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IMO 1979 LL ROM58 Origin: ROM Problem Prove that there exists a natural number k0 such that for every natural number k > k0 we may find a finite number of lines in the plane, not all parallel to one of them, that divide the plane exactly in k regions. Find k0.
IMO 1971 LL YUG53 Origin: YUG Problem Denote by xn(p) the multiplicity of the prime p in the canonical representation of the number n! as a product of primes. Prove that xn(p) n < p−1 and limn\to\infty xn(p) n p−1.
IMO 1966 LL USS6 Origin: USS Problem A convex planar polygon M with perimeter l and area S is given. Let M(R) be the set of all points in space that lie a distance at most R from a point of M. Show that the volume V (R) of this set equals V (R) = 4 3\piR3 + \pi 2 lR2 + 2SR.
IMO 1971 LL SWE41 Origin: SWE Problem Consider the set of grid points (m, n) in the plane, m, n inte- gers. Let \sigma be a finite subset and define S(\sigma) = (m,n)\in\sigma (100 −|m| −|n|). Find the maximum of S, taken over the set of all such subsets \sigma.
IMO 1970 LL SWE50 Origin: SWE Problem The area of a triangle is S and the sum of the lengths of its sides is L. Prove that 36S \leqL2\sqrt 3 and give a necessary and sufficient condition for equality.
IMO 1977 LL NET26 Origin: NET Problem Let p be a prime number greater than 5. Let V be the collection of all positive integers n that can be written in the form n = kp + 1 or n = kp −1 (k = 1, 2, . . .). A number n \inV is called indecomposable in V if it is impossible to find k, l \inV such that...
IMO 1977 LL POL32 Origin: POL Problem In a room there are nine men. Among every three of them there are two mutually acquainted. Prove that some four of them are mutually acquainted. Solution It is a known result that among six persons there are 3 mutually ac- quainted or 3 mutually unacquainted. By the condition of the problem the last case is excluded. If there is a man in...
IMO 1992 LL JAP41 Origin: JAP Problem Let S be a set of positive integers n1, n2, . . . , n6 and let n(f) denote the number n1nf(1) + n2nf(2) + \cdot \cdot \cdot + n6nf(6), where f is a permu- tation of {1, 2, . . ., 6}. Let Ω= {n(f) | f is a permutation of {1, 2, . . ., 6}}. Give an example of positive...
IMO 1969 LL MON45 Origin: MON Problem Given n points in the plane such that no three of them are collinear, prove that one can find at least n−3 convex quadrilaterals with their vertices at these points.
IMO 1992 LL COL11 Origin: COL Problem Let \varphi(n, m), m ̸= 1, be the number of positive integers less than or equal to n that are coprime with m. Clearly, \varphi(m, m) = \varphi(m), where \varphi(m) is Euler’s phi function. Find all integers m that satisfy the following inequality: \varphi(n, m) n \geq\varphi(m) m for every positive integer n.
IMO 1977 LL POL33 Origin: POL Problem A circle K centered at (0, 0) is given. Prove that for every vector (a1, a2) there is a positive integer n such that the circle K translated by the vector n(a1, a2) contains a lattice point (i.e., a point both of whose coordinates are integers). Solution Let r be the radius of K and s > \sqrt 2/r an integer. Consider the...
IMO 1972 LL MON23 Origin: MON Problem Does there exist a 2n-digit number a2na2n−1 . . . a1 (for an arbitrary n) for which the following equality holds: a2n . . . a1 = (an . . . a1)2?
IMO 1969 LL GBR25 Origin: GBR Problem Let a, b, x, y be positive integers such that a and b have no common divisor greater than 1. Prove that the largest number not ex- pressible in the form ax + by is ab −a −b. If N(k) is the largest number not expressible in the form ax + by in only k ways, find N(k).
IMO 1992 LL FRG19 Origin: FRG Problem Denote by an the greatest number that is not divisible by 3 and that divides n. Consider the sequence s0 = 0, sn = a1 + a2 + \cdot \cdot \cdot+ an, n \inN. Denote by A(n) the number of all sums sk (0 \leqk \leq3n, k \inN0) that are divisible by 3. Prove the formula A(n) = 3n−1 + 2 \cdot 3(n/2)−1...
IMO 1987 LL AUS2 Origin: AUS Problem Suppose we have a pack of 2n cards, in the order 1, 2, . . . , 2n. A perfect shuffle of these cards changes the order to n+1, 1, n+2, 2, . . ., n− 1, 2n, n; i.e., the cards originally in the first n positions have been moved to the places 2, 4, . . . , 2n, while the...
IMO 1974 LL ROM28 Origin: ROM Problem Let M be a finite set and P = {M1, M2, . . . , Mk} a partition of M (i.e., %k i=1 Mi = M, Mi ̸= \emptyset, Mi \capMj = \emptysetfor all i, j \in{1, 2, . . ., k}, i ̸= j). We define the following elementary operation on P: Choose i, j \in{1, 2, . . ., k}, such...
IMO 1985 LL FRG29 Origin: FRG Problem Call a four-digit number (xyzt)B in the number system with base B stable if (xyzt)B = (dcba)B −(abcd)B, where a \leqb \leqc \leqd are the digits of (xyzt)B in ascending order. Determine all stable numbers in the number system with base B. (FRG 2a) The same problem with B = 1985. (FRG 2b) With assumptions as in FRG 2, determine the number of...
IMO 1974 LL VIE48 Origin: VIE Problem Let a be a number different from zero. For all integers n define Sn = an + a−n. Prove that if for some integer k both Sk and Sk+1 are integers, then for each integer n the number Sn is an integer.
IMO 1983 LL LUX40 Origin: LUX Problem Four faces of tetrahedron ABCD are congruent triangles whose angles form an arithmetic progression. If the lengths of the sides of the triangles are a < b < c, determine the radius of the sphere circumscribed about the tetrahedron as a function on a, b, and c. What is the ratio c/a if R = a?
IMO 1987 LL FIN11 Origin: FIN Problem Let S \subset[0, 1] be a set of 5 points with {0, 1} \subsetS. The graph of a real function f : [0, 1] \to[0, 1] is continuous and increasing, and it is linear on every subinterval I in [0, 1] such that the endpoints but no interior points of I are in S. We want to compute, using a computer, the extreme...
IMO 1987 LL BEL9 Origin: BEL Problem In the set of 20 elements {1, 2, 3, 4, 5, 6, 7, 8, 9, 0, A, B, C, D, J, K, L, U, X, Y , Z} we have made a random sequence of 28 throws. What is the probability that the sequence CUBA JULY 1987 appears in this order in the sequence already thrown?
IMO 1979 LL FRA24 Origin: FRA Problem Let a and b be coprime integers, greater than or equal to 1. Prove that all integers n greater than or equal to (a −1)(b −1) can be written in the form: n = ua + vb, with (u, v) \inN \times N.
IMO 1986 LL FRA22 Origin: FRA Problem Let (an)n\inN be the sequence of integers defined recursively by a0 = 0, a1 = 1, an+2 = 4an+1 + an for n \geq0. Find the common divisors of a1986 and a6891.
IMO 1983 LL VIE73 Origin: VIE Problem Let ABC be a nonequilateral triangle. Prove that there exist two points P and Q in the plane of the triangle, one in the interior and one in the exterior of the circumcircle of ABC, such that the orthogonal projections of any of these two points on the sides of the triangle are vertices of an equilateral triangle.
IMO 1970 LL CZS21 Origin: CZS Problem Find necessary and sufficient conditions on given positive num- bers u, v for the following claim to be valid: there exists a right-angled triangle \triangleABC with CD = u, CE = v, where D, E are points of the segments AB such that AD = DE = EB = 1 3AB.
IMO 1986 LL MON54 Origin: MON Problem Find the least integer n with the following property: For any set V of 8 points in the plane, no three lying on a line, and for any set E of n line segments with endpoints in V , one can find a straight line intersecting at least 4 segments in E in interior points.
IMO 1989 LL INA43 Origin: INA Problem Let f(x) = a sin2 x+b sin x+c, where a, b, and c are real num- bers. Find all values of a, b, and c such that the following three conditions are satisfied simultaneously: (i) f(x) = 381 if sin x = 1/2. (ii) The absolute maximum of f(x) is 444. (iii) The absolute minimum of f(x) is 364.
IMO 1969 LL USS66 Origin: USS Problem (a) Prove that if 0 \leqa0 \leqa1 \leqa2, then (a0 + a1x −a2x2)2 \leq(a0 + a1 + a2)2 1 + 1 2x + 1 3x2 + 1 2x3 + x4 . (b) Formulate and prove the analogous result for polynomials of third degree.
IMO 1986 LL GRE40 Origin: GRE Problem Find the maximum value that the quantity 2m + 7n can have such that there exist distinct positive integers xi (1 \leqi \leqm), yj (1 \leqj \leq n) such that the xi’s are even, the yj’s are odd, and m i=1 xi +n j=1 yj = 1986.
IMO 1988 LL INA42 Origin: INA Problem (a) Four balls of radius 1 are mutually tangent, three resting an the floor and the fourth resting on the others. A tetrahedron, each of whose edges has length s, is circumscribed around the balls. Find the value of s. (b) Suppose that ABCD and EFGH are opposite faces of a rectangu- lar solid, with \angleDHC = 45◦and \angleFHB = 60◦. Find the...
IMO 1989 LL HKG33 Origin: HKG Problem Let n be a positive integer. Show that ( \sqrt 2 + 1)n = \sqrtm + \sqrtm −1 for some positive integer m.
IMO 1988 LL VIE93 Origin: VIE Problem Given a natural number n, find all polynomials P(x) of degree less than n satisfying the following condition: n i=0 P(i)(−1)i n i = 0.
IMO 1970 LL ROM47 Origin: ROM Problem Given a polynomial P(x) = ab(a −c)x3 + (a3 −a2c + 2ab2 −b2c + abc)x2 +(2a2b + b2c + a2c + b3 −abc)x + ab(b + c), where a, b, c ̸= 0, prove that P(x) is divisible by Q(x) = abx2 + (a2 + b2)x + ab and conclude that P(x0) is divisible by (a + b)3 for x0 = (a +...
IMO 1977 LL SWE41 Origin: SWE Problem A wheel consists of a fixed circular disk and a mobile circular ring. On the disk the numbers 1, 2, 3, . . ., N are marked, and on the ring N integers a1, a2, . . . , aN of sum 1 are marked (see the figure). The ring can be turned into N differ- ent positions in which the numbers on...
IMO 1979 LL FRA23 Origin: FRA Problem Consider the set E consisting of pairs of integers (a, b), with a \geq 1 and b \geq1, that satisfy in the decimal system the following properties: (i) b is written with three digits, as \alpha2\alpha1\alpha0, \alpha2 ̸= 0; (ii) a is written as \betap . . . \beta1\beta0 for some p; (iii) (a + b)2 is written as \betap . . ....
IMO 1976 LL GDR22 Origin: GDR Problem A regular pentagon A1A2A3A4A5 with side length s is given. At each point Ai a sphere Ki of radius s/2 is constructed. There are two spheres K1′ and K2′ eah of radius s/2 touching all the five spheres Ki. Decide whether K1′ and K2′ intersect each other, touch each other, or have no common points.
