some students join together to form several clubs (a student may belong to diﬀerent clubs). Some clubs join together to form several societies (a club may belong to diﬀerent societies). There are a total of k societies. Suppose that the following conditions hold: (i) Each pair of students are in exactly one club....

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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 1988 Shortlist ›

IMO 1988 SL 22

IMO 1988 SL 22 Origin: KOR Problem Let p be the product of two consecutive integers greater than Solution Set X = p i=1 xi and w.l.o.g. assume that X \geq0 (if (x1, . . . , xp) is a solution, then (−x1, . . . , −xp) is a solution too). Since x2 \geqx for all integers x, it follows that p i=1 x2 i \geqX. If the last...

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 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 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 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 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 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 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 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 1986 Shortlist ›

IMO 1986 SL 16

IMO 1986 SL 16 Origin: ISR Problem Let A, B be adjacent vertices of a regular n-gon in the plane and let O be its center. Now let the triangle ABO glide around the polygon in such a way that the points A and B move along the whole circumference of the polygon. Describe the ﬁgure traced by the vertex O. Solution Let Z be the center of the polygon....

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 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 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 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 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 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 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 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 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 1977 Shortlist ›

IMO 1977 SL 16

IMO 1977 SL 16 Origin: VIE Problem Let E be a set of n points in the plane (n \geq3) whose co- ordinates are integers such that any three points from E are vertices of a nondegenerate triangle whose centroid doesn’t have both coordinates integers. Determine the maximal n.

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL A6

IMO 1996 SL A6 Origin: IRE | Category: Algebra Problem Let $n$ be an even positive integer. Prove that there exists a positive integer $k$ such that $$k=f(x)(x+1)^n+g(x)(x^n+1)$$ for some polynomials $f(x)$, $g(x)$ having integer coefficients. If $k_0$ denotes the least such $k$, determine $k_0$ as a function of $n$. $A6'$ Let $n$ be an even positive integer. Prove that there exists a positive integer $k$ such that $$k=f(x)(x+1)^n+g(x)(x^n+1)$$ for...

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 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 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 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 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 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 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 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 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 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 1999 Shortlist ›

IMO 1999 SL N1

IMO 1999 SL N1 Origin: TWN | Category: Number Theory Problem Find all pairs of positive integers (x, p) such that p is a prime, x \leq2p , and xp−1 is a divisor of (p −1)x + 1. Solution Obviously (1, p) (where p is an arbitrary prime) and (2, 2) are solutions and the only solutions to the problem for x < 3 or p < 3. Let us...

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 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 2000 Shortlist ›

IMO 2000 SL N4

IMO 2000 SL N4 Origin: BRA | Category: Number Theory Problem Determine all triples of positive integers (a, m, n) such that am + 1 divides (a + 1)n. Solution Trivially all triples (a, 1, n) and (1, m, n) are solutions. Assume now that a > 1 and m > 1. If m is even, then am + 1 \equiv(−1)m + 1 \equiv2 (mod a + 1), which implies...

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 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 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 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 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 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 1997 Shortlist ›

IMO 1997 SL 2

IMO 1997 SL 2 Origin: CAN Problem Let R1, R2, . . . be the family of ﬁnite sequences of positive inte- gers deﬁned by the following rules: R1 = (1), and if Rn−1 = (x1, . . . , xs), then Rn = (1, 2, . . . , x1, 1, 2, . . ., x2, . . . , 1, 2, . . . , xs, n). For...

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 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 2003 Shortlist ›

IMO 2003 SL A6

IMO 2003 SL A6 Origin: USA | Category: Algebra Problem Let n be a positive integer and let (x1, . . . , xn), (y1, . . . , yn) be two sequences of positive real numbers. Suppose (z2, z3, . . . , z2n) is a sequence of positive real numbers such that z2 i+j \geqxiyj for all 1 \leqi, j \leqn. Let M = max{z2, . . ....

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 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 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 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 1977 Shortlist ›

IMO 1977 SL 9

IMO 1977 SL 9 Origin: HUN Problem For which positive integers n do there exist two polynomials f and g with integer coeﬃcients of n variables x1, x2, . . . , xn such that the following equality is satisﬁed: n i=1 xi f(x1, x2, . . . , xn) = g(x2 1, x2 2, . . . , x2 n)?

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 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 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 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 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 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 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 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 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 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 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 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 1998 Shortlist ›

IMO 1998 SL 22

IMO 1998 SL 22 Origin: UKR Problem A rectangular array of numbers is given. In each row and each column, the sum of all numbers is an integer. Prove that each nonintegral number x in the array can be changed into either \lceilx\rceilor \lfloorx\rfloorso that the row sums and column sums remain unchanged. (Note that \lceilx\rceilis the least integer greater than or equal to x, while \lfloorx\rflooris the greatest integer...

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 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 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 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 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 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 1987 Shortlist ›

IMO 1987 SL 8

IMO 1987 SL 8 Origin: HUN Problem (a) Let (m, k) = 1. Prove that there exist integers a1, a2, . . . , am and b1, b2, . . . , bk such that each product aibj (i = 1, 2, . . ., m; j = 1, 2, . . . , k) gives a diﬀerent residue when divided by mk. (b) Let (m, k) > 1. Prove...

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL A1

IMO 2003 SL A1 Origin: USA | Category: Algebra Problem Let aij, i = 1, 2, 3, j = 1, 2, 3, be real numbers such that aij is positive for i = j and negative for i ̸= j. Prove that there exist positive real numbers c1, c2, c3 such that the num- bers a11c1 + a12c2 + a13c3, a21c1 + a22c2 + a23c3, a31c1 + a32c2 + a33c3...

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 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 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 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 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 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 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 2004 Shortlist ›

IMO 2004 SL G3

IMO 2004 SL G3 Origin: KOR | Category: Geometry Problem Let O be the circumcenter of an acute-angled triangle ABC with \angleB < \angleC. The line AO meets the side BC at D. The circumcenters of the triangles ABD and ACD are E and F, respectively. Extend the sides BA and CA beyond A, and choose on the respective extension points G and H such that AG = AC and...

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 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 1997 Shortlist ›

IMO 1997 SL 4

IMO 1997 SL 4 Origin: IRN Problem An n \times n matrix with entries from {1, 2, . . ., 2n −1} is called a coveralls matrix if for each i the union of the ith row and the ith column contains 2n −1 distinct entries. Show that: (a) There exist no coveralls matrices for n = 1997. (b) Coveralls matrices exist for inﬁnitely many values of n. Solution (a)...

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 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 1983 Shortlist ›

IMO 1983 SL 2

IMO 1983 SL 2 Origin: BEL Problem Let n be a positive integer. Let \sigma(n) be the sum of the natural divisors d of n (including 1 and n). We say that an integer m \geq1 is superabundant (P.Erd¨os, 1944) if \forallk \in{1, 2, . . ., m −1}, \sigma(m) m \sigma(k) k . Prove that there exists an inﬁnity of superabundant numbers. Solution By deﬁnition, \sigma(n) = d|n...

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 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 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 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 2001 Shortlist ›

IMO 2001 SL A6

IMO 2001 SL A6 Origin: KOR | Category: Algebra Problem Prove that for all positive real numbers a, b, c, a \sqrt a2 + 8bc + a \sqrt b2 + 8ca + c \sqrt c2 + 8ab \geq1. Solution We shall determine a constant k > 0 such that a \sqrt a2 + 8bc \geq ak ak + bk + ck for all a, b, c > 0. (1) This...

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 1998 Shortlist ›

IMO 1998 SL 20

IMO 1998 SL 20 Origin: ARG Problem Prove that for each positive integer n, there exists a positive integer with the following properties: (i) It has exactly n digits. (ii) None of the digits is 0. (iii) It is divisible by the sum of its digits. Solution We ﬁrst consider the special case n = 3r. Then the simplest choice 10n−1 11 . . . 1 (n digits) works. This...

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 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 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 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 1997 Shortlist ›

IMO 1997 SL 16

IMO 1997 SL 16 Origin: BLR Problem In an acute-angled triangle ABC, let AD, BE be altitudes and AP, BQ internal bisectors. Denote by I and O the incenter and the cir- cumcenter of the triangle, respectively. Prove that the points D, E, and I are collinear if and only if the points P, Q, and O are collinear. Solution Let da(X), db(X), dc(X) denote the distances of a point...

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 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 1993 Shortlist ›

IMO 1993 SL 26

IMO 1993 SL 26 Origin: VIE Problem Let a, b, c, d be four nonnegative numbers satisfying a+b+c+d = Solution Set f(a, b, c, d) = abc + bcd + cda + dab −176 27 abcd = ab(c + d) + cd a + b −176 27 ab . If a + b −176 a b \leq0, by the arithmetic-geometric inequality we have f(a, b, c, d) \leqab(c...

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 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 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 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 1977 Shortlist ›

IMO 1977 SL 15

IMO 1977 SL 15 Origin: VIE Problem The length of a ﬁnite sequence is deﬁned as the number of terms of this sequence. Determine the maximal possible length of a ﬁnite sequence that satisﬁes the following condition: The sum of each seven successive terms is negative, and the sum of each eleven successive terms is positive.

Practice › Mathematics › IMO Shortlist › IMO 1981 Shortlist ›

IMO 1981 SL 13

IMO 1981 SL 13 Origin: ROM Problem Let P be a polynomial of degree n satisfying P(k) = n + 1 k −1 for k = 0, 1, . . ., n. Determine P(n + 1). Solution Lemma. For any polynomial P of degree at most n, n+1 i=0 (−1)i n + 1 i P(i) = 0. (1) Proof. We shall use induction on n. For n =...

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 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 1989 Shortlist ›

IMO 1989 SL 24

IMO 1989 SL 24 Origin: POL Problem For points A1, . . . , A5 on the sphere of radius 1, what is the maximum value that min1\leqi,j\leq5 AiAj can take? Determine all conﬁgura- tions for which this maximum is attained. (Or: determine the diameter of any set {A1, . . . , A5} for which this maximum is attained.) Solution Instead of Euclidean distance, we will use the angles...

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 1992 Shortlist ›

IMO 1992 SL 20

IMO 1992 SL 20 Origin: FRA Problem In the plane, let there be given a circle C, a line l tangent to C, and a point M on l. Find the locus of points P that have the following property: There exist two points Q and R on l such that M is the midpoint of QR and C is the incircle of PQR. Solution Denote by U the point...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL A9

IMO 1996 SL A9 Origin: POL | Category: Algebra Problem Let the sequence $a(n)$, $n = 1, 2, 3, \dots$, be generated as follows: $a(1) = 0$, and for $n > 1$, $$a(n) = a\bigl(\lfloor n/2 \rfloor\bigr) + (-1)^n \frac{n(n+1)}{2}.$$ (Here $\lfloor t \rfloor$ denotes the greatest integer $\leq t$.) (a) Determine the maximum and minimum value of $a(n)$ over $n \leq 1996$ and find all $n \leq 1996$ for...

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 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 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 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 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 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 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 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 1997 Shortlist ›

IMO 1997 SL 13

IMO 1997 SL 13 Origin: IND Problem In town A, there are n girls and n boys, and each girl knows each boy. In town B, there are n girls g1, g2, . . . , gn and 2n −1 boys b1, b2, . . ., b2n−1. The girl gi, i = 1, 2, . . ., n, knows the boys b1, b2, . . . , b2i−1, and no...

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 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 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 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 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 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 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 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 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 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 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 1997 Shortlist ›

IMO 1997 SL 14

IMO 1997 SL 14 Origin: IND Problem Let b, m, n be positive integers such that b > 1 and m ̸= n. Prove that if bm −1 and bn −1 have the same prime divisors, then b + 1 is a power of 2. Solution We use the following nonstandard notation: (1◦) for x, y \inN, x ∼y means that x and y have the same prime divisors; (2◦)...

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 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 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 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 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 1990 Shortlist ›

IMO 1990 SL 3

IMO 1990 SL 3 Origin: CZS Problem On a circle, 2n −1 (n \geq3) diﬀerent points are given. Find the minimal natural number N with the property that whenever N of the given points are colored black, there exist two black points such that the interior of one of the corresponding arcs contains exactly n of the given 2n −1 points. Solution A segment connecting two points which divides the...

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 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 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 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 2000 Shortlist ›

IMO 2000 SL G2

IMO 2000 SL G2 Origin: RUS | Category: Geometry Problem Two circles G1 and G2 intersect at M and N. Let AB be the line tangent to these circles at A and B, respectively, such that M lies closer to AB than N. Let CD be the line parallel to AB and passing through M, with C on G1 and D on G2. Lines AC and BD meet at E;...

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 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 1974 Shortlist ›

IMO 1974 SL 4

IMO 1974 SL 4 Problem I 4 (USS 4) The sum of the squares of ﬁve real numbers a1, a2, a3, a4, a5 equals 1. Prove that the least of the numbers (ai −aj)2, where i, j = 1, 2, 3, 4, 5 and i ̸= j, does not exceed 1/10. Solution Assume w.l.o.g. that a1 \leqa2 \leqa3 \leqa4 \leqa5. If m is the least value of |ai −aj|, i...

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 1968 Shortlist ›

IMO 1968 SL 19

IMO 1968 SL 19 Origin: ITA Problem We are given a fixed point on the circle of radius $1$, and going from this point along the circumference in the positive direction on curved distances $0$, $1$, $2$, $\ldots$ from it we obtain points with abscissas $n = 0, 1, 2, \ldots$ respectively. How many points among them should we take to ensure that some two of them are less than...

Practice › Mathematics › IMO Shortlist › IMO 1972 Shortlist ›

IMO 1972 SL 8

IMO 1972 SL 8 Origin: GBR Problem Let m and n be nonnegative integers. Prove that m!n!(m+ n)! divides (2m)!(2n)!. Solution Let f(m, n) = (2m)!(2n)! m!n!(m+n)!. Then it is directly shown that f(m, n) = 4f(m, n −1) −f(m + 1, n −1), and thus n may be successively reduced until one obtains f(m, n) = r crf(r, 0). Now f(r, 0) is a simple binomial coeﬃcient, and...

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 1995 Shortlist ›

IMO 1995 SL N8

IMO 1995 SL N8 Origin: GER | Category: Number Theory Problem Let p be an odd prime. Determine positive integers x and y for which x \leqy and \sqrt2p −\sqrtx −\sqrty is nonnegative and as small as possible. Solution Let u = \sqrt2p −\sqrtx −\sqrty and v = u(2\sqrt2p −u) = 2p −(\sqrt2p −u)2 = 2p −x −y −\sqrt4xy for x, y \inN, x \leqy. Obviously u \geq0 if and...

Practice › Mathematics › IMO Shortlist › IMO 2003 Shortlist ›

IMO 2003 SL G5

IMO 2003 SL G5 Origin: KOR | Category: Geometry Problem Let ABC be an isosceles triangle with AC = BC, whose incenter is I. Let P be a point on the circumcircle of the triangle AIB lying inside the triangle ABC. The lines through P parallel to CA and CB meet AB at D and E, respectively. The line through P parallel to AB meets CA and CB at F...

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 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 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 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 1984 Shortlist ›

IMO 1984 SL 16

IMO 1984 SL 16 Origin: POL Problem Let a, b, c, d be odd positive integers such that a < b < c < d, ad = bc, and a + d = 2k, b + c = 2m for some integers k and m. Prove that a = 1. Solution From the ﬁrst two conditions we can easily conclude that a + d > b + c (indeed, (d...

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 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 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 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 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 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 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 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 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 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 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 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 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 1996 Shortlist ›

IMO 1996 SL G4

IMO 1996 SL G4 Origin: USA | Category: Geometry Problem Let $\triangle ABC$ be an equilateral triangle and let $P$ be a point in its interior. Let the lines $AP$, $BP$, $CP$ meet the sides $BC$, $CA$, $AB$ in the points $A_1$, $B_1$, $C_1$ respectively. Prove that $$A_1B_1 \cdot B_1C_1 \cdot C_1A_1 \geq A_1B \cdot B_1C \cdot C_1A.$$ Solution By the law of cosines applied to $\triangle CA_1B_1$, we obtain...

Practice › Mathematics › IMO Shortlist › IMO 2004 Shortlist ›

IMO 2004 SL N6

IMO 2004 SL N6 Origin: IRE | Category: Number Theory Problem Given an integer n > 1, denote by Pn the product of all positive integers x less than n and such that n divides x2 −1. For each n > 1, ﬁnd the remainder of Pn on division by n. Solution Let Sn = {x \inN | x \leqn, n | x2 −1}. It is easy to check that...