IMO 1974 LL YUG50 Origin: YUG Problem Let m and n be natural numbers with m > n. Prove that 2(m −n)2(m2 −n2 + 1) \geq2m2 −2mn + 1.
IMO 1966 LL CZS28 Origin: CZS Problem Let there be given a circle with center S and radius 1 in the plane, and let ABC be an arbitrary triangle circumscribed about the circle such that SA \leqSB \leqSC. Find the loci of the vertices A, B, C.
IMO 1970 LL FRA24 Origin: FRA Problem Let n and p be two integers such that 2p \leqn. Prove the inequality (n −p)! p! \leq n + 1 n−2p . For which values does equality hold?
IMO 1982 LL POL38 Origin: POL Problem Numbers un,k (1 \leqk \leqn) are defined as follows: u1,1 = 1, un,k = n k − d|n, d|k, d>1 un/d,k/d (the empty sum is defined to be equal to zero). Prove that n | un,k for every natural number n and for every k (1 \leqk \leqn).
IMO 1969 LL BUL11 Origin: BUL Problem Let Z be a set of points in the plane. Suppose that there exists a pair of points that cannot be joined by a polygonal line not passing through any point of Z. Let us call such a pair of points unjoinable. Prove that for each real r > 0 there exists an unjoinable pair of points separated by distance r.
IMO 1984 LL ROM49 Origin: ROM Problem Let n > 1 and xi \inR for i = 1, . . . , n. Set Sk = xk 1 + xk 2 + \cdot \cdot \cdot + xk n for k \geq1. If S1 = S2 = \cdot \cdot \cdot = Sn+1, show that xi \in{0, 1} for every i = 1, 2, . . ., n.
IMO 1986 LL IRE46 Origin: IRE Problem We wish to construct a matrix with 19 rows and 86 columns, with entries xij \in{0, 1, 2} (1 \leqi \leq19, 1 \leqj \leq86), such that: (i) in each column there are exactly k terms equal to 0; (ii) for any distinct j, k \in{1, . . . , 86} there is i \in{1, . . . , 19} with xij + xik...
IMO 1979 LL ROM57 Origin: ROM Problem Let M be a set, and A, B, C given subsets of M. Find a necessary and sufficient condition for the existence of a set X \subsetM for which (X \cupA) \ (X \capB) = C. Describe all such sets X.
IMO 1989 LL IRE53 Origin: IRE Problem Let f(x) = (x −a1)(x −a2) \cdot \cdot \cdot (x −an) −2, where n \geq3 and a1, a2, . . . , an are distinct integers. Suppose that f(x) = g(x)h(x), where g(x), h(x) are both nonconstant polynomials with integer coeffi- cients. Prove that n = 3.
IMO 1985 LL TUR83 Origin: TUR Problem Let \Gammai, i = 0, 1, 2, . . ., be a circle of radius ri inscribed in an angle of measure 2\alpha such that each \Gammai is externally tangent to \Gammai+1 and ri+1 < ri. Show that the sum of the areas of the circles \Gammai is equal to the area of a circle of radius r = 1 2r0( \sqrt sin...
IMO 1984 LL LUX31 Origin: LUX Problem Let f1(x) = x3 +a1x2 +b1x+c1 = 0 be an equation with three positive roots \alpha > \beta > \gamma > 0. From the equation f1(x) = 0 one constructs the equation f2(x) = x3 + a2x2 + b2x + c2 = x(x + b1)2 −(a1x + c1)2 = 0. Continuing this process, we get equations f3, . . . , fn. Prove...
IMO 1972 LL ROM36 Origin: ROM Problem A finite number of parallel segments in the plane are given with the property that for any three of the segments there is a line intersecting each of them. Prove that there exists a line that intersects all the given segments.
IMO 1976 LL YUG51 Origin: YUG Problem Four swallows are catching a fly. At first, the swallows are at the four vertices of a tetrahedron, and the fly is in its interior. Their maximal speeds are equal. Prove that the swallows can catch the fly.
IMO 1966 LL POL15 Origin: POL Problem Points A, B, C, D lie on a circle such that AB is a diameter and CD is not. If the tangents at C and D meet at P while AC and BD meet at Q, show that PQ is perpendicular to AB.
IMO 1967 LL GDR13 Origin: GDR Problem Find whether among all quadrilaterals whose interiors lie inside a semicircle of radius r there exists one (or more) with maximal area. If so, determine their shape and area. Solution The maximum area is 3 \sqrt 3r2/4 (where r is the radius of the semicircle) and is attained in the case of a trapezoid with two vertices at the endpoints of the diameter...
IMO 1988 LL VIE90 Origin: VIE Problem Does there exist a number \alpha (0 < \alpha < 1) such that there is an infinite sequence {an} of positive numbers satisfying 1 + an+1 \leqan + \alpha nan, n = 1, 2, . . .?
IMO 1966 LL HUN19 Origin: HUN Problem Construct a triangle given the three exradii.
IMO 1983 LL BEL5 Origin: BEL Problem Consider the set Q2 of points in R2, both of whose coordinates are rational. (a) Prove that the union of segments with vertices from Q2 is the entire set R2. (b) Is the convex hull of Q2 (i.e., the smallest convex set in R2 that con- tains Q2) equal to R2?
IMO 1967 LL SWE51 Origin: SWE Problem A subset S of the set of integers 0, . . . , 99 is said to have property A if it is impossible to fill a crossword puzzle with 2 rows and 2 columns with numbers in S (0 is written as 00, 1 as 01, and so on). Determine the maximal number of elements in sets S with property A. Solution...
IMO 1985 LL BEL4 Origin: BEL Problem Let x, y, and z be real numbers satisfying x + y + z = xyz. Prove that x(1 −y2)(1 −z2) + y(1 −z2)(1 −x2) + z(1 −x2)(1 −y2) = 4xyz.
IMO 1967 LL SWE49 Origin: SWE Problem Let n and k be positive integers such that 1 \leqn \leqN + 1, 1 \leqk \leqN + 1. Show that min n̸=k | sin n −sin k| < 2 N . Solution Since sin 1, sin 2, . . . , sin(N +1) \in(−1, 1), two of these N +1 numbers have distance less than 2/N. Therefore | sin n −sin k|...
IMO 1985 LL MON51 Origin: MON Problem Let f1 = (a1, a2, . . . , an), n > 2, be a sequence of integers. From f1 one constructs a sequence fk of sequences as follows: if fk = (c1, c2, . . . , cn), then fk+1 = (ci1, ci2, ci3 + 1, ci4 + 1, . . . , cin + 1), where (ci1, ci2, . . ....
IMO 1970 LL USS55 Origin: USS Problem A turtle runs away from an UFO with a speed of 0.2 m/s. The UFO flies 5 meters above the ground, with a speed of 20 m/s. The UFO’s path is a broken line, where after flying in a straight path of length ℓ(in meters) it may turn through for any acute angle \alpha such that tan \alpha < ℓ 1000. When the...
IMO 1988 LL ICE40 Origin: ICE Problem A sequence of numbers an, n = 1, 2, . . ., is defined as follows: a1 = 1/2, and for each n \geq2, an = 2n −3 2n an−1. Prove that n k=1 ak < 1 for all n \geq1.
IMO 1972 LL ROM31 Origin: ROM Problem Find values of n \inN for which the fraction 3n−2 2n−3 is reducible.
IMO 1988 LL FRG17 Origin: FRG Problem Show that if n runs through all positive integers, f(n) = / n + \sqrt 3n + 1/2 runs through all positive integers skipping the terms of the sequence an = n2+2n .
IMO 1971 LL SWE40 Origin: SWE Problem Prove that 1 −1 1 −1 1 −1 \cdot \cdot \cdot 1 −1 n3 1 2, n = 2, 3, . . ..
IMO 1983 LL AUS2 Origin: AUS Problem Seventeen cities are served by four airlines. It is noted that there is direct service (without stops) between any two cities 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 of landings.
IMO 1969 LL CZS14 Origin: CZS Problem Let a and b be two positive real numbers. If x is a real solution of the equation x2 + px + q = 0 with real coefficients p and q such that |p| \leqa, |q| \leqb, prove that |x| \leq1 a + a2 + 4b . (1) Conversely, if x satisfies (1), prove that there exist real numbers p and q...
IMO 1982 LL AUS1 Origin: AUS Problem It is well known that the binomial coefficients n k n! k!(n−k)!, 0 \leqk \leqn, are positive integers. The factorial n! is defined inductively by 0! = 1, n! = n \cdot (n −1)! for n \geq1. (a) Prove that n+1 2n n is an integer for n \geq0. (b) Given a positive integer k, determine the smallest integer Ck with...
IMO 1985 LL GDR37 Origin: GDR Problem Prove that a triangle with angles \alpha, \beta, \gamma, circumradius R, and area A satisfies tan \alpha 2 + tan \beta 2 + tan \gamma 2 \leq9R2 4A .
IMO 1987 LL MOR44 Origin: MOR Problem Let \theta1, \theta2, . . . , \thetan be real numbers such that sin \theta1 + \cdot \cdot \cdot + sin \thetan = 0. Prove that | sin \theta1 + 2 sin \theta2 + \cdot \cdot \cdot + n sin \thetan| \leq n2 .
IMO 1971 LL NET32 Origin: NET Problem Two half-lines a and b, with the common endpoint O, make an acute angle \alpha. Let A on a and B on b be points such that OA = OB, and let b′ be the line through A parallel to b. Let \beta be the circle with center B and radius BO. We construct a sequence of half-lines c1, c2, c3, . ....
IMO 1979 LL FIN18 Origin: FIN Problem Show that for no integers a \geq1, n \geq1 is the sum 1 + 1 + a + 1 + 2a + \cdot \cdot \cdot + 1 + na an integer.
IMO 1989 LL POL79 Origin: POL Problem To each pair (x, y) of distinct elements of a finite set X a number f(x, y) equal to 0 or 1 is assigned in such a way that f(x, y) ̸= f(y, x) for all x, y (x ̸= y). Prove that exactly one of the following situations occurs: (i) X is the union of two disjoint nonempty subsets U, V such...
IMO 1969 LL NET48 Origin: NET Problem Let x1, x2, x3, x4, and x5 be positive integers satisfying x1 +x2 +x3 +x4 +x5 = 1000, x1 −x2 +x3 −x4 +x5 > 0, x1 +x2 −x3 +x4 −x5 > 0, −x1 +x2 +x3 −x4 +x5 > 0, x1 −x2 +x3 +x4 −x5 > 0, −x1 +x2 −x3 +x4 +x5 > 0. (a) Find the maximum of (x1 + x3)x2+x4. (b) In...
IMO 1970 LL POL41 Origin: POL Problem Let a cube of side 1 be given. Prove that there exists a point A on the surface S of the cube such that every point of S can be joined to A by a path on S of length not exceeding 2. Also prove that there is a point of S that cannot be joined with A by a path on S...
IMO 1988 LL NET68 Origin: NET Problem Let S be the set of all sequences {ai | 1 \leqi \leq7, ai = 0 or 1}. The distance between two elements {ai} and {bi} of S is defined as 7 i=1 |ai −bi|. Let T be a subset of S in which any two elements have a distance apart greater than or equal to 3. Prove that T contains at most...