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 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 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 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 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 1992 Shortlist ›

IMO 1992 SL 18

IMO 1992 SL 18 Origin: USA Problem Let [x] denote the greatest integer less than or equal to x. Pick any x1 in [0, 1) and deﬁne the sequence x1, x2, x3, . . . by xn+1 = 0 if xn = 0 and xn+1 = 1/xn −[1/xn] otherwise. Prove that x1 + x2 + \cdot \cdot \cdot + xn < F1 F2 F2 F3 \cdot \cdot \cdot + Fn...

Practice › Mathematics › IMO Shortlist › IMO 2002 Shortlist ›

IMO 2002 SL N2

IMO 2002 SL N2 Origin: ROM | Category: Number Theory Problem Let n \geq2 be a positive integer, with divisors 1 = d1 < d2 < \cdot \cdot \cdot < dk = n. Prove that d1d2 + d2d3 + \cdot \cdot \cdot + dk−1dk is always less than n2, and determine when it is a divisor of n2. Solution Set S = d1d2 + \cdot \cdot \cdot + dk−1dk. Since...

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 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 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 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 6

IMO 1991 SL 6 Origin: USS Problem Prove for each triangle ABC the inequality 4 < IA \cdot IB \cdot IC lAlBlC \leq8 27, where I is the incenter and lA, lB, lC are the lengths of the angle bisectors of ABC. Solution Let a, b, c be sides of the triangle. Let A1 be the intersection of line AI with BC. By the known fact, BA1 : A1C =...

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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 1996 Shortlist ›

IMO 1996 SL A5

IMO 1996 SL A5 Origin: ROM | Category: Algebra Problem Let $P(x)$ be the real polynomial function $P(x) = ax^3 + bx^2 + cx + d.$ Prove that if $|P(x)| \leq 1 \quad \text{for all } x \text{ such that } |x| \leq 1,$ then $|a| + |b| + |c| + |d| \leq 7.$ Solution Considering the polynomials $\pm P(\pm x)$ we may assume w.l.o.g. that $a, b \geq 0$....

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 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 1996 Shortlist ›

IMO 1996 SL A1

IMO 1996 SL A1 Origin: SLO | Category: Algebra Problem Let $a$, $b$, and $c$ be positive real numbers such that $abc=1$. Prove that $$\frac{ab}{a^5+b^5+ab} + \frac{bc}{b^5+c^5+bc} + \frac{ca}{c^5+a^5+ca} \leq 1.$$ When does equality hold? Solution We have $$a^5+b^5-a^2b^2(a+b) = (a^3-b^3)(a^2-b^2) \geq 0,$$ i.e. $$a^5+b^5 \geq a^2b^2(a+b).$$ Hence $$\frac{ab}{a^5+b^5+ab} \leq \frac{ab}{a^2b^2(a+b)+ab} = \frac{abc^2}{a^2b^2c^2(a+b)+abc^2} = \frac{c}{a+b+c}.$$ Now, the left-hand side of the inequality to be proved does not exceed $$\frac{c}{a+b+c} +...

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 1973 Shortlist ›

IMO 1973 SL 7

IMO 1973 SL 7 Origin: POL Problem Given a tetrahedron ABCD, let x = AB \cdot CD, y = AC \cdot BD, and z = AD \cdot BC. Prove that there exists a triangle with edges x, y, z. Solution The result follows immediately from Ptolemy’s inequality.

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 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 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 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 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 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 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 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 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 1987 Shortlist ›

IMO 1987 SL 13

IMO 1987 SL 13 Origin: GDR Problem Is it possible to put 1987 points in the Euclidean plane such that the distance between each pair of points is irrational and each three points determine a nondegenerate triangle with rational area? Solution We claim that the points Pi(i, i2), i = 1, 2, . . . , 1987, satisfy the conditions. In fact: (i) PiPj = (i −j)2 + (i2 −j2)2...

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 2003 Shortlist ›

IMO 2003 SL N5

IMO 2003 SL N5 Origin: KOR | Category: Number Theory Problem An integer n is said to be good if |n| is not the square of an integer. Determine all integers m with the following property: m can be represented in inﬁnitely many ways as a sum of three distinct good integers whose product is the square of an odd integer. Solution Suppose that m = u+v+w where u, v,...

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 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 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 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 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 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 1999 Shortlist ›

IMO 1999 SL A3

IMO 1999 SL A3 Origin: FIN | Category: Algebra Problem A game is played by n girls (n \geq2), everybody having a ball. Each of the n pairs of players, in an arbitrary order, exchange the balls they have at that moment. The game is called nice if at the end nobody has her own ball, and it is called tiresome if at the end everybody has her initial...

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 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 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 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 1995 Shortlist ›

IMO 1995 SL N3

IMO 1995 SL N3 Origin: CZE | Category: Number Theory Problem Determine all integers n > 3 such that there are n points A1, A2, . . . , An in the plane that satisfy the following two conditions si- multaneously: (a) No three lie on the same line. (b) There exist real numbers p1, p2, . . . , pn such that the area of \triangleAiAjAk is equal to...

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 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 1986 Shortlist ›

IMO 1986 SL 4

IMO 1986 SL 4 Origin: CZS Problem Let n be a positive integer and let p be a prime number, p > 3. Find at least 3(n+1) [easier version: 2(n+1)] sequences of positive integers x, y, z satisfying xyz = pn(x + y + z) that do not diﬀer only by permutation. Solution Let x = p\alphax′, y = p\betay′, z = p\gammaz′ with p ∤x′y′z′ and \alpha \geq\beta \geq\gamma....

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 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 1983 Shortlist ›

IMO 1983 SL 13

IMO 1983 SL 13 Origin: LUX Problem Let E be the set of 19833 points of the space R3 all three of whose coordinates are integers between 0 and 1982 (including 0 and 1982). A coloring of E is a map from E to the set {red, blue}. How many colorings of E are there satisfying the following property: The number of red vertices among the 8 vertices of any...

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 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 2003 Shortlist ›

IMO 2003 SL G4

IMO 2003 SL G4 Origin: ARM | Category: Geometry Problem Let \Gamma1, \Gamma2, \Gamma3, \Gamma4 be distinct circles such that \Gamma1, \Gamma3 are externally tangent at P, and \Gamma2, \Gamma4 are externally tangent at the same point P. Suppose that \Gamma1 and \Gamma2; \Gamma2 and \Gamma3; \Gamma3 and \Gamma4; \Gamma4 and \Gamma1 meet at A, B, C, D, respectively, and that all these points are diﬀerent from P. Prove that...

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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 2000 Shortlist ›

IMO 2000 SL A6

IMO 2000 SL A6 Origin: IRE | Category: Algebra Problem A nonempty set A of real numbers is called a B3-set if the conditions a1, a2, a3, a4, a5, a6 \inA and a1+a2+a3 = a4+a5+a6 imply that the sequences (a1, a2, a3) and (a4, a5, a6) are identical up to a permutation. Let A = {a0 = 0 < a1 < a2 < \cdot \cdot \cdot }, B = {b0...

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 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 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 1979 Shortlist ›

IMO 1979 SL 18

IMO 1979 SL 18 Origin: POL Problem Let m positive integers a1, . . . , am be given. Prove that there exist fewer than 2m positive integers b1, . . . , bn such that all sums of dis- tinct bk’s are distinct and all ai (i \leqm) occur among them. Solution Let us write all ai in binary representation. For S \subseteq{1, 2, . . ., m} let...

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 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 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 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 1996 Shortlist ›

IMO 1996 SL C2

IMO 1996 SL C2 Origin: UKR | Category: Combinatorics Problem An $(n-1) \times (n-1)$ square is divided into $(n-1)^2$ unit squares in the usual manner. Each of the $n^2$ vertices of these squares is to be colored red or blue. Find the number of different colorings such that each unit square has exactly two red vertices. (Two coloring schemes are regarded as different if at least one vertex is colored...

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 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 1979 Shortlist ›

IMO 1979 SL 21

IMO 1979 SL 21 Origin: USS Problem Let N be the number of integral solutions of the equation x2 −y2 = z3 −t3 satisfying the condition 0 \leqx, y, z, t \leq106, and let M be the number of integral solutions of the equation x2 −y2 = z3 −t3 + 1 satisfying the condition 0 \leqx, y, z, t \leq106. Prove that N > M. Solution Let f(n) be the...

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 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 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 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 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 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 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 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 1971 Shortlist ›

IMO 1971 SL 16

IMO 1971 SL 16 Origin: USS Problem Given a convex polyhedron P1 with 9 vertices A1, . . . , A9, let us denote by P2, P3, . . . , P9 the images of P1 under the translations mapping the vertex A1 to A2, A3, . . . , A9 respectively. Prove that among the polyhedra P1, . . . , P9 at least two have a common interior...

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 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 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 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 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 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 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 2002 Shortlist ›

IMO 2002 SL C7

IMO 2002 SL C7 Origin: NZL | Category: Combinatorics Problem Among a group of 120 people, some pairs are friends. A weak quartet is a set of four people containing exactly one pair of friends. What is the maximum possible number of weak quartets? Solution For a graph G on 120 vertices (i.e., people at the party), write q(G) for the number of weak quartets in G. Our solution will...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 28

IMO 1988 SL 28 Origin: GBR Problem The sequence {an} of integers is deﬁned by a1 = 2, a2 = 7, and −1 2 < an+1 − a2 n an−1 \leq1 2, for n \geq2. Prove that an is odd for all n > 1. Solution The sequence is uniquely determined by the conditions, and a1 = 2, a2 = 7, a3 = 25, a4 = 89, a5 = 317,...

Practice › Mathematics › IMO Shortlist › IMO 1983 Shortlist ›

IMO 1983 SL 15

IMO 1983 SL 15 Origin: POL Problem Decide whether there exists a set M of natural numbers satis- fying the following conditions: (i) For any natural number m > 1 there are a, b \inM such that a+b = m. (ii) If a, b, c, d \inM, a, b, c, d > 10 and a + b = c + d, then a = c or a = d. Solution...

Practice › Mathematics › IMO Shortlist › IMO 1987 Shortlist ›

IMO 1987 SL 15

IMO 1987 SL 15 Origin: FRG Problem Suppose x1, x2, . . . , xn are real numbers with x2 1 + x2 2 + \cdot \cdot \cdot + x2 n = 1. Prove that for any integer k > 1 there are integers ei not all 0 and with |ei| < k such that |e1x1 + e2x2 + \cdot \cdot \cdot + enxn| \leq(k −1)\sqrtn kn −1 . Solution...

Practice › Mathematics › IMO Shortlist › IMO 1983 Shortlist ›

IMO 1983 SL 4

IMO 1983 SL 4 Origin: BEL Problem On the sides of the triangle ABC, three similar isosceles tri- angles ABP (AP = PB), AQC (AQ = QC), and BRC (BR = RC) are constructed. The ﬁrst two are constructed externally to the triangle ABC, but the third is placed in the same half-plane determined by the line BC as the triangle ABC. Prove that APRQ is a parallelogram. Solution The...

Practice › Mathematics › IMO Shortlist › IMO 1982 Shortlist ›

IMO 1982 SL 8

IMO 1982 SL 8 Problem B2 (POL 4) A convex, closed ﬁgure lies inside a given circle. The ﬁgure is seen from every point of the circumference at a right angle (that is, the two rays drawn from the point and supporting the convex ﬁgure are perpendicular). Prove that the center of the circle is a center of symmetry of the ﬁgure. Solution Let F be the given ﬁgure. Consider...

Practice › Mathematics › IMO Shortlist › IMO 2001 Shortlist ›

IMO 2001 SL C7

IMO 2001 SL C7 Origin: FRA | Category: Combinatorics Problem A pile of n pebbles is placed in a vertical column. This conﬁguration is modiﬁed according to the following rules. A pebble can be moved if it is at the top of a column that contains at least two more pebbles than the column immediately to its right. (If there are no pebbles to the right, think of this as...

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 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 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 1991 Shortlist ›

IMO 1991 SL 14

IMO 1991 SL 14 Origin: POL Problem Let a, b, c be integers and p an odd prime number. Prove that if f(x) = ax2 + bx + c is a perfect square for 2p −1 consecutive integer values of x, then p divides b2 −4ac. Solution Suppose that f(x0), f(x0 + 1), . . . , f(x0 + 2p −2) are squares. If p | a and p ∤b,...

Practice › Mathematics › IMO Shortlist › IMO 1978 Shortlist ›

IMO 1978 SL 11

IMO 1978 SL 11 Origin: SWE Problem A function f : I \toR, deﬁned on an interval I, is called concave if f(\thetax + (1 −\theta)y) \geq\thetaf(x) + (1 −\theta)f(y) for all x, y \inI and 0 \leq\theta \leq1. Assume that the functions f1, . . . , fn, having all nonnegative values, are concave. Prove that the function (f1f2 . . . fn)1/n is concave. Solution Set F(x) =...

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 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 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 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 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 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 2004 Shortlist ›

IMO 2004 SL C4

IMO 2004 SL C4 Origin: POL | Category: Combinatorics Problem Consider a matrix of size n\timesn whose entries are real numbers of absolute value not exceeding 1, and the sum of all entries is 0. Let n be an even positive integer. Determine the least number C such that every such matrix necessarily has a row or a column with the sum of its entries not exceeding C in absolute...

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 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 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 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 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 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 1986 Shortlist ›

IMO 1986 SL 1

IMO 1986 SL 1 Origin: GBR Problem Find, with proof, all functions f deﬁned on the nonnegative real numbers and taking nonnegative real values such that (i) f[xf(y)]f(y) = f(x + y), (ii) f(2) = 0 but f(x) ̸= 0 for 0 \leqx < 2. Solution If w > 2, then setting in (i) x = w −2, y = 2, we get f(w) = f((w − 2)f(w))f(2) = 0....

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 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 1998 Shortlist ›

IMO 1998 SL 27

IMO 1998 SL 27 Origin: BLR Problem Ten points such that no three of them lie on a line are marked in the plane. Each pair of points is connected with a segment. Each of these segments is painted with one of k colors in such a way that for any k of the ten points, there are k segments each joining two of them with no two being painted...

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 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 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 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 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 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 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 1992 Shortlist ›

IMO 1992 SL 14

IMO 1992 SL 14 Origin: POL Problem For any positive integer x deﬁne g(x) = greatest odd divisor of x, f(x) = . x/2 + x/g(x), if x is even; 2(x+1)/2, if x is odd. Construct the sequence x1 = 1, xn+1 = f(xn). Show that the number 1992 appears in this sequence, determine the least n such that xn = 1992, and determine whether n is unique. Solution We...

Practice › Mathematics › IMO Shortlist › IMO 1977 Shortlist ›

IMO 1977 SL 13

IMO 1977 SL 13 Origin: POL Problem Let B be a set of k sequences each having n terms equal to 1 or −1. The product of two such sequences (a1, a2, . . . , an) and (b1, b2, . . . , bn) is deﬁned as (a1b1, a2b2, . . . , anbn). Prove that there exists a sequence (c1, c2, . . . , cn) such that...

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 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 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 2002 Shortlist ›

IMO 2002 SL A4

IMO 2002 SL A4 Origin: IND | Category: Algebra Problem Find all functions f from the reals to the reals such that (f(x) + f(z))(f(y) + f(t)) = f(xy −zt) + f(xt + yz) for all real x, y, z, t. Solution Putting x = z = 0 and t = y into the given equation gives 4f(0)f(y) = 2f(0) for all y. If f(0) ̸= 0, then we deduce...

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 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 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 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 1987 Shortlist ›

IMO 1987 SL 3

IMO 1987 SL 3 Origin: FIN Problem Does there exist a second-degree polynomial p(x, y) in two variables such that every nonnegative integer n equals p(k, m) for one and only one ordered pair (k, m) of nonnegative integers? Solution The answer: yes. Set p(k, m) = k + [1 + 2 + \cdot \cdot \cdot + (k + m)] = (k + m)2 + 3k + m . It...

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 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 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 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 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 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 1974 Shortlist ›

IMO 1974 SL 8

IMO 1974 SL 8 Problem II 2 (NET 3)IMO5 If a, b, c, d are arbitrary positive real numbers, ﬁnd all possible values of S = a a + b + d + b a + b + c + c b + c + d + d a + c + d. Solution It is clear that a a + b + c + d + b a +...

Practice › Mathematics › IMO Shortlist › IMO 1977 Shortlist ›

IMO 1977 SL 6

IMO 1977 SL 6 Origin: GDR Problem Let n be a positive integer. How many integer solutions (i, j, k, l), 1 \leqi, j, k, l \leqn, does the following system of inequalities have: 1 \leq−j + k + l \leqn 1 \leq i −k + l \leqn 1 \leq i −j + l \leqn 1 \leq i + j −k \leqn ?

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 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 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 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 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 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 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 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 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 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 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 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 16

IMO 1973 SL 16 Origin: CUB Problem Given a, \theta \inR, m \inN, and P(x) = x2m −2|a|mxm cos \theta+a2m, factorize P(x) as a product of m real quadratic polynomials. Solution First, we have P(x) = Q(x)R(x) for Q(x) = xm −|a|mei\theta and R(x) = xm −|a|me−i\theta, where eiϕ means of course cos ϕ + i sin ϕ. It remains to factor both Q and R. Suppose that Q(x) =...

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 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 2002 Shortlist ›

IMO 2002 SL N4

IMO 2002 SL N4 Origin: GER | Category: Number Theory Problem Is there a positive integer m such that the equation a + 1 b + 1 c + 1 abc = m a + b + c has inﬁnitely many solutions in positive integers a, b, c? Solution For a = b = c = 1 we obtain m = 12. We claim that the given equation has inﬁnitely...

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 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 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 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 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 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 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 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 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 1995 Shortlist ›

IMO 1995 SL G5

IMO 1995 SL G5 Origin: NZL | Category: Geometry Problem Let ABCDEF be a convex hexagon with AB = BC = CD, DE = EF = FA, and ∡BCD = ∡EFA = \pi/3 (that is, 60◦). Let G and H be two points interior to the hexagon such that angles AGB and DHE are both 2\pi/3 (that is, 120◦). Prove that AG + GB + GH + DH + HE...

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 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 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 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 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 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 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 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 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 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 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 1968 Shortlist ›

IMO 1968 SL 22

IMO 1968 SL 22 Origin: CZS Problem Find all positive integers $x$ for which $p(x) = x^2 - 10x - 22$, where $p(x)$ denotes the product of the digits of $x$.

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 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 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 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 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 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 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 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 1991 Shortlist ›

IMO 1991 SL 10

IMO 1991 SL 10 Origin: USA Problem Suppose G is a connected graph with n edges. Prove that it is possible to label the edges of G from 1 to n in such a way that in every vertex v of G with two or more incident edges, the set of numbers labeling those edges has no common divisor greater than 1. Solution We start at some vertex v0 and...

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 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 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 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 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 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 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 1973 Shortlist ›

IMO 1973 SL 8

IMO 1973 SL 8 Origin: ROM Problem Prove that there are exactly k [k/2] arrays a1, a2, . . . , ak+1 of nonnegative integers such that a1 = 0 and |ai−ai+1| = 1 for i = 1, 2, . . . , k. Solution Let fn be the required total number, and let fn(k) denote the number of sequences a1, . . . , an of nonnegative...

Practice › Mathematics › IMO Shortlist › IMO 1991 Shortlist ›

IMO 1991 SL 29

IMO 1991 SL 29 Origin: FIN Problem We call a set S on the real line R superinvariant if for any stretching A of the set by the transformation taking x to A(x) = x0 + a(x −x0) there exists a translation B, B(x) = x + b, such that the images of S under A and B agree; i.e., for any x \inS there is a y \inS such...

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 2002 Shortlist ›

IMO 2002 SL C6

IMO 2002 SL C6 Origin: POL | Category: Combinatorics Problem Let n be an even positive integer. Show that there is a permutation x1, x2, . . . , xn of 1, 2, . . . , n such that for every 1 \leqi \leqn the number xi+1 is one of 2xi, 2xi −1, 2xi −n, 2xi −n −1 (where we take xn+1 = x1). Solution Write n = 2m....

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL G3

IMO 1996 SL G3 Origin: GBR | Category: Geometry Problem Let $\triangle ABC$ be an acute-angled triangle with $BC > CA$. Let $O$ be the circumcenter, $H$ its orthocenter, and $F$ the foot of its altitude $CH$. Let the perpendicular to $OF$ at $F$ meet the side $CA$ at $P$. Prove that $\angle FHP = \angle BAC.$ Possible second part: What happens if $|BC| \leq |CA|$ (the triangle still being...

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 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 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 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 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 1995 Shortlist ›

IMO 1995 SL G1

IMO 1995 SL G1 Origin: BUL | Category: Geometry Problem Let A, B, C, and D be distinct points on a line, in that order. The circles with diameters AC and BD intersect at X and Y . O is an arbitrary point on the line XY but not on AD. CO intersects the circle with diameter AC again at M, and BO intersects the other circle again at N....

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 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 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 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 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 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 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 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 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 1973 Shortlist ›

IMO 1973 SL 10

IMO 1973 SL 10 Origin: SWE Problem Let a1, a2, . . . , an be positive numbers and q a given real number, 0 < q < 1. Find n real numbers b1, b2, . . . , bn that satisfy: (1) ak < bk for all k = 1, 2, . . ., n; (2) q < bk+1 bk < 1 q for all k = 1, 2,...

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 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 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 1991 Shortlist ›

IMO 1991 SL 5

IMO 1991 SL 5 Origin: SPA Problem In the triangle ABC, with ∡A = 60◦, a parallel IF to AC is drawn through the incenter I of the triangle, where F lies on the side AB. The point P on the side BC is such that 3BP = BC. Show that ∡BFP = ∡B/2. Solution Let P1 be the point on the side BC such that \angleBFP1 = \beta/2. Then...

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 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 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 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 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 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 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 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 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 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 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 1997 Shortlist ›

IMO 1997 SL 15

IMO 1997 SL 15 Origin: RUS Problem An inﬁnite arithmetic progression whose terms are positive in- tegers contains the square of an integer and the cube of an integer. Show that it contains the sixth power of an integer. Solution Let a + bt, t = 0, 1, 2, . . ., be a given arithmetic progression that contains a square and a cube (a, b > 0). We use...

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 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 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 1995 Shortlist ›

IMO 1995 SL N7

IMO 1995 SL N7 Origin: BLR | Category: Number Theory Problem Does there exist an integer n > 1 that satisﬁes the following condition? The set of positive integers can be partitioned into n nonempty subsets such that an arbitrary sum of n −1 integers, one taken from each of any n −1 of the subsets, lies in the remaining subset. Solution We shall show that there is no such...

Practice › Mathematics › IMO Shortlist › IMO 1979 Shortlist ›

IMO 1979 SL 20

IMO 1979 SL 20 Origin: SWE Problem Given the integer n > 1 and the real number a > 0 determine the maximum of n−1 i=1 xixi+1 taken over all nonnegative numbers xi with sum a. Solution Let xk = max{x1, x2, . . . , xn}. Then xixi+1 \leqxixk for i = 1, 2, . . . , k −1 and xixi+1 \leqxkxi+1 for i = k, . ....

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 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 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 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 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 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 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 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 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 1984 Shortlist ›

IMO 1984 SL 8

IMO 1984 SL 8 Origin: ROM Problem In a plane two diﬀerent points O and A are given. For each point X ̸= O of the plane denote by \alpha(X) the angle AOX measured in radians (0 \leq\alpha(X) < 2\pi) and by C(X) the circle with center O and radius OX + \alpha(X) OX . Suppose each point of the plane is colored by one of a ﬁnite number of...

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 1998 Shortlist ›

IMO 1998 SL 21

IMO 1998 SL 21 Origin: CAN Problem Let a0, a1, a2, . . . be an increasing sequence of nonnegative inte- gers such that every nonnegative integer can be expressed uniquely in the form ai + 2aj + 4ak, where i, j, k are not necessarily distinct. Determine a1998. Solution Such a sequence is obviously strictly increasing. We note that it must be unique. Indeed, given a0, a1, . ....

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 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 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 2003 Shortlist ›

IMO 2003 SL N7

IMO 2003 SL N7 Origin: BRA | Category: Number Theory Problem The sequence a0, a1, a2, . . . is deﬁned as follows: a0 = 2, ak+1 = 2a2 k −1 for k \geq0. Prove that if an odd prime p divides an, then 2n+3 divides p2 −1. Solution Deﬁne the sequence xk of positive reals by ak = cosh xk (cosh is the hyperbolic cosine deﬁned by cosh t...

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 2001 Shortlist ›

IMO 2001 SL C1

IMO 2001 SL C1 Origin: COL | Category: Combinatorics Problem Let A = (a1, a2, . . . , a2001) be a sequence of positive integers. Let m be the number of 3-element subsequences (ai, aj, ak) with 1 \leqi < j < k \leq2001 such that aj = ai + 1 and ak = aj + 1. Considering all such sequences A, ﬁnd the greatest value of m. Solution...

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 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 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 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 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 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 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 1968 Shortlist ›

IMO 1968 SL 24

IMO 1968 SL 24 Origin: MON Problem Find the number of all $n$-digit numbers for which some fixed digit stands only in the $i$th $(1 < i < n)$ place and the last $j$ digits are distinct.$^3$ Solution If the $i$th digit is $0$, then the result is $\frac{9^{k-j}9!}{(10-j)!}$ if $i > k-j$ and $\frac{9^{k-j-1}9!}{(9-j)!}$ otherwise. If the $i$th digit is not $0$, then the above results are multiplied by...

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 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 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 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 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 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 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 1973 Shortlist ›

IMO 1973 SL 12

IMO 1973 SL 12 Origin: SWE Problem Consider the two square matrices A = ⎡ ⎢⎢⎢⎢⎣ 1 −1 −1 1 −1 −1 1 −1 −1 −1 1 −1 1 −1 ⎤ ⎥⎥⎥⎥⎦ and B = ⎡ ⎢⎢⎢⎢⎣ 1 −1 −1 1 −1 1 −1 1 −1 −1 1 −1 1 −1 ⎤ ⎥⎥⎥⎥⎦ with entries 1 and −1. The following operations will be called elementary: (1) Changing signs of all...

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 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 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 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 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 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 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 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 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 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 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 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 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 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 1991 Shortlist ›

IMO 1991 SL 11

IMO 1991 SL 11 Origin: AUS Problem Prove that m=0 (−1)m 1991 −m 1991 −m m Solution To start with, observe that n−m n−m m = 1 n n−m m + n−m−1 m−1 . For n = 1, 2, . . . set Sn = [n/2] m=0(−1)mn−m m . Using the identity m k m−1 k + m−1 k−1 we obtain...

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 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 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 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 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 2001 Shortlist ›

IMO 2001 SL G1

IMO 2001 SL G1 Origin: UKR | Category: Geometry Problem Let A1 be the center of the square inscribed in acute triangle ABC with two vertices of the square on side BC. Thus one of the two re- maining vertices of the square is on side AB and the other is on AC. Points B1, C1 are deﬁned in a similar way for inscribed squares with two vertices on sides...

Practice › Mathematics › IMO Shortlist › IMO 1988 Shortlist ›

IMO 1988 SL 17

IMO 1988 SL 17 Origin: ISR Problem In the convex pentagon ABCDE, the sides BC, CD, DE have the same length. Moreover, each diagonal of the pentagon is parallel to a side (AC is parallel to DE, BD is parallel to AE, etc.). Prove that ABCDE is a regular pentagon. Solution Let AC and AD meet BE in R, S, respectively. Then by the conditions of the problem, \angleAEB =...

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 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 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 2000 Shortlist ›

IMO 2000 SL A4

IMO 2000 SL A4 Origin: GBR | Category: Algebra Problem The function F is deﬁned on the set of nonnegative integers and takes nonnegative integer values satisfying the following conditions: For every n \geq0, (i) F(4n) = F(2n) + F(n); (ii) F(4n + 2) = F(4n) + 1; (iii) F(2n + 1) = F(2n) + 1. Prove that for each positive integer m, the number of integers n with 0...

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 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 1975 Shortlist ›

IMO 1975 SL 15

IMO 1975 SL 15 Origin: USS Problem Is it possible to plot 1975 points on a circle with radius 1 so that the distance between any two of them is a rational number (distances have to be measured by chords)? Solution Assume that the center of the circle is at the origin O(0, 0), and that the points A1, A2, . . . , A1975 are arranged on the upper...

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 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 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 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 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 1986 Shortlist ›

IMO 1986 SL 18

IMO 1986 SL 18 Origin: TUR Problem Let AX, BY, CZ be three cevians concurrent at an inte- rior point D of a triangle ABC. Prove that if two of the quadrangles DY AZ, DZBX, DXCY are circumscribable, so is the third. Solution We shall use the following criterion for a quadrangle to be circumscribable. Lemma. The quadrangle AY DZ is circumscribable if and only if DB − DC =...

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 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 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 16

IMO 1968 SL 16 Origin: GBR Problem A polynomial $p(x) = a_0 x^k + a_1 x^{k-1} + \cdots + a_k$ with integer coefficients is said to be divisible by an integer $m$ if $p(x)$ is divisible by $m$ for all integers $x$. Prove that if $p(x)$ is divisible by $m$, then $k! a_0$ is also divisible by $m$. Also prove that if $a_0, k, m$ are nonnegative integers for which...

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 1992 Shortlist ›

IMO 1992 SL 1

IMO 1992 SL 1 Origin: AUS Problem Prove that for any positive integer m there exist an inﬁnite number of pairs of integers (x, y) such that (i) x and y are relatively prime; (ii) y divides x2 + m; (iii) x divides y2 + m. Solution Assume that a pair (x, y) with x < y satisﬁes the required conditions. We claim that the pair (y, x1) also satisﬁes...

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 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 2000 Shortlist ›

IMO 2000 SL C5

IMO 2000 SL C5 Origin: RUS | Category: Combinatorics Problem In the plane we have n rectangles with parallel sides. The sides of distinct rectangles lie on distinct lines. The boundaries of the rectangles cut the plane into connected regions. A region is nice if it has at least one of the vertices of the n rectangles on its boundary. Prove that the sum of the numbers of the vertices...

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 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 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 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 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 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 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 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 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 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 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 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 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 2004 Shortlist ›

IMO 2004 SL N1

IMO 2004 SL N1 Origin: BLR | Category: Number Theory Problem Let \tau(n) denote the number of positive divisors of the positive integer n. Prove that there exist inﬁnitely many positive integers a such that the equation \tau(an) = n does not have a positive integer solution n. Solution Setting m = an we reduce the given equation to m/\tau(m) = a. Let us show that for a = pp−1...

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 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 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 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 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 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 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 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 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 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 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 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 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 1987 Shortlist ›

IMO 1987 SL 1

IMO 1987 SL 1 Origin: AUS Problem Let f be a function that satisﬁes the following conditions: (i) If x > y and f(y) −y \geqv \geqf(x) −x, then f(z) = v + z, for some number z between x and y. (ii) The equation f(x) = 0 has at least one solution, and among the solutions of this equation, there is one that is not smaller than all the...

Practice › Mathematics › IMO Shortlist › IMO 1996 Shortlist ›

IMO 1996 SL G7

IMO 1996 SL G7 Origin: GBR | Category: Geometry Problem Let $\triangle ABC$ be an acute-angled triangle with circumcenter $O$ and circumradius $R$. Let $AO$ meet the circle $BOC$ again in $A'$, let $BO$ meet the circle $COA$ again in $B'$, and let $CO$ meet the circle $AOB$ again in $C'$. Prove that $$OA' \cdot OB' \cdot OC' \geq 8R^3.$$ When does equality hold? Solution Let $A_1$ be the point...

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 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 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 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 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 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 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 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 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 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 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 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 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 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 1999 Shortlist ›

IMO 1999 SL C7

IMO 1999 SL C7 Origin: IRE | Category: Combinatorics Problem Let p > 3 be a prime number. For each nonempty subset T of {0, 1, 2, 3, . . ., p−1} let E(T ) be the set of all (p−1)-tuples (x1, . . . , xp−1), where each xi \inT and x1 + 2x2 + \cdot \cdot \cdot + (p −1)xp−1 is divisible by p and let |E(T )|...

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 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 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 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 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 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 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 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 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 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...

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 1971 Shortlist ›

IMO 1971 SL 2

IMO 1971 SL 2 Origin: BUL Problem Prove that for every natural number m \geq1 there exists a ﬁnite set Sm of points in the plane satisfying the following condition: If A is any point in Sm, then there are exactly m points in Sm whose distance to A equals 1. Solution We will construct such a set Sm of 2m points. Take vectors u1, . . . , um...