IMO 1974 LL SWE36 Origin: SWE Problem Consider infinite diagrams D = ... ... ... n20 n21 n22 . . . n10 n11 n12 . . . n00 n01 n02 . . . where all but a finite number of the integers nij, i = 0, 1, 2, . . ., j = 0, 1, 2, . . ., are equal to 0. Three elements of a diagram are called...
IMO 1988 LL SWE80 Origin: SWE Problem Let S be an infinite set of integers containing zero and such that the distance between successive numbers never exceeds a given fixed number. Consider the following procedure: Given a set X of integers, we construct a new set consisting of all numbers x \pm s, where x belongs to X and s belongs to S. Starting from S0 = {0} we successively...
IMO 1977 LL GBR19 Origin: GBR Problem Given any integer m > 1 prove that there exist infinitely many positive integers n such that the last m digits of 5n are a sequence am, am−1, . . . , a1 = 5 (0 \leqaj < 10) in which each digit except the last is of opposite parity to its successor (i.e., if ai is even, then ai−1 is odd, and...
IMO 1984 LL MOR37 Origin: MOR Problem Denote by [x] the greatest integer not exceeding x. For all real k > 1, define two sequences: an(k) = [nk] and bn(k) = nk k −1 . If A(k) = {an(k) : n \inN} and B(k) = {bn(k) : n \inN}, prove that A(k) and B(k) form a partition of N if and only if k is irrational.
IMO 1985 LL SWE77 Origin: SWE Problem Two equilateral triangles are inscribed in a circle with radius r. Let A be the area of the set consisting of all points interior to both triangles. Prove that 2A \geqr2\sqrt 3.
IMO 1970 LL SWE49 Origin: SWE Problem For n \inN, let f(n) be the number of positive integers k \leqn that do not contain the digit 9. Does there exist a positive real number p such that f(n) n \geqp for all positive integers n?
IMO 1985 LL USA88 Origin: USA Problem Determine the range of w(w + x)(w + y)(w + z), where x, y, z, and w are real numbers such that x + y + z + w = x7 + y7 + z7 + w7 = 0.
IMO 1978 LL SWE33 Origin: SWE Problem A sequence (an)\infty of real numbers is called convex if 2an \leq an−1 +an+1 for all positive integers n. Let (bn)\infty 0 be a sequence of positive numbers and assume that the sequence (\alphanbn)\infty 0 is convex for any choice of \alpha > 0. Prove that the sequence (log bn)\infty 0 is convex.
IMO 1967 LL ROM45 Origin: ROM Problem (a) Solve the equation sin3 x + sin3 2\pi 3 + x sin3 4\pi 3 + x 3 4 cos 2x = 0. (b) Suppose the solutions are in the form of arcs AB of the trigonometric circle (where A is the beginning of arcs of the trigonometric circle), and P is a regular n-gon inscribed in the circle with one...
IMO 1989 LL IND49 Origin: IND Problem Let A, B denote two distinct fixed points in space. Let X, P denote variable points (in space), while K, N, n denote positive integers. Call (X, K, N, P) admissible if (N −K)PA + K \cdot PB \geqN \cdot PX. Call (X, K, N) admissible if (X, K, N, P) is admissible for all choices of P. Call (X, N) admissible if...
IMO 1992 LL TWN75 Origin: TWN Problem A sequence {an} of positive integers is defined by an = n + \sqrtn + 1 , n \inN. Determine the positive integers that occur in the sequence.
IMO 1974 LL USS43 Origin: USS Problem An (n2 +n+1)\times(n2 +n+1) matrix of zeros and ones is given. If no four ones are vertices of a rectangle, prove that the number of ones does not exceed (n + 1)(n2 + n + 1).
IMO 1969 LL BUL8 Origin: BUL Problem Find all functions f defined for all x that satisfy the condition xf(y) + yf(x) = (x + y)f(x)f(y), for all x and y. Prove that exactly two of them are continuous.
IMO 1969 LL HUN35 Origin: HUN Problem Prove that 1 + 1 23 + 1 33 + \cdot \cdot \cdot + 1 n3 < 5 4.
IMO 1967 LL POL41 Origin: POL Problem A line l is drawn through the intersection point H of the altitudes of an acute-angled triangle. Prove that the symmetric images la, lb, lc of l with respect to sides BC, CA, AB have one point in common, which lies on the circumcircle of ABC. Solution It is well known that the points K, L, M, symmetric to H with respect to...
IMO 1992 LL IRE32 Origin: IRE Problem Let Sn = {1, 2, . . ., n} and fn : Sn \toSn be defined inductively as follows: f1(1) = 1, fn(2j) = j (j = 1, 2, . . ., [n/2]) and (i) if n = 2k (k \geq1), then fn(2j −1) = fk(j) + k (j = 1, 2, . . ., k); (ii) if n = 2k + 1...
IMO 1966 LL USS55 Origin: USS Problem Given the vertex A and the centroid M of a triangle ABC, find the locus of vertices B such that all the angles of the triangle lie in the interval [40◦, 70◦].
IMO 1966 LL GDR27 Origin: GDR Problem We are given a circle K and a point P lying on a line g. Construct a circle that passes through P and touches K and g.
IMO 1969 LL SWE58 Origin: SWE Problem Six points P1, . . . , P6 are given in 3-dimensional space such that no four of them lie in the same plane. Each of the line segments PjPk is colored black or white. Prove that there exists one triangle PjPkPl whose edges are of the same color.
IMO 1987 LL GBR25 Origin: GBR Problem Numbers d(n, m), with m, n integers, 0 \leqm \leqn, ae defined by d(n, 0) = d(n, n) = 0 for all n \geq0 and md(n, m) = md(n −1, m) + (2n −m)d(n −1, m −1) for all 0 < m < n. Prove that all the d(n, m) are integers.
IMO 1974 LL BUL3 Origin: BUL Problem Let ABCD be an arbitrary quadrilateral. Let squares ABB1A2, BCC1B2, CDD1C2, DAA1D2 be constructed in the exterior of the quadrilateral. Furthermore, let AA1PA2 and CC1QC2 be parallelograms. For any arbitrary point P in the interior of ABCD, parallelograms RASC and RPTQ are constructed. Prove that these two parallelograms have two vertices in common.
IMO 1988 LL MEX61 Origin: MEX Problem Prove that the numbers A, B, and C are equal, where we define A as the number of ways that we can cover a 2 \times n rectangle with 2 \times 1 rectangles, B as the number of sequences of ones and twos that add up to n, and C as . m + m+1 \cdot \cdot \cdot + 2m 2m...
IMO 1978 LL BUL2 Origin: BUL Problem If f(x) = (x + 2x2 + \cdot \cdot \cdot + nxn)2 = a2x2 + a3x3 + \cdot \cdot \cdot + a2nx2n, prove that an+1 + an+2 + \cdot \cdot \cdot + a2n = n + 1 5n2 + 5n + 2 .
IMO 1985 LL CYP19 Origin: CYP Problem Solve the system of simultaneous equations \sqrtx −1/y − 2w + 3z = 1, x + 1/y2 −4w2 −9z2 = 3, x\sqrtx −1/y3 −8w3 + 27z3 = −5, x2 + 1/y4 −16w4 −81z4 = 15.
IMO 1989 LL FRA22 Origin: FRA Problem Let ABC be an equilateral triangle with side length equal to a natural number N. Consider the set S of all points M inside the triangle ABC such that −−\to AM = N (n−−\to AB + m−\to AC), where m, n are integers and 0 \leqm, n, m+n \leqN. Every point of S is colored in one of the three colors blue, white,...
IMO 1967 LL ITA26 Origin: ITA Problem Let ABCD be a regular tetrahedron. To an arbitrary point M on one edge, say CD, corresponds the point P = P(M), which is the intersection of two lines AH and BK, drawn from A orthogonally to BM and from B orthogonally to AM. What is the locus of P as M varies? Solution Let L be the midpoint of the edge AB....
IMO 1989 LL HKG34 Origin: HKG Problem Given an acute triangle find a point inside the triangle such that the sum of the distances from this point to the three vertices is the least.
IMO 1982 LL TUN43 Origin: TUN Problem (a) What is the maximal number of acute angles in a convex polygon? (b) Consider m points in the interior of a convex n-gon. The n-gon is partitioned into triangles whose vertices are among the n + m given points (the vertices of the n-gon and the given points). Each of the m points in the interior is a vertex of at least...
IMO 1992 LL THA70 Origin: THA Problem Let two circles A and B with unequal radii r and R, respec- tively, be tangent internally at the point A0. If there exists a sequence of distinct circles (Cn) such that each circle is tangent to both A and B, and each circle Cn+1 touches circle Cn at the point An, prove that \infty n=1 |An+1An| < 4\piRr R + r...
IMO 1974 LL SWE35 Origin: SWE Problem If p and q are distinct prime numbers, then there are integers x0 and y0 such that 1 = px0 + qy0. Determine the maximum value of b −a, where a and b are positive integers with the following property: If a \leqt \leqb, and t is an integer, then there are integers x and y with 0 \leqx \leqq −1 and 0...
IMO 1983 LL GDR34 Origin: GDR Problem In a plane are given n points Pi (i = 1, 2, . . . , n) and two angles \alpha and \beta. Over each of the segments PiPi=1 (Pn+1 = P1) a point Qi is constructed such that for all i: (i) upon moving from Pi to Pi+1, Qi is seen on the same side of PiPi+1, (ii) \anglePi+1PiQi = \alpha, (iii)...
IMO 1971 LL CUB12 Origin: CUB Problem A system of n numbers x1, x2, . . . , xn is given such that x1 = logxn−1 xn, x2 = logxn x1, . . . , xn = logxn−2 xn−1. Prove that $n k=1 xk = 1.
IMO 1987 LL AUS3 Origin: AUS Problem A town has a road network that consists entirely of one-way streets that are used for bus routes. Along these routes, bus stops have been set up. If the one-way signs permit travel from bus stop X to bus stop Y ̸= X, then we shall say Y can be reached from X. We shall use the phrase Y comes after X when...
IMO 1976 LL GDR19 Origin: GDR Problem For a positive integer n, let 6(n) be the natural number whose decimal representation consists of n digits 6. Let us define, for all natural numbers m, k with 1 \leqk \leqm, m k = 6(m) \cdot 6(m−1) \cdot \cdot \cdot 6(m−k+1) 6(1) \cdot 6(2) \cdot \cdot \cdot 6(k) . Prove that for all m, k, m k is a natural number whose...
IMO 1985 LL CAN12 Origin: CAN Problem Find the maximum value of sin2 \theta1 + sin2 \theta2 + \cdot \cdot \cdot + sin2 \thetan subject to the restrictions 0 \leq\thetai \leq\pi, \theta1 + \theta2 + \cdot \cdot \cdot + \thetan = \pi.
IMO 1986 LL BEL6 Origin: BEL Problem In an urn there are one ball marked 1, two balls marked 2, and so on, up to n balls marked n. Two balls are randomly drawn without replacement. Find the probability that the two balls are assigned the same number.
IMO 1985 LL SPA73 Origin: SPA Problem Let A1A2, B1B2, C1C2 be three equal segments on the three sides of an equilateral triangle. Prove that in the triangle formed by the lines B2C1, C2A1, A2B1, the segments B2C1, C2A1, A2B1 are proportional to the sides in which they are contained.