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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 1976 Shortlist ›

IMO 1976 SL 8

IMO 1976 SL 8 Origin: SWE Problem Let P be a polynomial with real coeﬃcients such that P(x) > 0 if x > 0. Prove that there exist polynomials Q and R with nonnegative coeﬃcients such that P(x) = Q(x) R(x) if x > 0. Solution Every polynomial with real coeﬃcients can be factored as a product of linear and quadratic polynomials with real coeﬃcients. Thus it suﬃces to prove...

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 1999 Shortlist ›

IMO 1999 SL G2

IMO 1999 SL G2 Origin: JAP | Category: Geometry Problem A circle is called a separator for a set of ﬁve points in a plane if it passes through three of these points, it contains a fourth point in its interior, and the ﬁfth point is outside the circle. Prove that every set of ﬁve points such that no three are collinear and no four are concyclic has exactly four...

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 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 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 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 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 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 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 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 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 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 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 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 › IMO Shortlist ›

IMO 2010 Shortlist

IMO 2010 Shortlist 31 problems across categories. Problem Category A1 Algebra A2 Algebra A3 Algebra A4 Algebra A5 Algebra A6 Algebra A7 Algebra A8 Algebra C1 Combinatorics C2 Combinatorics C3 Combinatorics C4 Combinatorics C5 Combinatorics C6 Combinatorics C7 Combinatorics C41 Combinatorics G1 Geometry G2 Geometry G3 Geometry G4 Geometry G5 Geometry G6 Geometry G7 Geometry G61 Geometry N1 Number Theory N2 Number Theory N3 Number Theory N4 Number Theory N5...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2007 Shortlist ›

IMO 2007 Shortlist G2

IMO 2007 Shortlist G2 Category: Geometry Problem Given an isosceles triangle ABC with AB = AC. The midpoint of side BC is denoted by M. Let X be a variable point on the shorter arc MA of the circumcircle of triangle ABM. Let T be the point in the angle domain BMA, for which ∠TMX = 90◦ and TX = BX. Prove that ∠MTB − ∠CTM does not depend on...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2008 Shortlist ›

IMO 2008 Shortlist C1

IMO 2008 Shortlist C1 Category: Combinatorics Problem In the plane we consider rectangles whose sides are parallel to the coordinate axes and have positive length. Such a rectangle will be called a box. Two boxes intersect if they have a common point in their interior or on their boundary. Find the largest n for which there exist n boxes B1,...,Bn such that Bi and Bj intersect if and only if...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist A6

IMO 2011 Shortlist A6 Category: Algebra Problem Let f be a function from the set of real numbers to itself that satisfies f(x + y) ≤ yf(x) + f(f(x)) for all real numbers x and y. Prove that f(x) = 0 for all x ≤ 0.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2010 Shortlist ›

IMO 2010 Shortlist A2

IMO 2010 Shortlist A2 Category: Algebra Problem Let the real numbers a,b,c,d satisfy the relations a b c d 6 and a2 b2 c2 d2 12. Prove that 36 ¨4pa3 b3 c3 d3 qpa4 b4 c4 d4 q¨48. (Ukraine)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2018 Shortlist ›

IMO 2018 Shortlist N3

IMO 2018 Shortlist N3 Category: Number Theory Problem Dene the sequen e a0,a1,a2,... by an “ 2n ` 2tn{2u . Prove that there are innitely many terms of the sequen e whi h an be expressed as a sum of (two or more) distin t terms of the sequen e, as well as innitely many of those whi h annot be expressed in su h a way. (Serbia)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2007 Shortlist ›

IMO 2007 Shortlist G3

IMO 2007 Shortlist G3 Category: Geometry Problem The diagonals of a trapezoid ABCD intersect at point P. Point Q lies between the parallel lines BC and AD such that ∠AQD = ∠CQB, and line CD separates points P and Q. Prove that ∠BQP = ∠DAQ. (Ukraine)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2020 Shortlist ›

IMO 2020 Shortlist A3

IMO 2020 Shortlist A3 Category: Algebra Problem Suppose that a, b, c, d are positive real numbers satisfying pa cqpb dq “ ac bd. Find the smallest possible value of a b b c c d d a . (Israel)

Practice › Mathematics › IMO › IMO Shortlist ›

IMO 2018 Shortlist

IMO 2018 Shortlist 28 problems across categories. Problem Category A1 Algebra A2 Algebra A3 Algebra A4 Algebra A5 Algebra A6 Algebra A7 Algebra C1 Combinatorics C2 Combinatorics C3 Combinatorics C4 Combinatorics C5 Combinatorics C6 Combinatorics C7 Combinatorics G1 Geometry G2 Geometry G3 Geometry G4 Geometry G5 Geometry G6 Geometry G7 Geometry N1 Number Theory N2 Number Theory N3 Number Theory N4 Number Theory N5 Number Theory N6 Number Theory N7...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2016 Shortlist ›

IMO 2016 Shortlist G3

IMO 2016 Shortlist G3 Category: Geometry Problem Let B = (−1,0) and C = (1,0) be fixed points on the coordinate plane. A nonempty, bounded subset S of the plane is said to be nice if (i) there is a point T in S such that for every point Q in S, the segment TQ lies entirely in S; and (ii) for any triangle P1P2P3, there exists a unique point...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2014 Shortlist ›

IMO 2014 Shortlist C1

IMO 2014 Shortlist C1 Category: Combinatorics Problem Let n points be given inside a rectangle R such that no two of them lie on a line parallel to one of the sides of R. The rectangle R is to be dissected into smaller rectangles with sides parallel to the sides of R in such a way that none of these rectangles contains any of the given points in its interior....

Practice › Mathematics › IMO › IMO Shortlist › IMO 2007 Shortlist ›

IMO 2007 Shortlist N3

IMO 2007 Shortlist N3 Category: Number Theory Problem Let X be a set of 10000 integers, none of them is divisible by 47. Prove that there exists a 2007-element subset Y of X such that a − b + c − d + e is not divisible by 47 for any a,b,c,d,e ∈ Y . (Netherlands)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2020 Shortlist ›

IMO 2020 Shortlist N5

IMO 2020 Shortlist N5 Category: Number Theory Problem Determine all functions f defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions: piq fpnq ‰ 0 for at least one n; piiq fpxyq “ fpxq ` fpyq for every positive integers x and y; piiiq there are infinitely many positive integers n such that fpkq “ fpn ´ kq for all k ă...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2014 Shortlist ›

IMO 2014 Shortlist G6

IMO 2014 Shortlist G6 Category: Geometry Problem Let ABC be a fixed acute-angled triangle. Consider some points E and F lying on the sides AC and AB, respectively, and let M be the midpoint of EF. Let the perpendicular bisector of EF intersect the line BC at K, and let the perpendicular bisector of MK intersect the lines AC and AB at S and T, respectively. We call the pair...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2019 Shortlist ›

IMO 2019 Shortlist G3

IMO 2019 Shortlist G3 Category: Geometry Problem In triangle ABC, let A1 and B1 be two points on sides BC and AC, and let P and Q be two points on segments AA1 and BB1, respectively, so that line PQ is parallel to AB. On ray PB1, beyond B1, let P1 be a point so that =PP1C “ =BAC. Similarly, on ray QA1, beyond A1, let Q1 be a point...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2010 Shortlist ›

IMO 2010 Shortlist C6

IMO 2010 Shortlist C6 Category: Combinatorics Problem Given a positive integer k and other two integers b ¡ w ¡ 1. There are two strings of pearls, a string of b black pearls and a string of w white pearls. The length of a string is the number of pearls on it. One cuts these strings in some steps by the following rules. In each step: (i) The strings are...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2008 Shortlist ›

IMO 2008 Shortlist A4

IMO 2008 Shortlist A4 Category: Algebra Problem For an integer m, denote by t(m) the unique number in {1,2,3} such that m+ t(m) is a multiple of 3. A function f : Z → Z satisfies f(−1) = 0, f(0) = 1, f(1) = −1 and f(2n m) = f(2n − t(m)) − f(m) for all integers m,n ≥ 0 with 2n m. Prove that f(3p) ≥ 0 holds for...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2022 Shortlist ›

IMO 2022 Shortlist G8

IMO 2022 Shortlist G8 Category: Geometry Problem Let AA1 BCC1 B1 be a convex cyclic hexagon such that AC is tangent to the incircle of the triangle A1 B1 C1 , and A1 C1 is tangent to the incircle of the triangle ABC. Let the lines AB and A1 B1 meet at X and let the lines BC and B1 C1 meet at Y . Prove that if XBY B1...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2021 Shortlist ›

IMO 2021 Shortlist C6

IMO 2021 Shortlist C6 Category: Combinatorics Problem A hunter and an invisible rabbit play a game on an infinite square grid. First the hunter fixes acolouringofthecellswithfinitelymanycolours. The rabbitthen secretly chooses a cell to start in. Every minute, the rabbit reports the colour of its current cell to the hunter, and then secretly moves to an adjacent cellthat ithas not visited before (two cellsare adjacent if they share a side). The...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2013 Shortlist ›

IMO 2013 Shortlist A5

IMO 2013 Shortlist A5 Category: Algebra Problem Let Zě0 be the set of all nonnegative integers. Find all the functions f : Zě0 Ñ Zě0 satisfying the relation fpfpfpnqqq “ fpn 1q 1 for all n P Zě0. (Serbia)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2009 Shortlist ›

IMO 2009 Shortlist G6

IMO 2009 Shortlist G6 Category: Geometry Problem Let the sides AD and BC of the quadrilateral ABCD (such that AB is not parallel to CD) intersect at point P. Points O1 and O2 are the circumcenters and points H1 and H2 are the orthocenters of triangles ABP and DCP, respectively. Denote the midpoints of segments O1H1 and O2H2 by E1 and E2, respectively. Prove that the perpendicular from E1 on...

Practice › Mathematics › IMO Shortlist ›

IMO 1989 Shortlist

IMO 1989 Shortlist 32 problems · Source: IMO Compendium Problems # Origin Problem 1 AUS Let ABC be a triangle. The bisector of angle A meets 2 AUS Ali Barber, the carpet merchant, has a rectangular piece of 3 AUS Ali Barber, the carpet merchant, has a rectangular piece of 4 BUL Prove that for every integer n > 1 the equation 5 COL Consider the polynomial p(x) = xn+nxn−1+a2xn−2...

Practice › Mathematics › IMO Shortlist ›

IMO 1977 Shortlist

IMO 1977 Shortlist 16 problems · Source: IMO Compendium Problems # Origin Problem 1 BUL Let f : N \toN be a function that satisﬁes the inequality 2 CZS A lattice point in the plane is a point both of whose coordinates 3 FRG Let a and b be natural numbers and let q and r be the 4 FRG Describe all closed bounded ﬁgures \Phi in the plane any...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2006 Shortlist ›

IMO 2006 Shortlist G8

IMO 2006 Shortlist G8 Category: Geometry Problem Let ABCD be a convex quadrilateral. A circle passing through the points A and D and a circle passing through the points B and C are externally tangent at a point P inside the quadrilateral. Suppose that ∠PAB + ∠PDC ≤ 90◦ and ∠PBA + ∠PCD ≤ 90◦ . Prove that AB + CD ≥ BC + AD. (Poland)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2022 Shortlist ›

IMO 2022 Shortlist C8

IMO 2022 Shortlist C8 Category: Combinatorics Problem Alice fills the fields of an n ˆ n board with numbers from 1 to n2 , each number being used exactly once. She then counts the total number of good paths on the board. A good path is a sequence of fields of arbitrary length (including 1) such that: (i) The first field in the sequence is one that is only adjacent...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2008 Shortlist ›

IMO 2008 Shortlist A5

IMO 2008 Shortlist A5 Category: Algebra Problem Let a,b,c,d be positive real numbers such that abcd = 1 and a + b + c + d > a b + b c + c d + d a . Prove that a + b + c + d < b a + c b + d c + a d .

Practice › Mathematics › IMO › IMO Shortlist › IMO 2024 Shortlist ›

IMO 2024 Shortlist C3

IMO 2024 Shortlist C3 Category: Combinatorics Problem Let n be a positive integer. There are 2n knights sitting at a round table. They consist of n pairs of partners, each pair of which wishes to shake hands. A pair can shake hands only when next to each other. Every minute, one pair of adjacent knights swaps places. Find the minimum number of exchanges of adjacent knights such that, regardless of...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2019 Shortlist ›

IMO 2019 Shortlist A3

IMO 2019 Shortlist A3 Category: Algebra Problem Let n ě 3 be a positive integer and let pa1,a2,...,anq be a strictly increasing sequence of n positive real numbers with sum equal to 2. Let X be a subset of t1,2,...,nu such that the value of ˇ ˇ ˇ ˇ ˇ 1 ´ ÿ iPX ai ˇ ˇ ˇ ˇ ˇ is minimised. Prove that there exists a strictly increasing sequence...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2021 Shortlist ›

IMO 2021 Shortlist N1

IMO 2021 Shortlist N1 Category: Number Theory Problem Determine all integers n ě 1 for which there exists a pair of positive integers pa,bq such that no cube of a prime divides a2 b 3 and ab 3b 8 a2 b 3 “n.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2006 Shortlist ›

IMO 2006 Shortlist G5

IMO 2006 Shortlist G5 Category: Geometry Problem In triangle ABC, let J be the centre of the excircle tangent to side BC at A1 and to the extensions of sides AC and AB at B1 and C1, respectively. Suppose that the lines A1B1 and AB are perpendicular and intersect at D. Let E be the foot of the perpendicular from C1 to line DJ. Determine the angles ∠BEA1 and ∠AEB1....

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist A4

IMO 2011 Shortlist A4 Category: Algebra Problem Determine all pairs (f,g) of functions from the set of positive integers to itself that satisfy fg(n)+1 (n) + gf(n) (n) = f(n + 1) − g(n + 1) + 1 for every positive integer n. Here, fk (n) means f(f(...f | {z } k (n)...)).

Practice › Mathematics › IMO › IMO Shortlist › IMO 2010 Shortlist ›

IMO 2010 Shortlist C2

IMO 2010 Shortlist C2 Category: Combinatorics Problem On some planet, there are 2N countries (N © 4). Each country has a flag N units wide and one unit high composed of N fields of size 11, each field being either yellow or blue. No two countries have the same flag. We say that a set of N flags is diverse if these flags can be arranged into an N N...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2021 Shortlist ›

IMO 2021 Shortlist C1

IMO 2021 Shortlist C1 Category: Combinatorics Problem Let S be an infiniteset of positive integers, such that there exist four pairwise distinct a,b,c,d PS with gcdpa,bq‰gcdpc,dq. Prove that there exist three pairwise distinct x,y,z PS such that gcdpx,yq“gcdpy,zq‰gcdpz,xq.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2006 Shortlist ›

IMO 2006 Shortlist A5

IMO 2006 Shortlist A5 Category: Algebra Problem Let a,b,c be the sides of a triangle. Prove that √ b + c − a √ b + √ c − √ a + √ c + a − b √ c + √ a − √ b + √ a + b − c √ a + √ b − √ c ≤ 3. (Korea)

Practice › Mathematics › IMO Shortlist ›

IMO 1971 Shortlist

IMO 1971 Shortlist 17 problems · Source: IMO Compendium Problems # Origin Problem 1 BUL Consider a sequence of polynomials P0(x), P1(x), P2(x), . . . , 2 BUL Prove that for every natural number m \geq1 there exists a 3 GDR Knowing that the system 4 GBR We are given two mutually tangent circles in the plane, with 5 HUN Let a, b, c, d, e be real numbers....

Practice › Mathematics › IMO › IMO Shortlist › IMO 2022 Shortlist ›

IMO 2022 Shortlist C2

IMO 2022 Shortlist C2 Category: Combinatorics Problem The Bank of Oslo issues coins made out of two types of metal: aluminium (denoted A) and copper (denoted C). Morgane has n aluminium coins, and n copper coins, and arranges her 2n coins in a row in some arbitrary initial order. Given a fixed positive integer k ď 2n, she repeatedly performs the following operation: identify the largest subsequence containing the k-th...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2023 Shortlist ›

IMO 2023 Shortlist A1

IMO 2023 Shortlist A1 Category: Algebra Problem Professor Oak is feeding his 100 Pokémon. Each Pokémon has a bowl whose capacity is a positive real number of kilograms. These capacities are known to Professor Oak. The total capacity of all the bowls is 100 kilograms. Professor Oak distributes 100 kilograms of food in such a way that each Pokémon receives a non-negative integer number of kilograms of food (which may...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2009 Shortlist ›

IMO 2009 Shortlist A3

IMO 2009 Shortlist A3 Category: Algebra Problem Determine all functions f from the set of positive integers into the set of positive integers such that for all x and y there exists a non degenerated triangle with sides of lengths x, f(y) and f(y + f(x) − 1).

Practice › Mathematics › IMO › IMO Shortlist › IMO 2019 Shortlist ›

IMO 2019 Shortlist G6

IMO 2019 Shortlist G6 Category: Geometry Problem Let I be the incentre of acute-angled triangle ABC. Let the incircle meet BC, CA, and AB at D, E, and F, respectively. Let line EF intersect the circumcircle of the triangle at P and Q, such that F lies between E and P. Prove that =DPA ` =AQD “ =QIP. (Slovakia)

Practice › Mathematics › IMO Shortlist ›

IMO 1992 Shortlist

IMO 1992 Shortlist 21 problems · Source: IMO Compendium Problems # Origin Problem 1 AUS Prove that for any positive integer m there exist an inﬁnite 2 CHN Let R+ be the set of all nonnegative real numbers. Given two 3 CHN The diagonals of a quadrilateral ABCD are perpendicular: 4 CHN Given nine points in space, no four of which are coplanar, 5 COL Let ABCD be a convex...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2021 Shortlist ›

IMO 2021 Shortlist N7

IMO 2021 Shortlist N7 Category: Number Theory Problem Let a1,a2,a3,... be an infinite sequence of positive integers such that an 2m divides an an m for allpositive integers n and m. Prove that this sequence is eventually periodic, i.e. there exist positive integers N and d such that an “an d for all n ąN.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2018 Shortlist ›

IMO 2018 Shortlist C1

IMO 2018 Shortlist C1 Category: Combinatorics Problem Let n ě 3 be an integer. Prove that there exists a set S of 2n positive integers satisfying the following property: For every m “ 2,3,...,n the set S an be partitioned into two subsets with equal sums of elements, with one of subsets of ardinality m. (I eland)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2016 Shortlist ›

IMO 2016 Shortlist G5

IMO 2016 Shortlist G5 Category: Geometry Problem Let D be the foot of perpendicular from A to the Euler line (the line passing through the circumcentre and the orthocentre) of an acute scalene triangle ABC. A circle ω with centre S passes through A and D, and it intersects sides AB and AC at X and Y respectively. Let P be the foot of altitude from A to BC, and...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2013 Shortlist ›

IMO 2013 Shortlist A1

IMO 2013 Shortlist A1 Category: Algebra Problem Let n be a positive integer and let a1,...,an´1 be arbitrary real numbers. Define the sequences u0,...,un and v0,...,vn inductively by u0 “ u1 “ v0 “ v1 “ 1, and uk 1 “ uk akuk´1, vk 1 “ vk an´kvk´1 for k “ 1,...,n ´ 1. Prove that un “ vn. (France)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2018 Shortlist ›

IMO 2018 Shortlist N4

IMO 2018 Shortlist N4 Category: Number Theory Problem Let a1, a2, ..., an, ... be a sequen e of positive integers su h that a1 a2 a2 a3 ¨¨¨ an´1 an an a1 is an integer for all n ě k, where k is some positive integer. Prove that there exists a positive integer m su h that an “ an`1 for all n ě m. (Mongolia)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2018 Shortlist ›

IMO 2018 Shortlist A6

IMO 2018 Shortlist A6 Category: Algebra Problem Let m,n ě 2 be integers. Let fpx1,...,xnq be a polynomial with real oe ients su h that fpx1,...,xnq “ Yx1 ... xn m ] for every x1,...,xn P 0,1,...,m ´ 1 ( . Prove that the total degree of f is at least n. (Brazil)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2009 Shortlist ›

IMO 2009 Shortlist G2

IMO 2009 Shortlist G2 Category: Geometry Problem Let ABC be a triangle with circumcenter O. The points P and Q are interior points of the sides CA and AB, respectively. The circle k passes through the midpoints of the segments BP, CQ, and PQ. Prove that if the line PQ is tangent to circle k then OP = OQ.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2010 Shortlist ›

IMO 2010 Shortlist G7

IMO 2010 Shortlist G7 Category: Geometry Problem Three circular arcs γ1, γ2, and γ3 connect the points A and C. These arcs lie in the same half-plane defined by line AC in such a way that arc γ2 lies between the arcs γ1 and γ3. Point B lies on the segment AC. Let h1, h2, and h3 be three rays starting at B, lying in the same half-plane, h2 being...

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IMO 2019 Shortlist C2

IMO 2019 Shortlist C2 Category: Combinatorics Problem You are given a set of n blocks, each weighing at least 1; their total weight is 2n. Prove that for every real number r with 0 ď r ď 2n ´ 2 you can choose a subset of the blocks whose total weight is at least r but at most r ` 2. (Thailand)

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IMO 2007 Shortlist G8

IMO 2007 Shortlist G8 Category: Geometry Problem Point P lies on side AB of a convex quadrilateral ABCD. Let ω be the incircle of triangle CPD, and let I be its incenter. Suppose that ω is tangent to the incircles of triangles APD and BPC at points K and L, respectively. Let lines AC and BD meet at E, and let lines AK and BL meet at F. Prove that...

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IMO 2011 Shortlist C4

IMO 2011 Shortlist C4 Category: Combinatorics Problem Determine the greatest positive integer k that satisfies the following property: The set of positive integers can be partitioned into k subsets A1,A2,...,Ak such that for all integers n ≥ 15 and all i ∈ {1,2,...,k} there exist two distinct elements of Ai whose sum is n.

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IMO 2019 Shortlist N8

IMO 2019 Shortlist N8 Category: Number Theory Problem Let a and b be two positive integers. Prove that the integer a2 ` R 4a2 b V is not a square. (Here rzs denotes the least integer greater than or equal to z.) (Russia) 10 Bath — UK, 11th–22nd July 2019

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IMO 2022 Shortlist G4

IMO 2022 Shortlist G4 Category: Geometry Problem Let ABC be an acute-angled triangle with AC ą AB, let O be its circumcentre, and let D be a point on the segment BC. The line through D perpendicular to BC intersects the lines AO, AC and AB at W, X and Y , respectively. The circumcircles of triangles AXY and ABC intersect again at Z ‰ A. Prove that if OW...

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IMO 2023 Shortlist N4

IMO 2023 Shortlist N4 Category: Number Theory Problem Let a1,a2,...,an,b1,b2,...,bn be 2n positive integers such that the n ` 1 products a1a2a3 ¨¨¨an, b1a2a3 ¨¨¨an, b1b2a3 ¨¨¨an, . . . b1b2b3 ¨¨¨bn form a strictly increasing arithmetic progression in that order. Determine the smallest positive integer that could be the common difference of such an arithmetic progression. (Canada)

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IMO 2015 Shortlist A4

IMO 2015 Shortlist A4 Category: Algebra Problem Find all functions f : R Ñ R satisfying the equation f x fpx yq ˘ fpxyq “ x fpx yq ` yfpxq for all real numbers x and y. (Albania)

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IMO 2011 Shortlist C7

IMO 2011 Shortlist C7 Category: Combinatorics Problem On a square table of 2011 by 2011 cells we place a finite number of napkins that each cover a square of 52 by 52 cells. In each cell we write the number of napkins covering it, and we record the maximal number k of cells that all contain the same nonzero number. Considering all possible napkin configurations, what is the largest value...

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IMO 2014 Shortlist C8

IMO 2014 Shortlist C8 Category: Combinatorics Problem A card deck consists of 1024 cards. On each card, a set of distinct decimal digits is written in such a way that no two of these sets coincide (thus, one of the cards is empty). Two players alternately take cards from the deck, one card per turn. After the deck is empty, each player checks if he can throw out one of...

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IMO 1990 Shortlist

IMO 1990 Shortlist 28 problems · Source: IMO Compendium Problems # Origin Problem 1 AUS The integer 9 can be written as a sum of two consecutive 2 CAN Given n countries with three representatives each, m commit- 3 CZS On a circle, 2n −1 (n \geq3) diﬀerent points are given. Find 4 CZS Assume that the set of all positive integers is decomposed into 5 FRA Given \triangleABC with...

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IMO 2019 Shortlist N6

IMO 2019 Shortlist N6 Category: Number Theory Problem Let H “ X i ? : i P Zą0 ( “ t1,2,4,5,7,...u, and let n be a positive integer. Prove that there exists a constant C such that, if A Ă t1,2,...,nu satisﬁes |A| ě C ? n, then there exist a,b P A such that a´b P H. (Here Zą0 is the set of positive integers, and tzu denotes the...

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IMO 2023 Shortlist A7

IMO 2023 Shortlist A7 Category: Algebra Problem Let N be a positive integer. Prove that there exist three permutations a1,a2,...,aN; b1,b2,...,bN; and c1,c2,...,cN of 1,2,...,N such that ? ak a bk ? ck ´ 2 ? N ă 2023 for every k “ 1,2,...,N. (China) Shortlisted problems 5

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IMO 2012 Shortlist N8

IMO 2012 Shortlist N8 Category: Number Theory Problem Prove that for every prime p > 100 and every integer r there exist two integers a and b such that p divides a2 b5 − r. Solution 1. Throughout the solution, all congruence relations are meant modulo p. Fix p, and let P = {0,1,...,p−1} be the set of residue classes modulo p. For every r ∈ P, let Sr =...

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IMO 1987 Shortlist

IMO 1987 Shortlist 23 problems · Source: IMO Compendium Problems # Origin Problem 1 AUS Let f be a function that satisﬁes the following conditions: 2 USA At a party attended by n married couples, each person talks 3 FIN Does there exist a second-degree polynomial p(x, y) in two 4 FRA Let ABCDEFGH be a parallelepiped with AE\parallelBF\parallelCG\parallelDH. 5 GBR Find, with proof, the point P in the interior...

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IMO 2021 Shortlist N4

IMO 2021 Shortlist N4 Category: Number Theory Problem Alice is given a rational number r ą 1 and a line with two points B ‰ R, where point R contains a red bead and point B contains a blue bead. Alice plays a solitaire game by performinga sequence of moves. In every move, she chooses a (not necessarily positive) integer k, and a bead to move. If that bead is...

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IMO 2010 Shortlist N5

IMO 2010 Shortlist N5 Category: Number Theory Problem Let N be the set of all positive integers. Find all functions f : N Ñ N such that the number fpmq n m fpnq is a square for all m,n PN. (U.S.A.)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2010 Shortlist ›

IMO 2010 Shortlist G4

IMO 2010 Shortlist G4 Category: Geometry Problem Let I be the incenter of a triangle ABC and Γ be its circumcircle. Let the line AI intersect Γ at a point D A. Let F and E be points on side BC and arc BDC respectively such that =BAF =CAE 1 =BAC. Finally, let G be the midpoint of the segment IF. Prove that the lines DG and EI...

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IMO 2019 Shortlist G4

IMO 2019 Shortlist G4 Category: Geometry Problem Let P be a point inside triangle ABC. Let AP meet BC at A1, let BP meet CA at B1, and let CP meet AB at C1. Let A2 be the point such that A1 is the midpoint of PA2, let B2 be the point such that B1 is the midpoint of PB2, and let C2 be the point such that C1 is...

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IMO 2022 Shortlist N5

IMO 2022 Shortlist N5 Category: Number Theory Problem For each 1 ď i ď 9 and T P N, define dipTq to be the total number of times the digit i appears when all the multiples of 1829 between 1 and T inclusive are written out in base 10. Show that there are infinitely many T P N such that there are precisely two distinct values among d1pTq,d2pTq,...,d9pTq. (United Kingdom)

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IMO 2022 Shortlist C5

IMO 2022 Shortlist C5 Category: Combinatorics Problem Let m,n ě 2 be integers, let X be a set with n elements, and let X1,X2,...,Xm be pairwise distinct non-empty, not necessary disjoint subsets of X. A function f : X Ñ t1,2,...,n ` 1u is called nice if there exists an index k such that ÿ xPXk fpxq ą ÿ xPXi fpxq for all i ‰ k. Prove that the number...

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IMO 2010 Shortlist A3

IMO 2010 Shortlist A3 Category: Algebra Problem Let x1,...,x100 be nonnegative real numbers such that xi xi 1 xi 2 ¨ 1 for all i 1,...,100 (we put x101 x1, x102 x2). Find the maximal possible value of the sum S 100 ¸ i1 xixi 2. (Russia) Answer. .

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IMO 2021 Shortlist N5

IMO 2021 Shortlist N5 Category: Number Theory Problem Prove that there are only finitely many quadruples pa,b,c,nq of positive integers such that n! “an´1 bn´1 cn´1 .

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IMO 2015 Shortlist C3

IMO 2015 Shortlist C3 Category: Combinatorics Problem For a finite set A of positive integers, we call a partition of A into two disjoint nonempty subsets A1 and A2 good if the least common multiple of the elements in A1 is equal to the greatest common divisor of the elements in A2. Determine the minimum value of n such that there exists a set of n positive integers with exactly...

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IMO 2017 Shortlist C7

IMO 2017 Shortlist C7 Category: Combinatorics Problem For any nite sets X and Y of positive integers, denote by fXpkq the kth smallest positive integer not in X, and let X ˚ Y “ X Y tfXpyq: y P Y u. Let A be a set of a ą 0 positive integers, and let B be a set of b ą 0 positive integers. Prove that if A ˚ B...

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IMO 2008 Shortlist G2

IMO 2008 Shortlist G2 Category: Geometry Problem Given trapezoid ABCD with parallel sides AB and CD, assume that there exist points E on line BC outside segment BC, and F inside segment AD, such that ∠DAE = ∠CBF. Denote by I the point of intersection of CD and EF, and by J the point of intersection of AB and EF. Let K be the midpoint of segment EF; assume it...

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IMO 2015 Shortlist N7

IMO 2015 Shortlist N7 Category: Number Theory Problem Let Zą0 denote the set of positive integers. For any positive integer k, a function f : Zą0 Ñ Zą0 is called k-good if gcd fpmq n,fpnq ` m ˘ ď k for all m ‰ n. Find all k such that there exists a k-good function. (Canada)

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IMO 1999 Shortlist

IMO 1999 Shortlist 27 problems · Source: IMO Compendium Algebra # Origin Problem A1 POL Let n \geq2 be a ﬁxed integer. Find the least constant C A2 RUS The numbers from 1 to n2 are randomly arranged in the cells A3 FIN A game is played by n girls (n \geq2), everybody having a ball. A4 BLR Prove that the set of positive integers cannot be partitioned A5 JAP...

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IMO 2007 Shortlist N6

IMO 2007 Shortlist N6 Category: Number Theory Problem Let k be a positive integer. Prove that the number (4k2 − 1)2 has a positive divisor of the form 8kn − 1 if and only if k is even. (United Kingdom)

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IMO 2016 Shortlist N6

IMO 2016 Shortlist N6 Category: Number Theory Problem Denote by N the set of all positive integers. Find all functions f : N → N such that for all positive integers m and n, the integer f(m) + f(n) − mn is nonzero and divides mf(m) + nf(n). 10 IMO 2016 Hong Kong

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IMO 2019 Shortlist

IMO 2019 Shortlist 32 problems across categories. Problem Category A1 Algebra A2 Algebra A3 Algebra A4 Algebra A5 Algebra A6 Algebra A7 Algebra C1 Combinatorics C2 Combinatorics C3 Combinatorics C4 Combinatorics C5 Combinatorics C6 Combinatorics C7 Combinatorics C8 Combinatorics C9 Combinatorics G1 Geometry G2 Geometry G3 Geometry G4 Geometry G5 Geometry G6 Geometry G7 Geometry G8 Geometry N1 Number Theory N2 Number Theory N3 Number Theory N4 Number Theory N5...

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IMO 2010 Shortlist G2

IMO 2010 Shortlist G2 Category: Geometry Problem Point P lies inside triangle ABC. Lines AP, BP, CP meet the circumcircle of ABC again at points K, L, M, respectively. The tangent to the circumcircle at C meets line AB at S. Prove that SC SP if and only if MK ML. (Poland)

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IMO 2015 Shortlist A1

IMO 2015 Shortlist A1 Category: Algebra Problem Suppose that a sequence a1,a2,... of positive real numbers satisfies ak 1 ě kak a2 k pk ´ 1q for every positive integer k. Prove that a1 a2 ¨¨¨ ` an ě n for every n ě 2. (Serbia)

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IMO 2015 Shortlist C2

IMO 2015 Shortlist C2 Category: Combinatorics Problem Let V be a finite set of points in the plane. We say that V is balanced if for any two distinct points A,B P V, there exists a point C P V such that AC “ BC. We say that V is center-free if for any distinct points A,B,C P V, there does not exist a point P P V such that...