IMO 1977 LL SWE40 Origin: SWE Problem The numbers 1, 2, 3, . . ., 64 are placed on a chessboard, one number in each square. Consider all squares on the chessboard of size 2 \times 2. Prove that there are at least three such squares for which the sum of the 4 numbers contained exceeds 100. Solution Let us divide the chessboard into 16 squares Q1, Q2, . ....
IMO 1970 LL NET34 Origin: NET Problem In connection with a convex pentagon ABCDE we consider the set of ten circles, each of which contains three of the vertices of the pentagon on its circumference. Is it possible that none of these circles contains the pentagon? Prove your answer.
IMO 1986 LL GBR32 Origin: GBR Problem Find, with proof, all solutions of the equation 1 x + 2 y −3 z = 1 in positive integers x, y, z.
IMO 1967 LL HUN20 Origin: HUN Problem In space, n points (n \geq3) are given. Every pair of points determines some distance. Suppose all distances are different. Connect every point with the nearest point. Prove that it is impossible to obtain a polygonal line in such a way. 1 Solution The statement so formulated is false. It would be true under the additional assumption that the polygonal line is closed....
IMO 1984 LL MON34 Origin: MON Problem One country has n cities and every two of them are linked by a railroad. A railway worker should travel by train exactly once through the entire railroad system (reaching each city exactly once). If it is impossible for worker to travel by train between two cities, he can travel by plane. What is the minimal number of flights that the worker will...
IMO 1967 LL ITA27 Origin: ITA Problem Which regular polygons can be obtained (and how) by cutting a cube with a plane? Solution Regular polygons with 3, 4, and 6 sides can be obtained by cutting a cube with a plane, as shown in the figure. A polygon with more than 6 sides cannot be obtained in such a way, for a cube has 6 faces. Also, if a pentagon...
IMO 1977 LL FRG11 Origin: FRG Problem Let n and z be integers greater than 1 and (n, z) = 1. Prove: (a) At least one of the numbers zi = 1+z+z2+\cdot \cdot \cdot+zi, i = 0, 1, . . . , n−1, is divisible by n. (b) If (z−1, n) = 1, then at least one of the numbers zi, i = 0, 1, . . ., n−2, is...
IMO 1979 LL HUN38 Origin: HUN Problem Prove the following statement: If a polynomial f(x) with real coefficients takes only nonnegative values, then there exists a positive integer n and polynomials g1(x), g2(x), . . . , gn(x) such that f(x) = g1(x)2 + g2(x)2 + \cdot \cdot \cdot + gn(x)2.
IMO 1983 LL BRA11 Origin: BRA Problem A boy at point A wants to get water at a circular lake and carry it to point B. Find the point C on the lake such that the distance walked by the boy is the shortest possible given that the line AB and the lake are exterior to each other.
IMO 1982 LL FIN24 Origin: FIN Problem Prove that if a person a has infinitely many descendants (chil- dren, their children, etc.), then a has an infinite sequence a0, a1, . . . of descendants (i.e., a = a0 and for all n \geq1, an+1 is always a child of an). It is assumed that no-one can have infinitely many children. Variant 1. Prove that if a has infinitely many...
IMO 1992 LL FIN15 Origin: FIN Problem Prove that there exist 78 lines in the plane such that they have exactly 1992 points of intersection.
IMO 1967 LL BUL4 Origin: BUL Problem Suppose medians ma and mb of a triangle are orthogonal. Prove that: (a) The medians of that triangle correspond to the sides of a right-angled triangle. (b) The inequality 5(a2 + b2 −c2) \geq8ab is valid, where a, b, and c are side lengths of the given triangle. Solution (a) Let ABCD be a parallelogram, and K, L the midpoints of segments BC...
IMO 1971 LL BUL9 Origin: BUL Problem The base of an inclined prism is a triangle ABC. The per- pendicular projection of B1, one of the top vertices, is the midpoint of BC. The dihedral angle between the lateral faces through BC and AB is \alpha, and the lateral edges of the prism make an angle \beta with the base. If r1, r2, r3 are exradii of a perpendicular section...
IMO 1967 LL SWE53 Origin: SWE Problem In making Euclidean constructions in geometry it is permit- ted to use a straightedge and compass. In the constructions considered in this question, no compasses are permitted, but the straightedge is as- sumed to have two parallel edges, which can be used for constructing two parallel lines through two given points whose distance is at least equal to the breadth of the ruler....
IMO 1983 LL USS72 Origin: USS Problem Prove that for all x1, x2, . . . , xn \inR the following inequality holds: n\geqi>j\geq1 cos2(xi −xj) \geqn(n −2) .
IMO 1988 LL VIE92 Origin: VIE Problem Let p \geq2 be a natural number. Prove that there exists an integer n0 such that n0 i=1 i p\sqrti + 1 > p.
IMO 1983 LL AUS3 Origin: AUS Problem (a) Given a tetrahedron ABCD and its four altitudes (i.e., lines through each vertex, perpendicular to the opposite face), assume that the altitude dropped from D passes through the orthocenter H4 of ∆ABC. Prove that this altitude DH4 intersects all the other three altitudes. (b) If we further know that a second altitude, say the one from vertex A to the face BCD,...
IMO 1971 LL CUB13 Origin: CUB Problem One Martian, one Venusian, and one Human reside on Pluton. One day they make the following conversation: Martian : I have spent 1/12 of my life on Pluton. Human : I also have. Venusian : Me too. Martian : But Venusian and I have spend much more time here than you, Human. Human : That is true. However, Venusian and I are of...
IMO 1987 LL BEL7 Origin: BEL Problem Let f : (0, +\infty) \toR be a function having the property that f(x) = f(1/x) for all x > 0. Prove that there exists a function u : [1, +\infty) \toR satisfying u x+1/x = f(x) for all x > 0.
IMO 1982 LL BRA12 Origin: BRA Problem Let there be 3399 numbers arbitrarily chosen among the first 6798 integers 1, 2, . . ., 6798 in such a way that none of them divides an- other. Prove that there are exactly 1982 numbers in {1, 2, . . ., 6798} that must end up being chosen.
IMO 1982 LL FRA29 Origin: FRA Problem Let f : R \toR be a continuous function. Suppose that the restriction of f to the set of irrational numbers is injective. What can we say about f? Answer the analogous question if f is restricted to rationals.
IMO 1988 LL USS87 Origin: USS Problem All the irreducible positive rational numbers such that the prod- uct of the numerator and the denominator is less than 1988 are written in increasing order. Prove that any two adjacent fractions a/b and c/d, a/b < c/d, satisfy the equation bc −ad = 1.
IMO 1969 LL BEL1 Origin: BEL Problem A parabola P1 with equation x2 −2py = 0 and parabola P2 with equation x2 + 2py = 0, p > 0, are given. A line t is tangent to P2. Find the locus of pole M of the line t with respect to P1.
IMO 1986 LL SWE68 Origin: SWE Problem Consider the equation x4 + ax3 + bx2 + ax + 1 = 0 with real coefficients a, b. Determine the number of distinct real roots and their multiplicities for various values of a and b. Display your result graphically in the (a, b) plane.
IMO 1984 LL BUL8 Origin: BUL Problem In the plane of a given triangle A1A2A3 determine (with proof) a straight line l such that the sum of the distances from A1, A2, and A3 to l is the least possible.
IMO 1985 LL CYP18 Origin: CYP Problem The circles (R, r) and (P, \rho), where r > \rho, touch externally at A. Their direct common tangent touches (R, r) at B and (P, \rho) at C. The line RP meets the circle (P, \rho) again at D and the line BC at E. If |BC| = 6|DE|, prove that: (a) the lengths of the sides of the triangle RBE are...
IMO 1979 LL SWE62 Origin: SWE Problem T is a given triangle with vertices P1, P2, P3. Consider an arbi- trary subdivision of T into finitely many subtriangles such that no vertex of a subtriangle lies strictly between two vertices of another subtriangle. To each vertex V of the subtriangles there is assigned a number n(V ) according to the following rules: (i) If V = Pi, then n(V )...
IMO 1970 LL AUT2 Origin: AUT Problem Prove that the two last digits of 999 and 9999 in decimal representation are equal.
IMO 1969 LL USS68 Origin: USS Problem Given 5 points in the plane, no three of which are collinear, prove that we can choose 4 points among them that form a convex quadrilateral.
IMO 1966 LL BUL23 Origin: BUL Problem Three faces of a tetrahedron are right triangles, while the fourth is not an obtuse triangle. (a) Prove that a necessary and sufficient condition for the fourth face to be a right triangle is that at some vertex exactly two angles are right. (b) Prove that if all the faces are right triangles, then the volume of the tetrahedron equals one -sixth the...
IMO 1984 LL USA61 Origin: USA Problem A fair coin is tossed repeatedly until there is a run of an odd number of heads followed by a tail. Determine the expected number of tosses.
IMO 1984 LL SPA55 Origin: SPA Problem Let a, b, c be natural numbers such that a+b+c = 2pq(p30+q30), p > q being two given positive integers. (a) Prove that k = a3 + b3 + c3 is not a prime number. (b) Prove that if a \cdot b \cdot c is maximum, then 1984 divides k.
IMO 1989 LL POR84 Origin: POR Problem Let a, b, c, r, and s be real numbers. Show that if r is a root of ax2 +bx+c = 0 and s is a root of −ax2 +bx+c = 0, then a 2x2 +bx+c = 0 has a root between r and s.
IMO 1985 LL MON50 Origin: MON Problem From each of the vertices of a regular n-gon a car starts to move with constant speed along the perimeter of the n-gon in the same direction. Prove that if all the cars end up at a vertex A at the same time, then they never again meet at any other vertex of the n-gon. Can they meet again at A?
IMO 1978 LL GDR25 Origin: GDR Problem Consider a polynomial P(x) = ax2 + bx + c with a > 0 that has two real roots x1, x2. Prove that the absolute values of both roots are less than or equal to 1 if and only if a + b + c \geq0, a −b + c \geq0, and a −c \geq0.
IMO 1986 LL ROM65 Origin: ROM Problem Let A1A2A3A4 be a quadrilateral inscribed in a circle C. Show that there is a point M on C such that MA1 −MA2 + MA3 −MA4 = 0.
IMO 1969 LL FRA23 Origin: FRA Problem Consider the integer d = ab−1 c , where a, b, and c are positive integers and c \leqa. Prove that the set G of integers that are between 1 and d and relatively prime to d (the number of such integers is denoted by ϕ(d)) can be partitioned into n subsets, each of which consists of b elements. What can be said...
IMO 1974 LL SWE32 Origin: SWE Problem Let a1, a2, . . . , an be n real numbers such that 0 < a \leqak \leqb for k = 1, 2, . . ., n. If m1 = 1 n(a1 + a2 + \cdot \cdot \cdot + an) and m2 = 1 n(a2 1 + a2 2 + \cdot \cdot \cdot + a2 n), prove that m2 \leq(a+b)2 4ab m2...
IMO 1970 LL FRA26 Origin: FRA Problem Consider a finite set of vectors in space {a1, a2, . . . , an} and the set E of all vectors of the form x = \lambda1a1 +\lambda2a2 +\cdot \cdot \cdot+\lambdanan, where \lambdai are nonnegative numbers. Let F be the set consisting of all the vectors in E and vectors parallel to a given plane P. Prove that there exists a set...