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IMO 2007 Shortlist N7

IMO 2007 Shortlist N7 Category: Number Theory Problem For a prime p and a positive integer n, denote by νp(n) the exponent of p in the prime factorization of n!. Given a positive integer d and a finite set {p1,...,pk} of primes. Show that there are infinitely many positive integers n such that d νpi (n) for all 1 ≤ i ≤ k. (India)

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IMO 2008 Shortlist C4

IMO 2008 Shortlist C4 Category: Combinatorics Problem Let n and k be fixed positive integers of the same parity, k ≥ n. We are given 2n lamps numbered 1 through 2n; each of them can be on or off. At the beginning all lamps are off. We consider sequences of k steps. At each step one of the lamps is switched (from off to on or from on to off)....

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IMO 2016 Shortlist C8

IMO 2016 Shortlist C8 Category: Combinatorics Problem Let n be a positive integer. Determine the smallest positive integer k with the following property: it is possible to mark k cells on a 2n × 2n board so that there exists a unique partition of the board into 1 × 2 and 2 × 1 dominoes, none of which contains two marked cells. Shortlisted problems 7

Practice › Mathematics › IMO › IMO Shortlist › IMO 2024 Shortlist ›

IMO 2024 Shortlist N5

IMO 2024 Shortlist N5 Category: Number Theory Problem Let S be a finite nonempty set of prime numbers. Let 1 “ b1 ă b2 ă ¨¨¨ be the sequence of all positive integers whose prime divisors all belong to S. Prove that, for all but finitely many positive integers n, there exist positive integers a1, a2, ..., an such that a1 b1 a2 b2 ¨¨¨ an bn “ R b1...

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IMO 2009 Shortlist N3

IMO 2009 Shortlist N3 Category: Number Theory Problem Let f be a non-constant function from the set of positive integers into the set of positive integers, such that a − b divides f(a) − f(b) for all distinct positive integers a,b. Prove that there exist infinitely many primes p such that p divides f(c) for some positive integer c.

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IMO 2007 Shortlist A6

IMO 2007 Shortlist A6 Category: Algebra Problem Let a1,a2,...,a100 be nonnegative real numbers such that a2 1 +a2 2 +...+a2 100 = 1. Prove that a2 1a2 + a2 2a3 + ... + a2 100a1 < . (Poland)

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IMO 2023 Shortlist C4

IMO 2023 Shortlist C4 Category: Combinatorics Problem Let n ě 2 be a positive integer. Paul has a 1 ˆ n2 rectangular strip consisting of n2 unit squares, where the ith square is labelled with i for all 1 ď i ď n2 . He wishes to cut the strip into several pieces, where each piece consists of a number of consecutive unit squares, and then translate (without rotating or...

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IMO 2006 Shortlist N6

IMO 2006 Shortlist N6 Category: Number Theory Problem Let a > b > 1 be relatively prime positive integers. Define the weight of an integer c, denoted by w(c), to be the minimal possible value of |x| + |y| taken over all pairs of integers x and y such that ax + by = c. An integer c is called a local champion if w(c) ≥ w(c ± a) and...

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IMO 2020 Shortlist N7

IMO 2020 Shortlist N7 Category: Number Theory Problem Let S be a set consisting of n ě 3 positive integers, none of which is a sum of two other distinct members of S. Prove that the elements of S may be ordered as a1,a2,...,an so that ai does not divide ai´1 ai 1 for all i “ 2,3,...,n ´ 1. (Ukraine) Shortlisted problems 11 12 Saint-Petersburg — Russia, 18th–28th September...

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IMO 2024 Shortlist C2

IMO 2024 Shortlist C2 Category: Combinatorics Problem Let n be a positive integer. The integers 1, 2, 3, ..., n2 are to be written in the cells of an n ˆ n board such that each integer is written in exactly one cell and each cell contains exactly one integer. For every integer d with d | n, the d-division of the board is the division of the board into...

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IMO 1994 Shortlist

IMO 1994 Shortlist 24 problems · Source: IMO Compendium Algebra # Origin Problem A1 USA Let a0 = 1994 and an+1 = A2 FRA Let m and n be positive integers. The set A = {a1, a2, . . . , A3 GBR Let S be the set of real numbers greater than −1. Find A4 MON Let R denote the set of all real numbers and R+ the subset...

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IMO 2007 Shortlist A3

IMO 2007 Shortlist A3 Category: Algebra Problem Let n be a positive integer, and let x and y be positive real numbers such that xn +yn = 1. Prove that n X k=1 1 + x2k 1 + x4k ! n X k=1 1 + y2k 1 + y4k ! < (1 − x)(1 − y) . (Estonia)

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IMO 2006 Shortlist N2

IMO 2006 Shortlist N2 Category: Number Theory Problem For x ∈ (0,1) let y ∈ (0,1) be the number whose nth digit after the decimal point is the (2n )th digit after the decimal point of x. Show that if x is rational then so is y. (Canada)

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IMO 2013 Shortlist N5

IMO 2013 Shortlist N5 Category: Number Theory Problem Fix an integer k ě 2. Two players, called Ana and Banana, play the following game of numbers: Initially, some integer n ě k gets written on the blackboard. Then they take moves in turn, with Ana beginning. A player making a move erases the number m just written on the blackboard and replaces it by some number m1 with k ď...

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IMO 2018 Shortlist A7

IMO 2018 Shortlist A7 Category: Algebra Problem Find the maximal value of S “ 3 c a b 7 c b c 7 3 c c d 7 c d a 7 , where a,b,c,d are nonnegative real numbers whi h satisfy a b c d “ 100. (Taiwan) 4 Cluj-Napo a Romania, 314 July 2018 Combinatori s

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IMO 2009 Shortlist G1

IMO 2009 Shortlist G1 Category: Geometry Problem Let ABC be a triangle with AB = AC. The angle bisectors of A and B meet the sides BC and AC in D and E, respectively. Let K be the incenter of triangle ADC. Suppose that ∠BEK = 45◦ . Find all possible values of ∠BAC.

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IMO 2013 Shortlist N3

IMO 2013 Shortlist N3 Category: Number Theory Problem Prove that there exist infinitely many positive integers n such that the largest prime divisor of n4 n2 1 is equal to the largest prime divisor of pn 1q4 pn 1q2 1. (Belgium)

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IMO 2017 Shortlist N7

IMO 2017 Shortlist N7 Category: Number Theory Problem Say that an ordered pair px,yq of integers is an irredu ible latti e point if x and y are relatively prime. For any nite set S of irredu ible latti e points, show that there is a homogenous polynomial in two variables, fpx,yq, with integer oe ients, of degree at least 1, su h that fpx,yq “ 1 for ea h...

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IMO 2016 Shortlist C4

IMO 2016 Shortlist C4 Category: Combinatorics Problem Find all positive integers n for which we can fill in the entries of an n × n table with the following properties: • each entry can be one of I, M and O; • in each row and each column, the letters I, M and O occur the same number of times; and • in any diagonal whose number of entries is...

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IMO 2018 Shortlist A4

IMO 2018 Shortlist A4 Category: Algebra Problem Let a0,a1,a2,... be a sequen e of real numbers su h that a0 “ 0, a1 “ 1, and for every n ě 2 there exists 1 ď k ď n satisfying an “ an´1 ¨¨¨ an´k k . Find the maximal possible value of a2018 ´ a2017. (Belgium)

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IMO 2012 Shortlist A4

IMO 2012 Shortlist A4 Category: Algebra Problem Let f and g be two nonzero polynomials with integer coefficients and degf > degg. Suppose that for infinitely many primes p the polynomial pf + g has a rational root. Prove that f has a rational root. Solution 1. Since degf > degg, we have |g(x)/f(x)| < 1 for sufficiently large x; more precisely, there is a real number R such that...

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IMO 2006 Shortlist N4

IMO 2006 Shortlist N4 Category: Number Theory Problem Let P be a polynomial of degree n > 1 with integer coefficients and let k be any positive integer. Consider the polynomial Q(x) = P(P(...P(P(x))...)), with k pairs of parentheses. Prove that Q has no more than n integer fixed points, i.e. integers satisfying the equation Q(x) = x. (Romania)

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IMO 2016 Shortlist A3

IMO 2016 Shortlist A3 Category: Algebra Problem Find all integers n ⩾ 3 with the following property: for all real numbers a1,a2,...,an and b1,b2,...,bn satisfying |ak| + |bk| = 1 for 1 ⩽ k ⩽ n, there exist x1,x2,...,xn, each of which is either −1 or 1, such that n X k=1 xkak + n X k=1 xkbk ⩽ 1.

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IMO 2008 Shortlist

IMO 2008 Shortlist 26 problems across categories. Problem Category A1 Algebra A2 Algebra A3 Algebra A4 Algebra A5 Algebra A6 Algebra A7 Algebra C1 Combinatorics C2 Combinatorics C3 Combinatorics C4 Combinatorics C5 Combinatorics C6 Combinatorics G1 Geometry G2 Geometry G3 Geometry G4 Geometry G5 Geometry G6 Geometry G7 Geometry N1 Number Theory N2 Number Theory N3 Number Theory N4 Number Theory N5 Number Theory N6 Number Theory

Practice › Mathematics › IMO › IMO Shortlist › IMO 2007 Shortlist ›

IMO 2007 Shortlist C1

IMO 2007 Shortlist C1 Category: Combinatorics Problem Let n > 1 be an integer. Find all sequences a1,a2,...,an2+n satisfying the following conditions: (a) ai ∈ {0,1} for all 1 ≤ i ≤ n2 n; (b) ai+1 + ai+2 + ... + ai+n < ai+n+1 + ai+n+2 + ... + ai+2n for all 0 ≤ i ≤ n2 − n. (Serbia)

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IMO 2021 Shortlist G6

IMO 2021 Shortlist G6 Category: Geometry Problem Determine all integers n ě 3 satisfying the following property: every convex n-gon whose sides all have length 1 contains an equilateral triangle of side length 1. (Every polygon is assumed to contain its boundary.) Shortlisted problems 9

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IMO 2013 Shortlist C7

IMO 2013 Shortlist C7 Category: Combinatorics Problem Let n ě 2 be an integer. Consider all circular arrangements of the numbers 0,1,...,n; the n 1 rotations of an arrangement are considered to be equal. A circular arrangement is called beautiful if, for any four distinct numbers 0 ď a,b,c,d ď n with a c “ b d, the chord joining numbers a and c does not intersect the chord joining...

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IMO 2015 Shortlist C6

IMO 2015 Shortlist C6 Category: Combinatorics Problem Let S be a nonempty set of positive integers. We say that a positive integer n is clean if it has a unique representation as a sum of an odd number of distinct elements from S. Prove that there exist infinitely many positive integers that are not clean. (U.S.A.)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist N5

IMO 2011 Shortlist N5 Category: Number Theory 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, n with f(m) ≤ f(n) the number f(n) is divisible by f(m).

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IMO 2016 Shortlist C6

IMO 2016 Shortlist C6 Category: Combinatorics Problem There are n ⩾ 3 islands in a city. Initially, the ferry company offers some routes between some pairs of islands so that it is impossible to divide the islands into two groups such that no two islands in different groups are connected by a ferry route. After each year, the ferry company will close a ferry route between some two islands X...

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IMO 2006 Shortlist N5

IMO 2006 Shortlist N5 Category: Number Theory Problem Find all integer solutions of the equation x7 − 1 x − 1 = y5 − 1. (Russia)

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IMO 2011 Shortlist N1

IMO 2011 Shortlist N1 Category: Number Theory Problem For any integer d > 0, let f(d) be the smallest positive integer that has exactly d positive divisors (so for example we have f(1) = 1, f(5) = 16, and f(6) = 12). Prove that for every integer k ≥ 0 the number f(2k ) divides f(2k+1 ).

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IMO 2012 Shortlist C5

IMO 2012 Shortlist C5 Category: Combinatorics Problem The columns and the rows of a 3n × 3n square board are numbered 1,2,...,3n. Every square (x,y) with 1 ≤ x,y ≤ 3n is colored asparagus, byzantium or citrine according as the modulo 3 remainder of x+y is 0, 1 or 2 respectively. One token colored asparagus, byzantium or citrine is placed on each square, so that there are 3n2 tokens of...

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IMO 2012 Shortlist N2

IMO 2012 Shortlist N2 Category: Number Theory Problem Find all triples (x,y,z) of positive integers such that x ≤ y ≤ z and x3 (y3 z3 ) = 2012(xyz + 2). Solution. First note that x divides 2012·2 = 23 ·503. If 503 | x then the right-hand side of the equation is divisible by 5033 , and it follows that 5032 | xyz + 2. This is false as...

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IMO 2014 Shortlist G5

IMO 2014 Shortlist G5 Category: Geometry Problem Let ABCD be a convex quadrilateral with =B “ =D “ 90˝ . Point H is the foot of the perpendicular from A to BD. The points S and T are chosen on the sides AB and AD, respectively, in such a way that H lies inside triangle SCT and =SHC ´ =BSC “ 90˝ , =THC ´ =DTC “ 90˝ . Prove...

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IMO 2009 Shortlist A6

IMO 2009 Shortlist A6 Category: Algebra Problem Suppose that s1,s2,s3,... is a strictly increasing sequence of positive integers such that the subsequences ss1, ss2, ss3, ... and ss1+1, ss2+1, ss3+1, ... are both arithmetic progressions. Prove that s1,s2,s3,... is itself an arithmetic progression.

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IMO 2020 Shortlist C2

IMO 2020 Shortlist C2 Category: Combinatorics Problem In a regular 100-gon, 41 vertices are colored black and the remaining 59 vertices are colored white. Prove that there exist 24 convex quadrilaterals Q1, ..., Q24 whose corners are vertices of the 100-gon, so that • the quadrilaterals Q1, ..., Q24 are pairwise disjoint, and • every quadrilateral Qi has three corners of one color and one corner of the other color....

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IMO 1985 Shortlist

IMO 1985 Shortlist 22 problems · Source: IMO Compendium Problems # Origin Problem 1 MON Given a set M of 1985 positive integers, none of which 2 BRA A polyhedron has 12 faces and is such that: 3 NET The weight w(p) of a polynomial p, p(x) = n 4 AUS Each of the numbers in the set N = {1, 2, 3, . . ., n −1}, 5 ROM...

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IMO 2020 Shortlist N3

IMO 2020 Shortlist N3 Category: Number Theory Problem Let n be an integer with n ě 2. Does there exist a sequence pa1,...,anq of positive integers with not all terms being equal such that the arithmetic mean of every two terms is equal to the geometric mean of some (one or more) terms in this sequence? (Estonia)

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IMO 2014 Shortlist C6

IMO 2014 Shortlist C6 Category: Combinatorics Problem We are given an infinite deck of cards, each with a real number on it. For every real number x, there is exactly one card in the deck that has x written on it. Now two players draw disjoint sets A and B of 100 cards each from this deck. We would like to define a rule that declares one of them a...

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IMO 1976 Shortlist

IMO 1976 Shortlist 12 problems · Source: IMO Compendium Problems # Origin Problem 1 BUL Let ABC be a triangle with bisectors AA1, BB1, CC1 (A1 \in 2 BUL Let a0, a1, . . . , an, an+1 be a sequence of real numbers satisfying 3 CZS In a convex quadrangle with area 32 cm2, the sum of the 4 — (GBR 1a)IMO6 For all positive integral n, un+1 =...

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IMO 2018 Shortlist A3

IMO 2018 Shortlist A3 Category: Algebra Problem Given any set S of positive integers, show that at least one of the following two assertions holds: (1) There exist distin t nite subsets F and G of S su h that ř xPF 1{x “ ř xPG 1{x; (2) There exists a positive rational number r ă 1 su h that ř xPF 1{x ‰ r for all nite subsets F...

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IMO 2009 Shortlist C2

IMO 2009 Shortlist C2 Category: Combinatorics Problem For any integer n ≥ 2, let N(n) be the maximal number of triples (ai,bi,ci), i = 1,...,N(n), consisting of nonnegative integers ai,bi and ci such that the following two conditions are satis- fied: (1) ai + bi + ci = n for all i = 1,...,N(n), (2) If i 6= j, then ai 6= aj,bi 6= bj and ci 6= cj. Determine...

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IMO 2024 Shortlist C8

IMO 2024 Shortlist C8 Category: Combinatorics Problem Let n be a positive integer. Given an nˆn board, the unit cell in the top left corner is initially coloured black, and the other cells are coloured white. We then apply a series of colouring operations to the board. In each operation, we choose a 2 ˆ 2 square with exactly one cell coloured black and we colour the remaining three cells...

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IMO 2016 Shortlist N1

IMO 2016 Shortlist N1 Category: Number Theory Problem For any positive integer k, denote the sum of digits of k in its decimal representation by S(k). Find all polynomials P(x) with integer coefficients such that for any positive integer n ⩾ 2016, the integer P(n) is positive and S(P(n)) = P(S(n)).

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IMO 2019 Shortlist G5

IMO 2019 Shortlist G5 Category: Geometry Problem Let ABCDE be a convex pentagon with CD “ DE and =EDC ‰ 2 ¨ =ADB. Suppose that a point P is located in the interior of the pentagon such that AP “ AE and BP “ BC. Prove that P lies on the diagonal CE if and only if areapBCDq areapADEq “ areapABDq areapABPq. (Hungary)

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IMO 2020 Shortlist G4

IMO 2020 Shortlist G4 Category: Geometry Problem In the plane, there are n ě 6 pairwise disjoint disks D1,D2,...,Dn with radii R1 ě R2 ě ... ě Rn. For every i “ 1,2,...,n, a point Pi is chosen in disk Di. Let O be an arbitrary point in the plane. Prove that OP1 OP2 ... OPn ě R6 R7 ... Rn. (A disk is assumed to contain its boundary.) (Iran)

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IMO 2023 Shortlist A5

IMO 2023 Shortlist A5 Category: Algebra Problem Let a1,a2,...,a2023 be positive integers such that • a1,a2,...,a2023 is a permutation of 1,2,...,2023, and • |a1 ´ a2|,|a2 ´ a3|,...,|a2022 ´ a2023| is a permutation of 1,2,...,2022. Prove that max ` a1,a2023 ˘ ě 507. (Australia) 4 Chiba, Japan, 2nd–13th July 2023

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IMO 2009 Shortlist G7

IMO 2009 Shortlist G7 Category: Geometry Problem Let ABC be a triangle with incenter I and let X, Y and Z be the incenters of the triangles BIC, CIA and AIB, respectively. Let the triangle XY Z be equilateral. Prove that ABC is equilateral too.

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IMO 2015 Shortlist

IMO 2015 Shortlist 29 problems across categories. Problem Category A1 Algebra A2 Algebra A3 Algebra A4 Algebra A5 Algebra A6 Algebra C1 Combinatorics C2 Combinatorics C3 Combinatorics C4 Combinatorics C5 Combinatorics C6 Combinatorics C7 Combinatorics G1 Geometry G2 Geometry G3 Geometry G4 Geometry G5 Geometry G6 Geometry G7 Geometry G8 Geometry N1 Number Theory N2 Number Theory N3 Number Theory N4 Number Theory N5 Number Theory N6 Number Theory N7...

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IMO 2016 Shortlist C7

IMO 2016 Shortlist C7 Category: Combinatorics Problem Let n ⩾ 2 be an integer. In the plane, there are n segments given in such a way that any two segments have an intersection point in the interior, and no three segments intersect at a single point. Jeff places a snail at one of the endpoints of each of the segments and claps his hands n−1 times. Each time when he...

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IMO 2016 Shortlist A8

IMO 2016 Shortlist A8 Category: Algebra Problem Determine the largest real number a such that for all n ⩾ 1 and for all real numbers x0,x1,...,xn satisfying 0 = x0 < x1 < x2 < ··· < xn, we have x1 − x0 + x2 − x1 ··· + xn − xn−1 ⩾ a Ç x1 x2 ··· + n + 1 xn å . Shortlisted problems 5

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IMO 2023 Shortlist C2

IMO 2023 Shortlist C2 Category: Combinatorics Problem Determine the maximal length L of a sequence a1,...,aL of positive integers satisfying both the following properties: • every term in the sequence is less than or equal to 22023 , and • there does not exist a consecutive subsequence ai,ai 1,...,aj (where 1 ď i ď j ď L) with a choice of signs si,si 1,...,sj P t1,´1u for which siai si...

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IMO 2017 Shortlist N6

IMO 2017 Shortlist N6 Category: Number Theory Problem Find the smallest positive integer n, or show that no su h n exists, with the following property: there are innitely many distin t n-tuples of positive rational numbers pa1,a2,...,anq su h that both a1 a2 ¨¨¨ an and a1 a2 ¨¨¨ an are integers. (Singapore)

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IMO 2019 Shortlist C3

IMO 2019 Shortlist C3 Category: Combinatorics Problem Let n be a positive integer. Harry has n coins lined up on his desk, each showing heads or tails. He repeatedly does the following operation: if there are k coins showing heads and k ą 0, then he ﬂips the kth coin over; otherwise he stops the process. (For example, the process starting with THT would be THT Ñ HHT Ñ HTT...

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IMO 2012 Shortlist N3

IMO 2012 Shortlist N3 Category: Number Theory Problem Determine all integers m ≥ 2 such that every n with m ≤ n ≤ m divides the binomial coefficient n m−2n . Solution. The integers in question are all prime numbers. First we check that all primes satisfy the condition, and even a stronger one. Namely, if p is a prime then every n with 1 ≤ n ≤ p...

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IMO 2019 Shortlist G2

IMO 2019 Shortlist G2 Category: Geometry Problem Let ABC be an acute-angled triangle and let D, E, and F be the feet of altitudes from A, B, and C to sides BC, CA, and AB, respectively. Denote by ωB and ωC the incircles of triangles BDF and CDE, and let these circles be tangent to segments DF and DE at M and N, respectively. Let line MN meet circles ωB...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2008 Shortlist ›

IMO 2008 Shortlist G3

IMO 2008 Shortlist G3 Category: Geometry Problem Let ABCD be a convex quadrilateral and let P and Q be points in ABCD such that PQDA and QPBC are cyclic quadrilaterals. Suppose that there exists a point E on the line segment PQ such that ∠PAE = ∠QDE and ∠PBE = ∠QCE. Show that the quadrilateral ABCD is cyclic.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2010 Shortlist ›

IMO 2010 Shortlist G6

IMO 2010 Shortlist G6 Category: Geometry Problem The vertices X, Y , Z of an equilateral triangle XY Z lie respectively on the sides BC, CA, AB of an acute-angled triangle ABC. Prove that the incenter of triangle ABC lies inside triangle XY Z.

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IMO 2009 Shortlist C7

IMO 2009 Shortlist C7 Category: Combinatorics Problem Variant 1. A grasshopper jumps along the real axis. He starts at point 0 and makes 2009 jumps to the right with lengths 1,2,...,2009 in an arbitrary order. Let M be a set of 2008 positive integers less than 1005 · 2009. Prove that the grasshopper can arrange his jumps in such a way that he never lands on a point from M....

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IMO 2021 Shortlist A8

IMO 2021 Shortlist A8 Category: Algebra Problem Determine all functions f : R ÑR that satisfy fpaq´fpbq ˘ fpbq´fpcq ˘ fpcq´fpaq ˘ “fpab2 bc2 ca2 q´fpa2 b b2 c `c2 aq for all real numbers a,b,c. 6 Saint-Petersburg — Russia, 16th–24th July 2021

Practice › Mathematics › IMO › IMO Shortlist › IMO 2008 Shortlist ›

IMO 2008 Shortlist G7

IMO 2008 Shortlist G7 Category: Geometry Problem Let ABCD be a convex quadrilateral with AB 6= BC. Denote by ω1 and ω2 the incircles of triangles ABC and ADC. Suppose that there exists a circle ω inscribed in angle ABC, tangent to the extensions of line segments AD and CD. Prove that the common external tangents of ω1 and ω2 intersect on ω.

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IMO 2019 Shortlist C9

IMO 2019 Shortlist C9 Category: Combinatorics Problem For any two diﬀerent real numbers x and y, we deﬁne Dpx,yq to be the unique integer d satisfying 2d ď |x ´ y| ă 2d`1 . Given a set of reals F, and an element x P F, we say that the scales of x in F are the values of Dpx,yq for y P F with x ‰ y. Let k...

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IMO 2021 Shortlist A6

IMO 2021 Shortlist A6 Category: Algebra Problem Let A be a finite set of (not necessarily positive) integers, and let m ě2 be an integer. Assume that there exist non-empty subsets B1,B2,B3,...,Bm of A whose elements add up to the sums m1 ,m2 ,m3 ,...,mm , respectively. Prove that A contains at least m{2 elements.

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IMO 2012 Shortlist G4

IMO 2012 Shortlist G4 Category: Geometry Problem Let ABC be a triangle with AB 6= AC and circumcenter O. The bisector of ∠BAC intersects BC at D. Let E be the reflection of D with respect to the midpoint of BC. The lines through D and E perpendicular to BC intersect the lines AO and AD at X and Y respectively. Prove that the quadrilateral BXCY is cyclic. Solution. The...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2007 Shortlist ›

IMO 2007 Shortlist A7

IMO 2007 Shortlist A7 Category: Algebra Problem Let n > 1 be an integer. In the space, consider the set S = (x,y,z) | x,y,z ∈ {0,1,...,n}, x + y + z > 0 . Find the smallest number of planes that jointly contain all (n+1)3 −1 points of S but none of them passes through the origin. (Netherlands)

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IMO 2020 Shortlist G2

IMO 2020 Shortlist G2 Category: Geometry Problem Let ABCD be a convex quadrilateral. Suppose that P is a point in the interior of ABCD such that =PAD : =PBA : =DPA “ 1 : 2 : 3 “ =CBP : =BAP : =BPC. The internal bisectors of angles ADP and PCB meet at a point Q inside the triangle ABP. Prove that AQ “ BQ. (Poland)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2007 Shortlist ›

IMO 2007 Shortlist N4

IMO 2007 Shortlist N4 Category: Number Theory Problem For every integer k ≥ 2, prove that 23k divides the number 2k+1 2k − 2k 2k−1 (1) but 23k+1 does not. (Poland)

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IMO 2016 Shortlist

IMO 2016 Shortlist 32 problems across categories. Problem Category A1 Algebra A2 Algebra A3 Algebra A4 Algebra A5 Algebra A6 Algebra A7 Algebra A8 Algebra C1 Combinatorics C2 Combinatorics C3 Combinatorics C4 Combinatorics C5 Combinatorics C6 Combinatorics C7 Combinatorics C8 Combinatorics G1 Geometry G2 Geometry G3 Geometry G4 Geometry G5 Geometry G6 Geometry G7 Geometry G8 Geometry N1 Number Theory N2 Number Theory N3 Number Theory N4 Number Theory N5...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2007 Shortlist ›

IMO 2007 Shortlist C7

IMO 2007 Shortlist C7 Category: Combinatorics Problem Let α < 3 − √ be a positive real number. Prove that there exist positive integers n and p > α · 2n for which one can select 2p pairwise distinct subsets S1, ..., Sp, T1, ..., Tp of the set {1,2,...,n} such that Si ∩ Tj 6= ∅ for all 1 ≤ i,j ≤ p. (Austria)

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IMO 2016 Shortlist C3

IMO 2016 Shortlist C3 Category: Combinatorics Problem Let n be a positive integer relatively prime to 6. We paint the vertices of a regular n-gon with three colours so that there is an odd number of vertices of each colour. Show that there exists an isosceles triangle whose three vertices are of different colours.

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IMO 2024 Shortlist G8

IMO 2024 Shortlist G8 Category: Geometry Problem Let ABC be a triangle with AB ă AC ă BC, and let D be a point in the interior of segment BC. Let E be a point on the circumcircle of triangle ABC such that A and E lie on opposite sides of line BC and =BAD “ =EAC. Let I, IB, IC, JB, and JC be the incentres of triangles ABC,...

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IMO 2021 Shortlist C4

IMO 2021 Shortlist C4 Category: Combinatorics Problem ThekingdomofAnisotropyconsistsofn cities. Foreverytwocitiesthereexistsexactly one direct one-way road between them. We say that a path from X to Y is a sequence of roads such that one can move from X to Y along this sequence without returning to an already visited city. A collection of paths is called diverse if no road belongs to two or more paths in the collection. Let A...

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IMO 2019 Shortlist N5

IMO 2019 Shortlist N5 Category: Number Theory Problem Let a be a positive integer. We say that a positive integer b is a-good if an b ˘ ´ 1 is divisible by an 1 for all positive integers n with an ě b. Suppose b is a positive integer such that b is a-good, but b 2 is not a-good. Prove that b 1 is prime. (Netherlands)

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IMO 2017 Shortlist C2

IMO 2017 Shortlist C2 Category: Combinatorics Problem Let n be a positive integer. Dene a hameleon to be any sequen e of 3n letters, with exa tly n o urren es of ea h of the letters a, b, and c. Dene a swap to be the transposition of two adja ent letters in a hameleon. Prove that for any hameleon X, there exists a hameleon Y su h that...

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IMO 2017 Shortlist C5

IMO 2017 Shortlist C5 Category: Combinatorics Problem A hunter and an invisible rabbit play a game in the Eu lidean plane. The hunter's starting point H0 oin ides with the rabbit's starting point R0. In the nth round of the game (n ě 1), the following happens. (1) First the invisible rabbit moves se retly and unobserved from its urrent point Rn´1 to some new point Rn with Rn´1Rn “...

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IMO 2016 Shortlist G2

IMO 2016 Shortlist G2 Category: Geometry Problem Let ABC be a triangle with circumcircle Γ and incentre I. Let M be the midpoint of side BC. Denote by D the foot of perpendicular from I to side BC. The line through I per- pendicular to AI meets sides AB and AC at F and E respectively. Suppose the circumcircle of triangle AEF intersects Γ at a point X other than...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2013 Shortlist ›

IMO 2013 Shortlist C1

IMO 2013 Shortlist C1 Category: Combinatorics Problem Let n be a positive integer. Find the smallest integer k with the following property: Given any real numbers a1,...,ad such that a1 a2 ¨¨¨ ` ad “ n and 0 ď ai ď 1 for i “ 1,2,...,d, it is possible to partition these numbers into k groups (some of which may be empty) such that the sum of the numbers in...

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IMO 2014 Shortlist A5

IMO 2014 Shortlist A5 Category: Algebra Problem Consider all polynomials Ppxq with real coefficients that have the following property: for any two real numbers x and y one has |y2 ´ Ppxq| ď 2|x| if and only if |x2 ´ Ppyq| ď 2|y|. Determine all possible values of Pp0q. (Belgium)

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IMO 2023 Shortlist C3

IMO 2023 Shortlist C3 Category: Combinatorics Problem Let n be a positive integer. We arrange 1 2 ¨¨¨ ` n circles in a triangle with n rows, such that the ith row contains exactly i circles. The following figure shows the case n “ 6. n “ 6 In this triangle, a ninja-path is a sequence of circles obtained by repeatedly going from a circle to one of the two...

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IMO 2011 Shortlist C5

IMO 2011 Shortlist C5 Category: Combinatorics Problem Let m be a positive integer and consider a checkerboard consisting of m by m unit squares. At the midpoints of some of these unit squares there is an ant. At time 0, each ant starts moving with speed 1 parallel to some edge of the checkerboard. When two ants moving in opposite directions meet, they both turn 90◦ clockwise and continue moving...

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IMO 2018 Shortlist G4

IMO 2018 Shortlist G4 Category: Geometry Problem A point T is hosen inside a triangle ABC. Let A1, B1, and C1 be the ree tions of T in BC, CA, and AB, respe tively. Let Ω be the ir um ir le of the triangle A1B1C1. The lines A1T, B1T, and C1T meet Ω again at A2, B2, and C2, respe tively. Prove that the lines AA2, BB2, and CC2...

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IMO 2022 Shortlist A5

IMO 2022 Shortlist A5 Category: Algebra Problem Find all positive integers n ě 2 for which there exist n real numbers a1 ă ¨¨¨ ă an and a real number r ą 0 such that the 1 npn´1q differences aj ´ai for 1 ď i ă j ď n are equal, in some order, to the numbers r1 ,r2 ,...,r npn´1q . (Czech Republic)

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IMO 2009 Shortlist C6

IMO 2009 Shortlist C6 Category: Combinatorics Problem On a 999×999 board a limp rook can move in the following way: From any square it can move to any of its adjacent squares, i.e. a square having a common side with it, and every move must be a turn, i.e. the directions of any two consecutive moves must be perpendicular. A non- intersecting route of the limp rook consists of a...