IMO 1988 LL GRE26 Origin: GRE Problem Let AB and CD be two perpendicular chords of a circle with center O and radius r, and let X, Y, Z, W denote in cyclical order the four parts into which the disk is thus divided. Find the maximum and minimum of the quantity A(Z) A(Y )+A(W), where A(U) denotes the area of U.
IMO 1966 LL CZS43 Origin: CZS Problem Five points in a plane are given, no three of which are collinear. Every two of them are joined by a segment, colored either red or gray, so that no three segments form a triangle colored in one color. (a) Prove that (1) every point is a vertex of exactly two red and two gray segments, and (2) the red segments form a...
IMO 1984 LL MOR38 Origin: MOR Problem Determine all continuous functions f such that \forall(x, y) \inR2 f(x + y)f(x −y) = (f(x)f(y))2 .
IMO 1977 LL USS51 Origin: USS Problem Several segments, which we shall call white, are given, and the sum of their lengths is 1. Several other segments, which we shall call black, are given, and the sum of their lengths is 1. Prove that every such system of segments can be distributed on the segment that is 1.51 long in the following way: Segments of the same color are disjoint,...
IMO 1987 LL ROM55 Origin: ROM Problem Two moving bodies M1, M2 are displaced uniformly on two coplanar straight lines. Describe the union of all straight lines M1M2.
IMO 1970 LL AUT3 Origin: AUT Problem Prove that for a, b \inN, a!b! divides (a + b)!.
IMO 1976 LL USS42 Origin: USS Problem For a point O inside a triangle ABC, denote by A1, B1, C1 the respective intersection points of AO, BO, CO with the corresponding sides. Let n1 = AO A1O, n2 = BO B1O, n3 = CO C1O . What possible values of n1, n2, n3 can all be positive integers?
IMO 1971 LL YUG55 Origin: YUG Problem Prove that the polynomial x4 + \lambdax3 + µx2 + \nux + 1 has no real roots if \lambda, µ, \nu are real numbers satisfying |\lambda| + |µ| + |\nu| \leq \sqrt 2.
IMO 1983 LL LUX43 Origin: LUX Problem Given a square ABCD, let P, Q, R, and S be four variable points on the sides AB, BC, CD, and DA, respectively. Determine the positions of the points P, Q, R, and S for which the quadrilateral PQRS is a parallelogram, a rectangle, a square, or a trapezoid.
IMO 1989 LL VIE107 Origin: VIE Problem Let E be the set of all triangles whose only points with integer coordinates (in the Cartesian coordinate system in space), in its interior or on its sides, are its three vertices, and let f be the function of area of a triangle. Determine the set of values f(E) of f.
IMO 1978 LL CUB6 Origin: CUB Problem Prove that for all X > 1 there exists a triangle whose sides have lengths P1(X) = X4+X3+2X2+X+1, P2(X) = 2X3+X2+2X+1, and P3(X) = X4−1. Prove that all these triangles have the same greatest angle and calculate it.
IMO 1974 LL NET21 Origin: NET Problem Let M be a nonempty subset of Z+ such that for every element x in M, the numbers 4x and [\sqrtx] also belong to M. Prove that M = Z+.
IMO 1989 LL GRE29 Origin: GRE Problem Let L denote the set of all lattice points of the plane (points with integral coordinates). Show that for any three points A, B, C of L there is a fourth point D, different from A, B, C, such that the interiors of the segments AD, BD, CD contain no points of L. Is the statement true if one considers four points of...
IMO 1983 LL CAN18 Origin: CAN Problem Let b \geq2 be a positive integer. (a) Show that for an integer N, written in base b, to be equal to the sum of the squares of its digits, it is necessary either that N = 1 or that N have only two digits. (b) Give a complete list of all integers not exceeding 50 that, relative to some base b, are...
IMO 1984 LL MON35 Origin: MON Problem Prove that there exist distinct natural numbers m1, m2, . . . , mk satisfying the conditions \pi−1984 < 25 − 1 m1 1 m2 \cdot \cdot \cdot + 1 mk < \pi−1960 where \pi is the ratio between circle and its diameter.
IMO 1984 LL FRA19 Origin: FRA Problem Let ABC be an isosceles triangle with right angle at point A. Find the minimum of the function F given by F(M) = BM + CM − \sqrt 3AM.
IMO 1979 LL CZS16 Origin: CZS Problem Let Q be a square with side length 6. Find the smallest integer n such that in Q there exists a set S of n points with the property that any square with side 1 completely contained in Q contains in its interior at least one point from S.
IMO 1969 LL SWE63 Origin: SWE Problem Prove that there are infinitely many positive integers that cannot be expressed as the sum of squares of three positive integers.
IMO 1971 LL HUN23 Origin: HUN Problem Find all integer solutions of the equation x2 + y2 = (x −y)3.
IMO 1974 LL GBR17 Origin: GBR Problem Show that there exists a set S of 15 distinct circles on the surface of a sphere, all having the same radius and such that 5 touch exactly 5 others, 5 touch exactly 4 others, and 5 touch exactly 3 others.
IMO 1974 LL BUL5 Origin: BUL Problem A straight cone is given inside a rectangular parallelepiped B, with the apex at one of the vertices, say T , of the parallelepiped, and the base touching the three faces opposite to T . Its axis lies at the long diagonal through T . If V1 and V2 are the volumes of the cone and the parallelepiped respectively, prove that V1 \leq...
IMO 1989 LL POR83 Origin: POR Problem Poldavia is a strange kingdom. Its currency unit is the bourbaki and there exist only two types of coins: gold ones and silver ones. Each gold coin is worth n bourbakis and each silver coin is worth m bourbakis (n and m are positive integers). Using gold and solver coins, it is possible to obtain sums such as 10000 bourbakis, 1875 bourbakis, 3072...
IMO 1989 LL VIE111 Origin: VIE Problem Find the greatest number c such that for all natural numbers n, {n \sqrt 2} \geqc n (where {n \sqrt 2} = n \sqrt 2 −[n \sqrt 2]; [x] is the integer part of x). For this number c, find all natural numbers n for which {n \sqrt 2} = c n.
IMO 1970 LL BUL15 Origin: BUL Problem Given a triangle ABC, let R be the radius of its circumcir- cle, O1, O2, O3 the centers of its exscribed circles, and q the perimeter of \triangleO1O2O3. Prove that q \leq6 \sqrt 3 R.
IMO 1966 LL BUL22 Origin: BUL Problem Assume that two parallelograms P, P ′ of equal areas have sides a, b and a′, b′ respectively such that a′ \leqa \leqb \leqb′ and a segment of length b′ can be placed inside P. Prove that P and P ′ can be partitioned into four pairwise congruent parts.
IMO 1966 LL ROM30 Origin: ROM Problem If n is a natural number, prove that (a) log10(n + 1) > 10n + log10 n; (b) log n! > 3n 1 2 + 1 3 + \cdot \cdot \cdot + 1 n −1 .
IMO 1966 LL POL24 Origin: POL Problem There are n \geq2 people in a room. Prove that there exist two among them having equal numbers of friends in that room. (Friendship is always mutual.)
IMO 1989 LL USA103 Origin: USA Problem An accurate 12-hour analog clock has an hour hand, a minute hand, and a second hand that are aligned at 12:00 o’clock and make one revolution in 12 hours, 1 hour, and 1 minute, respectively. It is well known, and not difficult to prove, that there is no time when the three hands are equally spaced around the clock, with each separating angle...
IMO 1984 LL CAN15 Origin: CAN Problem Consider all the sums of the form 1985 k=1 ekk5 = \pm15 \pm 25 \pm \cdot \cdot \cdot \pm 19855, where ek = \pm1. What is the smallest nonnegative value attained by a sum of this type?
IMO 1987 LL FIN10 Origin: FIN Problem In a Cartesian coordinate system, the circle C1 has center O1(−2, 0) and radius 3. Denote the point (1, 0) by A and the origin by O. Prove that there is a constant c > 0 such that for every X that is exterior to C1, OX −1 \geqc min{AX, AX2}. Find the largest possible c.
IMO 1984 LL MOR39 Origin: MOR Problem Let ABC be an isosceles triangle, AB = AC, \angleA = 20◦. Let D be a point on AB, and E a point on AC such that \angleACD = 20◦and \angleABE = 30◦. What is the measure of the angle \angleCDE?
IMO 1988 LL FRG16 Origin: FRG Problem Show that if n runs through all positive integers, f(n) = n + n/3 + 1/2 runs through all positive integers skipping the terms of the sequence an = 3n2 −2n.
IMO 1984 LL FRA18 Origin: FRA Problem Let c be the inscribed circle of the triangle ABC, d a line tan- gent to c which does not pass through the vertices of triangle ABC. Prove the existence of points A1, B1, C1, respectively, on the lines BC, CA, AB satisfying the following two properties: (i) Lines AA1, BB1, and CC1 are parallel. (ii) Lines AA1, BB1, and CC1 meet d...
IMO 1992 LL USA78 Origin: USA Problem Let Fn be the nth Fibonacci number, defined by F1 = F2 = 1 and Fn = Fn−1 + Fn−2 for n > 2. Let A0, A1, A2, . . . be a sequence of points on a circle of radius 1 such that the minor arc from Ak−1 to Ak runs clockwise and such that µ(Ak−1Ak) = 4F2k+1 F 2 2k+1 +...
IMO 1982 LL GBR33 Origin: GBR Problem A sequence (un) of integers is defined for n \geq0 by u0 = 0, u1 = 1, and un −2un−1 + (1 −c)un−2 = 0 (n \geq2), where c is a fixed integer independent of n. Find the least value of c for which both of the following statements are true: (i) If p is a prime less than or equal to P,...
IMO 1989 LL SWE94 Origin: SWE Problem Prove that a < b implies that a3 −3a \leqb3 −3b + 4. When does equality occur?
IMO 1970 LL NET32 Origin: NET Problem Let there be given an acute angle \angleAOB = 3\alpha, where OA = OB. The point A is the center of a circle with radius OA. A line s parallel to OA passes through B. Inside the given angle a variable line t is drawn through O. It meets the circle in O and C and the given line s in D, where...
IMO 1971 LL GDR18 Origin: GDR Problem Let a1, a2, . . . , an be positive numbers, mg = (a1a2 \cdot \cdot \cdot an)1/n their geometric mean, and ma = (a1 + a2 + \cdot \cdot \cdot + an)/n their arithmetic mean. Prove that (1 + mg)n \leq(1 + a1) \cdot \cdot \cdot (1 + an) \leq(1 + ma)n.
IMO 1970 LL ROM48 Origin: ROM Problem Let a polynomial p(x) with integer coefficients take the value 5 for five different integer values of x. Prove that p(x) does not take the value 8 for any integer x.
IMO 1992 LL NET52 Origin: NET Problem Let n be an integer > 1. In a circular arrangement of n lamps L0, . . . , Ln−1, each one of which can be either ON or OFF, we start with the situation that all lamps are ON, and then carry out a sequence of steps, Step0, Step1, . . . . If Lj−1 (j is taken mod n) is ON,...