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IMO 2017 Shortlist N5

IMO 2017 Shortlist N5 Category: Number Theory Problem Find all pairs pp,qq of prime numbers with p ą q for whi h the number pp qqp q pp ´ qqp´q ´ 1 pp qqp´qpp ´ qqp q ´ 1 is an integer. (Japan) Shortlisted problems 11

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IMO 2012 Shortlist A7

IMO 2012 Shortlist A7 Category: Algebra Problem We say that a function f : Rk → R is a metapolynomial if, for some positive integers m and n, it can be represented in the form f(x1,...,xk) = max i=1,...,m min j=1,...,n Pi,j(x1,...,xk) where Pi,j are multivariate polynomials. Prove that the product of two metapolynomials is also a metapolynomial. Solution. We use the notation f(x) = f(x1,...,xk) for x = (x1,...,xk)...

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IMO 2016 Shortlist N7

IMO 2016 Shortlist N7 Category: Number Theory Problem Let n be an odd positive integer. In the Cartesian plane, a cyclic polygon P with area S is chosen. All its vertices have integral coordinates, and all squares of its side lengths are divisible by n. Prove that 2S is an integer divisible by n.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2012 Shortlist ›

IMO 2012 Shortlist N1

IMO 2012 Shortlist N1 Category: Number Theory Problem Call admissible a set A of integers that has the following property: If x,y ∈ A (possibly x = y) then x2 kxy + y2 ∈ A for every integer k. Determine all pairs m,n of nonzero integers such that the only admissible set containing both m and n is the set of all integers. Solution. A pair of integers m,n fulfills...

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IMO 2023 Shortlist C6

IMO 2023 Shortlist C6 Category: Combinatorics Problem Let N be a positive integer, and consider an N ˆ N grid. A right-down path is a sequence of grid cells such that each cell is either one cell to the right of or one cell below the previous cell in the sequence. A right-up path is a sequence of grid cells such that each cell is either one cell to the...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2008 Shortlist ›

IMO 2008 Shortlist A6

IMO 2008 Shortlist A6 Category: Algebra Problem Let f : R → N be a function which satisfies f x + f(y) = f y + f(x) for all x,y ∈ R. (1) Prove that there is a positive integer which is not a value of f.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2018 Shortlist ›

IMO 2018 Shortlist A2

IMO 2018 Shortlist A2 Category: Algebra Problem Find all positive integers n ě 3 for whi h there exist real numbers a1,a2,...,an, an 1 “ a1, an 2 “ a2 su h that aiai 1 1 “ ai`2 for all i “ 1,2,...,n. (Slovakia)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2008 Shortlist ›

IMO 2008 Shortlist C2

IMO 2008 Shortlist C2 Category: Combinatorics Problem For every positive integer n determine the number of permutations (a1,a2,...,an) of the set {1,2,...,n} with the following property: 2(a1 + a2 + ··· + ak) is divisible by k for k = 1,2,...,n.

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IMO 2011 Shortlist N2

IMO 2011 Shortlist N2 Category: Number Theory Problem Consider a polynomial P(x) = (x + d1)(x + d2) · ... · (x + d9), where d1, d2,..., d9 are nine distinct integers. Prove that there exists an integer N such that for all integers x ≥ N the number P(x) is divisible by a prime number greater than 20.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2020 Shortlist ›

IMO 2020 Shortlist G7

IMO 2020 Shortlist G7 Category: Geometry Problem Let P be a point on the circumcircle of an acute-angled triangle ABC. Let D, E, and F be the reflections of P in the midlines of triangle ABC parallel to BC, CA, and AB, respectively. Denote by ωA, ωB, and ωC the circumcircles of triangles ADP, BEP, and CFP, respectively. Denote by ω the circumcircle of the triangle formed by the perpendicular...

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IMO 2022 Shortlist A8

IMO 2022 Shortlist A8 Category: Algebra Problem For a positive integer n, an n-sequence is a sequence pa0,...,anq of non-negative integers satisfying the following condition: if i and j are non-negative integers with i j ď n, then ai aj ď n and aai aj “ ai j. Let fpnq be the number of n-sequences. Prove that there exist positive real numbers c1, c2 and λ such that c1λn ă...

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IMO 2020 Shortlist A7

IMO 2020 Shortlist A7 Category: Algebra Problem Let n and k be positive integers. Prove that for a1,...,an P r1,2k s one has n ÿ i“1 ai a a2 1 ... a2 i ď 4 ? kn. (Iran)

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IMO 2014 Shortlist N8

IMO 2014 Shortlist N8 Category: Number Theory Problem For every real number x, let }x} denote the distance between x and the nearest integer. Prove that for every pair pa,bq of positive integers there exist an odd prime p and a positive integer k satisfying › › › › a pk › › › › › › › › b pk › › › › › › › › a...

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IMO 2019 Shortlist A6

IMO 2019 Shortlist A6 Category: Algebra Problem A polynomial Ppx,y,zq in three variables with real coeﬃcients satisﬁes the identities Ppx,y,zq “ Ppx,y,xy ´ zq “ Ppx,zx ´ y,zq “ Ppyz ´ x,y,zq. Prove that there exists a polynomial Fptq in one variable such that Ppx,y,zq “ Fpx2 y2 z2 ´ xyzq. (Russia)

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IMO 2014 Shortlist C9

IMO 2014 Shortlist C9 Category: Combinatorics Problem There are n circles drawn on a piece of paper in such a way that any two circles intersect in two points, and no three circles pass through the same point. Turbo the snail slides along the circles in the following fashion. Initially he moves on one of the circles in clockwise direction. Turbo always keeps sliding along the current circle until he...

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IMO 2017 Shortlist N4

IMO 2017 Shortlist N4 Category: Number Theory Problem Call a rational number short if it has nitely many digits in its de imal expansion. For a positive integer m, we say that a positive integer t is m-tasti if there exists a number c P t1,2,3,...,2017u su h that 10t ´ 1 c ¨ m is short, and su h that 10k ´ 1 c ¨ m is not short...

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IMO 2022 Shortlist N7

IMO 2022 Shortlist N7 Category: Number Theory 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 (modulo rotation and reflection) to place the elements of S around a circle such that the product of any two neighbors is of the form x2 x k for some positive integer x. (U.S.A.)

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IMO 2019 Shortlist A2

IMO 2019 Shortlist A2 Category: Algebra Problem Let u1, u2, ..., u2019 be real numbers satisfying u1 u2 ¨¨¨ u2019 “ 0 and u2 1 u2 2 ¨¨¨ u2 2019 “ 1. Let a “ minpu1,u2,...,u2019q and b “ maxpu1,u2,...,u2019q. Prove that ab ď ´ . (Germany)

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IMO 2018 Shortlist G6

IMO 2018 Shortlist G6 Category: Geometry Problem A onvex quadrilateral ABCD satises AB ¨ CD “ BC ¨ DA. A point X is hosen inside the quadrilateral so that =XAB “ =XCD and =XBC “ =XDA. Prove that =AXB` =CXD “ 180˝ . (Poland)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist C2

IMO 2011 Shortlist C2 Category: Combinatorics Problem Suppose that 1000 students are standing in a circle. Prove that there exists an integer k with 100 ≤ k ≤ 300 such that in this circle there exists a contiguous group of 2k students, for which the first half contains the same number of girls as the second half.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2008 Shortlist ›

IMO 2008 Shortlist C5

IMO 2008 Shortlist C5 Category: Combinatorics Problem Let S = {x1,x2,...,xk+ } be a (k + )-element set of real numbers contained in the interval [0,1]; k and are positive integers. A k-element subset A ⊂ S is called nice if k X xi∈A xi − X xj∈S\A xj ≤ k + 2k . Prove that the number of nice subsets is at least k + k + k...

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IMO 2022 Shortlist

IMO 2022 Shortlist 33 problems across categories. Problem Category A1 Algebra A2 Algebra A3 Algebra A4 Algebra A5 Algebra A6 Algebra A7 Algebra A8 Algebra C1 Combinatorics C2 Combinatorics C3 Combinatorics C4 Combinatorics C5 Combinatorics C6 Combinatorics C7 Combinatorics C8 Combinatorics C9 Combinatorics G1 Geometry G2 Geometry G3 Geometry G4 Geometry G5 Geometry G6 Geometry G7 Geometry G8 Geometry N1 Number Theory N2 Number Theory N3 Number Theory N4 Number...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2006 Shortlist ›

IMO 2006 Shortlist A4

IMO 2006 Shortlist A4 Category: Algebra Problem Prove the inequality X i<j aiaj ai + aj ≤ n 2(a1 + a2 + ··· + an) X i<j aiaj for positive real numbers a1,a2,...,an. (Serbia)

Practice › Mathematics › IMO › IMO Shortlist › IMO 2009 Shortlist ›

IMO 2009 Shortlist C1

IMO 2009 Shortlist C1 Category: Combinatorics Problem Consider 2009 cards, each having one gold side and one black side, lying in parallel on a long table. Initially all cards show their gold sides. Two players, standing by the same long side of the table, play a game with alternating moves. Each move consists of choosing a block of 50 consecutive cards, the leftmost of which is showing gold, and turning...

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IMO 2004 Shortlist

IMO 2004 Shortlist 30 problems · Source: IMO Compendium Algebra # Origin Problem A1 KOR Let n \geq3 be an integer and t1, t2, . . . , tn positive real A2 ROM An inﬁnite sequence a0, a1, a2, . . . of real numbers satisﬁes A3 CAN Does there exist a function s: Q … such that if x A4 KOR Find all polynomials P(x) with real coeﬃcients that...

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IMO 2018 Shortlist C7

IMO 2018 Shortlist C7 Category: Combinatorics Problem Consider 2018 pairwise rossing ir les no three of whi h are on urrent. These ir les subdivide the plane into regions bounded by ir ular edges that meet at verti es. Noti e that there are an even number of verti es on ea h ir le. Given the ir le, alternately olour the verti es on that ir le red and...

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IMO 2016 Shortlist C2

IMO 2016 Shortlist C2 Category: Combinatorics Problem Find all positive integers n for which all positive divisors of n can be put into the cells of a rectangular table under the following constraints: • each cell contains a distinct divisor; • the sums of all rows are equal; and • the sums of all columns are equal.

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IMO 2021 Shortlist N6

IMO 2021 Shortlist N6 Category: Number Theory Problem Determine all integers n ě 2 with the following property: every n pairwise distinct integers whose sum is not divisible by n can be arranged in some order a1, a2, ..., an so that n divides 1 ¨a1 2 ¨a2 ¨¨¨`n ¨an.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2011 Shortlist ›

IMO 2011 Shortlist G1

IMO 2011 Shortlist G1 Category: Geometry Problem Let ABC be an acute triangle. Let ω be a circle whose center L lies on the side BC. Suppose that ω is tangent to AB at B′ and to AC at C′ . Suppose also that the circumcenter O of the triangle ABC lies on the shorter arc B′ C′ of ω. Prove that the circumcircle of ABC and ω meet at...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2009 Shortlist ›

IMO 2009 Shortlist N6

IMO 2009 Shortlist N6 Category: Number Theory Problem Let k be a positive integer. Show that if there exists a sequence a0,a1,... of integers satisfying the condition an = an−1 + nk n for all n ≥ 1, then k − 2 is divisible by 3.

Practice › Mathematics › IMO › IMO Shortlist › IMO 2014 Shortlist ›

IMO 2014 Shortlist A4

IMO 2014 Shortlist A4 Category: Algebra Problem Determine all functions f : Z Ñ Z satisfying f fpmq n ˘ fpmq “ fpnq fp3mq ` 2014 for all integers m and n. (Netherlands) Shortlisted problems 5

Practice › Mathematics › IMO › IMO Shortlist › IMO 2021 Shortlist ›

IMO 2021 Shortlist G5

IMO 2021 Shortlist G5 Category: Geometry Problem Let ABCD be a cyclic quadrilateral whose sides have pairwise different lengths. Let O be the circumcentre of ABCD. The internal anglebisectors of =ABC and =ADC meet AC at B1 and D1, respectively. Let OB be the centre of the circle which passes through B and is tangent to AC at D1. Similarly,let OD be the centre of the circle which passes through...

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IMO 2023 Shortlist N8

IMO 2023 Shortlist N8 Category: Number Theory Problem Let Zą0 be the set of positive integers. Determine all functions f : Zą0 Ñ Zą0 such that fbfpaq pa 1q “ pa 1qfpbq holds for all a,b P Zą0, where fk pnq “ fpfp¨¨¨fpnq¨¨¨qq denotes the composition of f with itself k times. (Taiwan) 10 Chiba, Japan, 2nd–13th July 2023 Shortlisted problems – solutions 11

Practice › Mathematics › IMO › IMO Shortlist › IMO 2006 Shortlist ›

IMO 2006 Shortlist C7

IMO 2006 Shortlist C7 Category: Combinatorics Problem Consider a convex polyhedron without parallel edges and without an edge parallel to any face other than the two faces adjacent to it. Call a pair of points of the polyhedron antipodal if there exist two parallel planes passing through these points and such that the polyhedron is contained between these planes. Let A be the number of antipodal pairs of vertices, and...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2009 Shortlist ›

IMO 2009 Shortlist N2

IMO 2009 Shortlist N2 Category: Number Theory Problem A positive integer N is called balanced, if N = 1 or if N can be written as a product of an even number of not necessarily distinct primes. Given positive integers a and b, consider the polynomial P defined by P(x) = (x + a)(x + b). (a) Prove that there exist distinct positive integers a and b such that all...

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IMO 2023 Shortlist C1

IMO 2023 Shortlist C1 Category: Combinatorics Problem Let m and n be positive integers greater than 1. In each unit square of an mˆn grid lies a coin with its tail-side up. A move consists of the following steps: select a 2 ˆ 2 square in the grid; flip the coins in the top-left and bottom-right unit squares; flip the coin in either the top-right or bottom-left unit square. Determine...

Practice › Mathematics › IMO › IMO Shortlist › IMO 2014 Shortlist ›

IMO 2014 Shortlist A1

IMO 2014 Shortlist A1 Category: Algebra Problem Let z0 ă z1 ă z2 ă ¨¨¨ be an infinite sequence of positive integers. Prove that there exists a unique integer n ě 1 such that zn ă z0 z1 ¨¨¨ zn n ď zn 1. (Austria)

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IMO 1968 SL 21

IMO 1981 SL 7

IMO 1996 SL C4

IMO 1981 SL 2

IMO 1972 SL 6

IMO 2001 SL C3

IMO 1986 SL 21

IMO 1978 SL 16

IMO 1996 SL C1

IMO 1985 SL 15

IMO 1976 SL 12

IMO 1992 SL 17

IMO 1993 SL 22

IMO 1998 SL 1

IMO 1988 SL 1

IMO 1983 SL 19

IMO 1998 SL 19

IMO 1990 SL 11

IMO 1999 SL G6

IMO 1985 SL 2

IMO 1999 SL N6

IMO 1989 SL 25

IMO 1996 SL N5

IMO 1977 SL 7

IMO 2001 SL N3

IMO 1998 SL 13

IMO 2001 SL N5

IMO 1981 SL 1

IMO 1989 SL 28

IMO 1992 SL 9

IMO 1986 SL 14

IMO 1985 SL 5

IMO 1984 SL 12

IMO 2004 SL G6

IMO 1992 SL 7

IMO 1992 SL 15

IMO 1991 SL 25

IMO 1968 SL 10

IMO 2002 SL N6

IMO 1997 SL 19

IMO 1994 SL C3

IMO 2003 SL A2

IMO 1999 SL C6

IMO 2004 SL A4

IMO 2001 SL G8

IMO 1978 SL 7

IMO 1989 SL 15

IMO 2004 SL G7

IMO 1991 SL 16

IMO 1998 SL 12

IMO 1984 SL 6

IMO 1988 SL 23

IMO 1972 SL 3

IMO 1998 SL 10

IMO 2004 SL N5

IMO 1987 SL 11

IMO 1973 SL 6

IMO 1968 SL 13

IMO 2000 SL C3

IMO 1973 SL 17

IMO 1987 SL 10

IMO 1989 SL 19

IMO 1990 SL 14

IMO 1986 SL 5

IMO 1984 SL 19

IMO 1997 SL 12

IMO 1978 SL 1

IMO 1994 SL C7

IMO 2003 SL C4

IMO 1976 SL 1

IMO 1988 SL 15

IMO 1984 SL 3

IMO 1989 SL 16

IMO 1976 SL 6

IMO 1990 SL 9

IMO 1993 SL 19

IMO 1987 SL 16

IMO 1999 SL A6

IMO 1971 SL 15