IMO 1992 LL ROM64 Origin: ROM Problem For any positive integer n consider all representations n = a1 + \cdot \cdot \cdot + ak, where a1 > a2 > \cdot \cdot \cdot > ak > 0 are integers such that for all i \in{1, 2, . . ., k −1}, the number ai is divisible by ai+1. Find the longest such representation of the number 1992.
IMO 1985 LL AUS2 Origin: AUS Problem We are given a triangle ABC and three rectangles R1, R2, R3 with sides parallel to two fixed perpendicular directions and such that their union covers the sides AB, BC, and CA; i.e., each point on the perimeter of ABC is contained in or on at least one of the rectangles. Prove that all points inside the triangle are also covered by the...
IMO 1972 LL GBR18 Origin: GBR Problem We have p players participating in a tournament, each player playing against every other player exactly once. A point is scored for each victory, and there are no draws. A sequence of nonnegative integers s1 \leqs2 \leqs3 \leq\cdot \cdot \cdot \leqsp is given. Show that it is possible for this sequence to be a set of final scores of the players in the...
IMO 1986 LL MOR56 Origin: MOR Problem Let A1A2A3A4A5A6 be a hexagon inscribed into a circle with center O. Consider the circular arc with endpoints A1, A6 not containing
IMO 1984 LL USS67 Origin: USS Problem With the medians of an acute-angled triangle another triangle is constructed. If R and Rm are the radii of the circles circumscribed about the first and the second triangle, respectively, prove that Rm > 5 6R.
IMO 1983 LL BRA12 Origin: BRA Problem The number 0 or 1 is to be assigned to each of the n vertices of a regular polygon. In how many different ways can this be done (if we consider two assignments that can be obtained one from the other through rotation in the plane of the polygon to be identical)?
IMO 1966 LL HUN20 Origin: HUN Problem We are given three equal rectangles with the same center in three mutually perpendicular planes, with the long sides also mutually perpendicular. Consider the polyhedron with vertices at the vertices of these rectangles. (a) Find the volume of this polyhedron; (b) can this polyhedron be regular, and under what conditions?
IMO 1989 LL INA45 Origin: INA Problem The expressions a + b + c, ab + ac + bc, and abc are called the elementary symmetric expressions on the three letters a, b, c; symmetric because if we interchange any two letters, say a and c, the expressions remain algebraically the same. The common degree of its terms is called the order of the expression. Let Sk(n) denote the elementary...
IMO 1987 LL ROM57 Origin: ROM Problem The bisectors of the angles B, C of a triangle ABC intersect the opposite sides in B′, C′ respectively. Prove that the straight line B′C′ intersects the inscribed circle in two different points.
IMO 1982 LL AUS2 Origin: AUS Problem Given a finite number of angular regions A1, . . . , Ak in a plane, each Ai being bounded by two half-lines meeting at a vertex and provided with a + or −sign, we assign to each point P of the plane and not on a bounding half-line the number k −l, where k is the number of + regions and l...
IMO 1988 LL HKG30 Origin: HKG Problem Find the total number of different integers that the function f(x) = [x] + [2x] + 5x [3x] + [4x] takes for 0 \leqx \leq100.
IMO 1982 LL USS50 Origin: USS Problem Let O be the midpoint of the axis of a right circular cylinder. Let A and B be diametrically opposite points of one base, and C a point of the other base circle that does not belong to the plane OAB. Prove that the sum of dihedral angles of the trihedral OABC is equal to 2\pi.
IMO 1988 LL INA41 Origin: INA Problem (a) Let ABC be a triangle with AB = 12 and AC = 16. Suppose M is the midpoint of side BC and points E and F are chosen on sides AC and AB respectively, and suppose that the lines EF and AM intersect at G. If AE = 2AF then find the ratio EG/GF. (b) Let E be a point external to...
IMO 1977 LL CZS8 Origin: CZS Problem A hexahedron ABCDE is made of two regular congruent tetra- hedra ABCD and ABCE. Prove that there exists only one isometry Z that maps points A, B, C, D, E onto B, C, A, E, D, respectively. Find all points X on the surface of hexahedron whose distance from Z(X) is minimal. Solution There is exactly one point satisfying the given condition on...
IMO 1978 LL TUR39 Origin: TUR Problem A is a 2m-digit positive integer each of whose digits is 1. B is an m-digit positive integer each of whose digits is 4. Prove that A+B + 1 is a perfect square.
IMO 1986 LL GRE39 Origin: GRE Problem Let S be a k-element set. (a) Find the number of mappings f : S \toS such that (i) f(x) ̸= x for x \inS, (ii) f(f(x)) = x for x \inS. (b) The same with the condition (i) left out.
IMO 1969 LL FRA21 Origin: FRA Problem A right-angled triangle OAB has its right angle at the point B. An arbitrary circle with center on the line OB is tangent to the line OA. Let AT be the tangent to the circle different from OA (T is the point of tangency). Prove that the median from B of the triangle OAB intersects AT at a point M such that MB...
IMO 1982 LL GDR35 Origin: GDR Problem If the inradius of a triangle is half of its circumradius, prove that the triangle is equilateral.
IMO 1966 LL GDR25 Origin: GDR Problem Show that tan 7◦30′ = \sqrt 6 + \sqrt 2 − \sqrt 3 −2.
IMO 1992 LL SPA67 Origin: SPA Problem In a triangle, a symmedian is a line through a vertex that is symmetric to the median with the respect to the internal bisector (all relative to the same vertex). In the triangle ABC, the median ma meets BC at A′ and the circumcircle again at A1. The symmedian sa meets BC at M and the circumcircle again at A2. Given that the...
IMO 1972 LL GBR19 Origin: GBR Problem Let S be a subset of the real numbers with the following properties: (i) If x \inS and y \inS, then x −y \inS; (ii) If x \inS and y \inS, then xy \inS; (iii) S contains an exceptional number x′ such that there is no number y in S satisfying x′y + x′ + y = 0; (iv) If x \inS and...
IMO 1976 LL SWE34 Origin: SWE Problem Let {an}\infty and {bn}\infty be two sequences determined by the recursion formulas an+1 = an + bn, bn+1 = 3an + bn, n = 0, 1, 2, . . ., and the initial values a0 = b0 = 1. Prove that there exists a uniquely determined constant c such that n|can−bn| < 2 for all nonnegative integers n.
IMO 1969 LL USS65 Origin: USS Problem Prove that for a > b2, ! a −b a + b a −b \sqrt a + \cdot \cdot \cdot = a −3 4b2 −1 2b.
IMO 1992 LL KOR47 Origin: KOR Problem Find the largest integer not exceeding $1992 n=1 3n+2 3n+1.
IMO 1986 LL ROM63 Origin: ROM Problem Let AA′, BB′, CC′ be the bisectors of the angles of a triangle ABC (A′ \inBC, B′ \inCA, C′ \inAB). Prove that each of the lines A′B′, B′C′, C′A′ intersects the incircle in two points.
IMO 1982 LL CAN18 Origin: CAN Problem You are given an algebraic system admitting addition and multiplication for which all the laws of ordinary arithmetic are valid except commutativity of multiplication. Show that (a + ab−1a)−1 + (a + b)−1 = a−1, where x−1 is the element for which x−1x = xx−1 = e, where e is the element of the system such that for all a the equality ea...
IMO 1985 LL BRA8 Origin: BRA Problem Let K be a convex set in the xy-plane, symmetric with respect to the origin and having area greater than 4. Prove that there exists a point (m, n) ̸= (0, 0) in K such that m and n are integers.
IMO 1967 LL MON30 Origin: MON Problem Given m+n numbers ai (i = 1, 2, . . . , m), bj (j = 1, 2, . . ., n), determine the number of pairs (ai, bj) for which |i −j| \geqk, where k is a nonnegative integer. Solution We assume w.l.o.g. that m \leqn. Let r and s be the numbers of pairs for which i −j \geqk and of...
IMO 1985 LL IRE39 Origin: IRE Problem Given a triangle ABC and external points X, Y , and Z such that ∡BAZ = ∡CAY , ∡CBX = ∡ABZ, and ∡ACY = ∡BCX, prove that AX, BY , and CZ are concurrent.
IMO 1966 LL GDR2 Origin: GDR Problem Given n positive real numbers a1, a2, . . . , an such that a1a2 \cdot \cdot \cdot an = 1, prove that (1 + a1)(1 + a2) \cdot \cdot \cdot (1 + an) \geq2n.
IMO 1977 LL USS47 Origin: USS Problem A square ABCD is given. A line passing through A intersects CD at Q. Draw a line parallel to AQ that intersects the boundary of the square at points M and N such that the area of the quadrilateral AMNQ is maximal. Solution M lies on AB and N lies on BC. If CQ \leq2CD/3, then BM = CQ/2. If CQ > 2CD/3,...
IMO 1984 LL POL45 Origin: POL Problem Let X be an arbitrary nonempty set contained in the plane and let sets A1, A2, . . . , Am and B1, B2, . . . , Bn be its images under parallel translations. Let us suppose that A1 \cupA2 \cup\cdot \cdot \cdot \cupAm \subsetB1 \cupB2 \cup\cdot \cdot \cdot \cupBn and that the sets A1, A2, . . . , Am are...
IMO 1974 LL VIE49 Origin: VIE Problem Determine an equation of third degree with integral coefficients having roots sin \pi 14, sin 5\pi 14 and sin −3\pi 14 .
IMO 1984 LL FRG21 Origin: FRG Problem (1) Start with a white balls and b black balls. (2) Draw one ball at random. (3) If the ball is white, then stop. Otherwise, add two black balls and go to step 2. Let S be the number of draws before the process terminates. For the cases a = b = 1 and a = b = 2 only, find an =...
IMO 1967 LL GDR16 Origin: GDR Problem Prove the following statement: If r1 and r2 are real numbers whose quotient is irrational, then any real number x can be approximated arbitrarily well by numbers of the form zk1,k2 = k1r1+k2r2, k1, k2 integers; i.e., for every real number x and every positive real number p two integers k1 and k2 can be found such that |x −(k1r1 + k2r2)| <...
IMO 1972 LL SWE38 Origin: SWE Problem Congruent rectangles with sides m (cm) and n (cm) are given (m, n positive integers). Characterize the rectangles that can be constructed from these rectangles (in the fashion of a jigsaw puzzle). (The number of rectangles is unbounded.)
IMO 1986 LL FIN20 Origin: FIN Problem For any angle \alpha with 0 < \alpha < 180◦, we call a closed convex planar set an \alpha-set if it is bounded by two circular arcs (or an arc and a line segment) whose angle of intersection is \alpha. Given a (closed) triangle T , find the greatest \alpha such that any two points in T are contained in an \alpha-set S...
IMO 1983 LL ROM56 Origin: ROM Problem Consider the expansion (1 + x + x2 + x3 + x4)496 = a0 + a1x + \cdot \cdot \cdot + a1984x1984. (a) Determine the greatest common divisor of the coefficients a3, a8, a13, . . . , a1983. (b) Prove that 10340 < a992 < 10347.
IMO 1979 LL BEL2 Origin: BEL Problem For a finite set E of cardinality n \geq3, let f(n) denote the maximum number of 3-element subsets of E, any two of them having exactly one common element. Calculate f(n).
IMO 1979 LL CZS14 Origin: CZS Problem Let S be a set of n2 + 1 closed intervals (n a positive integer). Prove that at least one of the following assertions holds: (i) There exists a subset S′ of n + 1 intervals from S such that the inter- section of the intervals in S′ is nonempty. (ii) There exists a subset S′′ of n + 1 intervals from S...
IMO 1970 LL USS57 Origin: USS Problem Let the numbers 1, 2, . . . , n2 be written in the cells of an n \times n square board so that the entries in each column are arranged increasingly. What are the smallest and greatest possible sums of the numbers in the kth row? (k a positive integer, 1 \leqk \leqn.)
IMO 1985 LL FRA25 Origin: FRA Problem Find eight positive integers n1, n2, . . . , n8 with the follow- ing property: For every integer k, −1985 \leqk \leq1985, there are eight integers \alpha1, \alpha2, . . . , \alpha8, each belonging to the set {−1, 0, 1}, such that k = 8 i=1 \alphaini.
IMO 1970 LL BUL13 Origin: BUL Problem A triangle ABC is given. Each side of ABC is divided into equal parts, and through each of the division points are drawn lines parallel to AB, BC, and CA, thus cutting ABC into small triangles. To each of the vertices of these triangles is assigned 1, 2, or 3, so that: (1) to A, B, C are assigned 1, 2 and 3...
IMO 1978 LL FRA18 Origin: FRA Problem Given a natural number n, prove that the number M(n) of points with integer coordinates inside the circle (O(0, 0), \sqrtn) satisfies \pin −5\sqrtn + 1 < M(n) < \pin + 4\sqrtn + 1.
IMO 1969 LL USS64 Origin: USS Problem Prove that for a natural number n > 2, (n!)! > n[(n −1)!]n!.
IMO 1989 LL MON68 Origin: MON Problem If 0 < k \leq1 and ai are positive real numbers, i = 1, 2, . . . , n, prove that a1 a2 + \cdot \cdot \cdot + an k \cdot \cdot \cdot + an a1 + \cdot \cdot \cdot + an−1 k \geq n (n −1)k .
IMO 1966 LL CZS11 Origin: CZS Problem Does there exist an integer z that can be written in two different ways as z = x! + y!, where x, y are natural numbers with x \leqy?
IMO 1967 LL SWE52 Origin: SWE Problem In the plane a point O and a sequence of points P1, P2, P3, . . . are given. The distances OP1, OP2, OP3, . . . are r1, r2, r3, . . . , where r1 \leq r2 \leqr3 \leq\cdot \cdot \cdot . Let \alpha satisfy 0 < \alpha < 1. Suppose that for every n the distance from the point Pn...
IMO 1986 LL ISR49 Origin: ISR Problem Let C1, C2 be circles of radius 1/2 tangent to each other and both tangent internally to a circle C of radius 1. The circles C1 and C2 are the first two terms of an infinite sequence of distinct circles Cn defined as follows: Cn+2 is tangent externally to Cn and Cn+1 and internally to C. Show that the radius of each Cn...
IMO 1978 LL CZS11 Origin: CZS Problem Find all natural numbers n < 1978 with the following property: If m is a natural number, 1 < m < n, and (m, n) = 1 (i.e., m and n are relatively prime), then m is a prime number.
IMO 1970 LL NET33 Origin: NET Problem The vertices of a given square are clockwise lettered A, B, C, D. On the side AB is situated a point E such that AE = AB/3. Starting from an arbitrarily chosen point P0 on segment AE and go- ing clockwise around the perimeter of the square, a series of points P0, P1, P2, . . . is marked on the perimeter such...
IMO 1989 LL KOR60 Origin: KOR Problem A real-valued function f on Q satisfies the following conditions for arbitrary \alpha, \beta \inQ: (i) f(0) = 0, (ii) f(\alpha) > 0 if \alpha ̸= 0, (iii) f(\alpha\beta) = f(\alpha)f(\beta), (iv) f(\alpha + \beta) \leqf(\alpha) + f(\beta), (v) f(m) \leq1989 for all m \inZ. Prove that f(\alpha + \beta) = max{f(\alpha), f(\beta)} if f(\alpha) ̸= f(\beta). Here, Z, Q denote the sets...
IMO 1987 LL FRA17 Origin: FRA Problem Consider the number \alpha obtained by writing one after another the decimal representations of 1, 1987, 19872, . . . to the right the decimal point. Show that \alpha is irrational.
IMO 1986 LL FRA24 Origin: FRA Problem Two families of parallel lines are given in the plane, consisting of 15 and 11 lines, respectively. In each family, any two neighboring lines are at a unit distance from one another; the lines of the first family are perpendicular to the lines of the second family. Let V be the set of 165 intersection points of the lines under consideration. Show that...
IMO 1979 LL GRE34 Origin: GRE Problem Notice that in the fraction 16 64 we can perform a simplification as 1̸6 ̸64 = 1 4 obtaining a correct equality. Find all fractions whose numer- ators and denominators are two-digit positive integers for which such a simplification is correct.
IMO 1971 LL NET30 Origin: NET Problem Prove that the system of equations 2yz + x −y −z = a, 2xz −x + y −z = a, 2xy −x −y + z = a, a being a parameter, cannot have five distinct solutions. For what values of a does this system have four distinct integer solutions?
IMO 1987 LL VIE73 Origin: VIE Problem Let f(x) be a periodic function of period T > 0 defined over R. Its first derivative is continuous on R. Prove that there exist x, y \in[0, T ) such that x ̸= y and f(x)f ′(y) = f(y)f ′(x).
IMO 1978 LL FRA15 Origin: FRA Problem Prove that for every positive integer n coprime to 10 there exists a multiple of n that does not contain the digit 1 in its decimal representation.
IMO 1985 LL NET57 Origin: NET Problem The solid S is defined as the intersection of the six spheres with the six edges of a regular tetrahedron T , with edge length 1, as diameters. Prove that S contains two points at a distance \sqrt 6. (NET 1a) Using the same assumptions, prove that no pair of points in S has a distance larger than \sqrt 6.
IMO 1988 LL SPA77 Origin: SPA Problem Consider h + 1 chessboards. Number the squares of each board from 1 to 64 in such a way that when the perimeters of any two boards of the collection are brought into coincidence in any possible manner, no two squares in the same position have the same number. What is the maximum value of h?
IMO 1992 LL IRN36 Origin: IRN Problem Find all rational solutions of a2 + c2 + 17(b2 + d2) = 21, ab + cd = 2.
IMO 1972 LL ROM32 Origin: ROM Problem If n1, n2, . . . , nk are natural numbers and n1+n2+\cdot \cdot \cdot+nk = n, show that max n1+\cdot\cdot\cdot+nk=n n1n2 \cdot \cdot \cdot nk = (t + 1)rtk−r, where t = [n/k] and r is the remainder of n upon division by k; i.e., n = tk + r, 0 \leqr \leqk −1.
IMO 1967 LL ROM42 Origin: ROM Problem Decompose into real factors the expression 1 −sin5 x−cos5 x. Solution E = (1−sin x)(1−cos x)[3+2(sin x+cos x)+2 sin x cos x+sin x cos x(sin x+ cos x)].
IMO 1992 LL TUR72 Origin: TUR Problem In a school six different courses are taught: mathematics, physics, biology, music, history, geography. The students were required to rank these courses according to their preferences, where equal preferences were allowed. It turned out that: (i) mathematics was ranked among the most preferred courses by all stu- dents; (ii) no student ranked music among the least preferred ones; (iii) all students preferred history...
IMO 1967 LL ROM46 Origin: ROM Problem If x, y, z are real numbers satisfying the relations x+y+z = 1 and arctanx + arctan y + arctan z = \pi/4, prove that x2n+1 + y2n+1 + z2n+1 = 1 for all positive integers n. Solution Let us set arctan x = a, arctan y = b, arctanz = c. Then tan(a+b) = x+y 1−xy and tan(a + b + c)...
IMO 1978 LL GBR22 Origin: GBR Problem Two nonzero integers x, y (not necessarily positive) are such that x + y is a divisor of x2 + y2, and the quotient x2+y2 x+y is a divisor of
IMO 1978 LL VIE51 Origin: VIE Problem Find the relations among the angles of the triangle ABC whose altitude AH and median AM satisfy \angleBAH = \angleCAM.
IMO 1992 LL IND30 Origin: IND Problem Let Pn = (19 + 92)(192 + 922) \cdot \cdot \cdot (19n + 92n) for each positive integer n. Determine, with proof, the least positive integer m, if it exists, for which Pm is divisible by 3333.
IMO 1983 LL KUW39 Origin: KUW Problem If \alpha is the real root of the equation E(x) = x3 −5x −50 = 0 such that xn+1 = (5xn + 50)1/3 and x1 = 5, where n is a positive integer, prove that: (a) x3 n+1 −\alpha3 = 5(xn −\alpha) (b) \alpha < xn+1 < xn
IMO 1966 LL ROM9 Origin: ROM Problem Find x such that sin 3x cos(60◦−4x) + 1 sin(60◦−7x) −cos(30◦+ x) + m = 0, where m is a fixed real number.
IMO 1979 LL ISR45 Origin: ISR Problem For any positive integer n we denote by F(n) the number of ways in which n can be expressed as the sum of three different positive integers, without regard to order. Thus, since 10 = 7+2+1 = 6+3+1 = 5 + 4 + 1 = 5 + 3 + 2, we have F(10) = 4. Show that F(n) is even if n \equiv2...
IMO 1966 LL HUN18 Origin: HUN Problem Solve the equation sin x + cos x = 1 p, where p is a real parameter. Discuss for which values of p the equation has at least one real solution and determine the number of solutions in [0, 2\pi) for a given p.
IMO 1971 LL USS48 Origin: USS Problem A sequence of real numbers x1, x2, . . . , xn is given such that xi+1 = xi + 30000 1 −x2 i , i = 1, 2, . . ., and x1 = 0. Can n be equal to 50000 if xn < 1?
IMO 1971 LL AUT3 Origin: AUT Problem Let a, b, c be positive real numbers, 0 < a \leqb \leqc. Prove that for any positive real numbers x, y, z the following inequality holds: (ax + by + cz) x a + y b + z c \leq(x + y + z)2 (a + c)2 4ac .
IMO 1987 LL POL49 Origin: POL Problem In the coordinate system in the plane we consider a convex polygon W and lines given by equations x = k, y = m, where k and m are integers. The lines determine a tiling of the plane with unit squares. We say that the boundary of W intersects a square if the boundary contains an interior point of the square. Prove that...
IMO 1989 LL CZS15 Origin: CZS Problem A sequence a1, a2, a3, . . . is defined recursively by a1 = 1 and a2k+j = −aj (j = 1, 2, . . . , 2k). Prove that this sequence is not periodic.
IMO 1985 LL USA85 Origin: USA Problem Let CD be a diameter of circle K. Let AB be a chord that is parallel to CD. The line segment AE, with E on K, is parallel to CB; F is the point of intersection of line segments AB and DE. The line segment FG, with G on DC, extended is parallel to CB. Is GA tangent to K at point A?
IMO 1989 LL HKG36 Origin: HKG Problem Prove the identity 1+ 1 2 −2 3 + 1 4 + 1 5 −2 6 +\cdot \cdot \cdot+ 1 478 + 1 479 −2 480 = 2 k=0 (161 + k)(480 −k).
IMO 1983 LL BUL13 Origin: BUL Problem Let p be a prime number and a1, a2, . . . , a(p+1)/2 different nat- ural numbers less than or equal to p. Prove that for each natural number r less than or equal to p, there exist two numbers (perhaps equal) ai and aj such that p \equivaiaj(mod r).
IMO 1987 LL GBR27 Origin: GBR Problem Find, with proof, the smallest real number C with the following property: For every infinite sequence {xi} of positive real numbers such that x1 + x2 + \cdot \cdot \cdot + xn \leqxn+1 for n = 1, 2, 3, . . ., we have \sqrtx1 + \sqrtx2 + \cdot \cdot \cdot + \sqrtxn \leqc\sqrtx1 + x2 + \cdot \cdot \cdot + xn for...
IMO 1982 LL USS51 Origin: USS Problem Let n numbers x1, x2, . . . , xn be chosen in such a way that 1 \geqx1 \geqx2 \geq\cdot \cdot \cdot \geqxn \geq0. Prove that (1 + x1 + x2 + \cdot \cdot \cdot + xn)\alpha \leq1 + x\alpha 1 + 2\alpha−1x\alpha 2 + \cdot \cdot \cdot + n\alpha−1x\alpha n if 0 \leq\alpha \leq1.
IMO 1970 LL FRA27 Origin: FRA Problem Find a natural number n such that for all prime numbers p, n is divisible by p if and only if n is divisible by p −1.
IMO 1983 LL BRA10 Origin: BRA Problem Which of the numbers 1, 2, . . ., 1983 has the largest number of divisors?
IMO 1969 LL GDR31 Origin: GDR Problem Find the number of permutations a1, . . . , an of the set {1, 2, . . ., n} such that |ai −ai+1| ̸= 1 for all i = 1, 2, . . ., n −1. Find a recurrence formula and evaluate the number of such permutations for n \leq6.
IMO 1988 LL FRG15 Origin: FRG Problem Let 1 \leqk < n. Consider all finite sequences of positive integers with sum n. Find T (n, k), the total number of terms of size k in all of these sequences.
IMO 1966 LL USS48 Origin: USS Problem Find all positive numbers p for which the equation x2+px+3p = 0 has integral roots.
IMO 1987 LL MOR45 Origin: MOR Problem Let us consider a variable polygon with 2n sides (n \inN) in a fixed circle such that 2n −1 of its sides pass through 2n −1 fixed points lying on a straight line ∆. Prove that the last side also passes through a fixed point lying on ∆.
IMO 1967 LL BUL1 Origin: BUL Problem Prove that all numbers in the sequence 107811 , 110778111 , 111077781111 , . . . are perfect cubes. Solution Let us denote the nth term of the given sequence by an. Then an = 1 103n+3 −102n+3 7102n+2 −10n+1 10n+2 −1 = 1 27(103n+3 −3 \cdot 102n+2 + 3 \cdot 10n+1 −1) = 10n+1 −1 3 .
IMO 1989 LL TUR101 Origin: TUR Problem Let ABC be an equilateral triangle and \Gamma the semicircle drawn exteriorly to the triangle, having BC as diameter. Show that if a line passing through A trisects BC, it also trisects the arc \Gamma.
IMO 1985 LL TUR82 Origin: TUR Problem Find all cubic polynomials x3 + ax2 + bx + c admitting the rational numbers a, b, and c as roots.
IMO 1967 LL CZS12 Origin: CZS Problem Given a segment AB of the length 1, define the set M of points in the following way: it contains the two points A, B, and also all points obtained from A, B by iterating the following rule: (∗) for every pair of points X, Y in M, the set M also contains the point Z of the segment XY for which Y...
IMO 1987 LL FIN13 Origin: FIN Problem A be an infinite set of positive integers such that every n \inA is the product of at most 1987 prime numbers. Prove that there is an infinite set B \subsetA and a number p such that the greatest common divisor of any two distinct numbers in B is b.
IMO 1986 LL USS77 Origin: USS Problem Find all integers x, y, z that satisfy x3 + y3 + z3 = x + y + z = 8.
IMO 1987 LL GBR23 Origin: GBR Problem A lampshade is part of the surface of a right circular cone whose axis is vertical. Its upper and lower edges are two horizontal circles. Two points are selected on the upper smaller circle and four points on the lower larger circle. Each of these six points has three of the others that are its nearest neighbors at a distance d from it....
IMO 1986 LL GBR31 Origin: GBR Problem Let P and Q be distinct points in the plane of a triangle ABC such that AP : AQ = BP : BQ = CP : CQ. Prove that the line PQ passes through the circumcenter of the triangle.
IMO 1984 LL GBR26 Origin: GBR Problem A cylindrical container has height 6 cm and radius 4 cm. It rests on a circular hoop, also of radius 4 cm, fixed in a horizontal plane with its axis vertical and with each circular rim of the cylinder touching the hoop at two points. The cylinder is now moved so that each of its circular rims still touches the hoop in two...
IMO 1967 LL BUL5 Origin: BUL Problem Solve the system x2 + x −1 = y, y2 + y −1 = z, z2 + z −1 = x. Solution If one of x, y, z is equal to 1 or −1, then we obtain solutions (−1, −1, −1) and (1, 1, 1). We claim that these are the only solutions to the system. Let f(t) = t2 + t −1....
IMO 1987 LL GBR24 Origin: GBR Problem Prove that if the equation x4 + ax3 + bx + c = 0 has all its roots real, then ab \leq0.
IMO 1978 LL USA43 Origin: USA Problem If p is a prime greater than 3, show that at least one of the numbers p2 , 4 p2 , . . . , p−2 p2 is expressible in the form 1 x + 1 y, where x and y are positive integers.
IMO 1985 LL FRA27 Origin: FRA Problem Let O be a point on the oriented Euclidean plane and (i, j) a directly oriented orthonormal basis. Let C be the circle of radius 1, centered at O. For every real number t and nonnegative integer n let Mn be the point on C for which ⟨i, −−−\to OMn⟩= cos 2nt (or −−−\to OMn = cos 2nti+sin 2ntj). Let k \geq2 be...
IMO 1970 LL NET35 Origin: NET Problem Find for every value of n a set of numbers p for which the fol- lowing statement is true: Any convex n-gon can be divided into p isosceles triangles. Alternative version. The same about division into p polygons with axis of symmetry.
IMO 1983 LL BEL7 Origin: BEL Problem Find all numbers x \inZ for which the number x4 + x3 + x2 + x + 1 is a perfect square.
IMO 1988 LL SIN73 Origin: SIN Problem In a group of n people each one knows exactly three others. They are seated around a table. We say that the seating is perfect if everyone knows the two sitting by their sides. Show that if there is a perfect seating S for the group, then there is always another perfect seating that cannot be obtained from S by rotation or reflection.
IMO 1989 LL POR82 Origin: POR Problem Solve in the set of real numbers the equation 3x3 −[x] = 3, where [x] denotes the integer part of x.
IMO 1966 LL CZS40 Origin: CZS Problem For a positive real number p, find all real solutions to the equation x2 + 2px −p2 − x2 −2px −p2 = 1.
IMO 1966 LL CZS42 Origin: CZS Problem Let a1, a2, . . . , an (n \geq2) be a sequence of integers. Show that there is a subsequence ak1, ak2, . . . , akm, where 1 \leqk1 < k2 < \cdot \cdot \cdot < km \leq n, such that a2 k1 + a2 k2 + \cdot \cdot \cdot + a2 km is divisible by n.
IMO 1979 LL SWE61 Origin: SWE Problem Let a1 \leqa2 \leq\cdot \cdot \cdot \leqan and b1 \leqb2 \leq\cdot \cdot \cdot \leqbn be two sequences such that m k=1 ak \geqm k=1 bk for all m \leqn with equality for m = n. Let f be a convex function defined on the real numbers. Prove that n k=1 f(ak) \leq n k=1 f(bk).
IMO 1967 LL MON32 Origin: MON Problem Determine the volume of the body obtained by cutting the ball of radius R by the trihedron with vertex in the center of that ball if its dihedral angles are \alpha, \beta, \gamma. Solution Let us denote by V the volume of the given body, and by Va, Vb, Vc the volumes of the parts of the given ball that lie inside the...
IMO 1986 LL ISR48 Origin: ISR Problem Let P be a convex 1986-gon in the plane. Let A, D be interior points of two distinct sides of P and let B, C be two distinct interior points of the line segment AD. Starting with an arbitrary point Q1 on the boundary of P, define recursively a sequence of points Qn as follows: given Qn extend the directed line segment QnB...
IMO 1988 LL FRA13 Origin: FRA Problem Let T be a triangle with inscribed circle C. A square with sides of length a is circumscribed about the same circle C. Show that the total length of the parts of the edges of the square interior to the triangle T is at least 2a.
IMO 1974 LL YUG52 Origin: YUG Problem A fox stands in the center of the field which has the form of an equilateral triangle, and a rabbit stands at one of its vertices. The fox can move through the whole field, while the rabbit can move only along the border of the field. The maximal speeds of the fox and rabbit are equal to u and v, respectively. Prove that:...
IMO 1976 LL POL31 Origin: POL Problem Into every lateral face of a quadrangular pyramid a circle is inscribed. The circles inscribed into adjacent faces are tangent (have one point in common). Prove that the points of contact of the circles with the base of the pyramid lie on a circle.
IMO 1989 LL MON70 Origin: MON Problem Three mutually nonparallel lines li (i = 1, 2, 3) are given in a plane. The lines li determine a triangle and reflections fi with axes on lines li. Prove that for every point of the plane, there exists a finite composition of the reflections fi that maps that point to a point interior to the triangle.
IMO 1979 LL SWE59 Origin: SWE Problem Determine the maximum value of x2y2z2w when x, y, z, w \geq0 and 2x + xy + z + yzw = 1.
IMO 1992 LL TWN76 Origin: TWN Problem Given any triangle ABC and any positive integer n, we say that n is a decomposable number for triangle ABC if there exists a de- composition of the triangle ABC into n subtriangles with each subtriangle similar to \triangleABC. Determine the positive integers that are decomposable numbers for every triangle.
IMO 1992 LL FRG18 Origin: FRG Problem Fibonacci numbers are defined as follows: F1 = F2 = 1, Fn+2 = Fn+1+Fn, n \geq1. Let an be the number of words that consist of n letters 0 or 1 and contain no two letters 1 at distance two from each other. Express an in terms of Fibonacci numbers.
IMO 1987 LL GRE32 Origin: GRE Problem Solve the equation 28x = 19y + 87z, where x, y, z are integers.
IMO 1989 LL INA44 Origin: INA Problem Let A and B be fixed distinct points on the X axis, none of which coincides with the origin O(0, 0), and let C be a point on the Y axis of an orthogonal Cartesian coordinate system. Let g be a line through the origin O(0, 0) and perpendicular to the line AC. Find the locus of the point of intersection of the...
IMO 1967 LL BUL6 Origin: BUL Problem Solve the system |x + y| + |1 −x| = 6, |x + y + 1| + |1 −y| = 4. Solution The given system has two solutions: (−2, −1) and (−14/3, 13/3).
IMO 1989 LL ISR58 Origin: ISR Problem Let P1(x), P2(x), . . . , Pn(x) be polynomials with real coefficients. Show that there exist real polynomials Ar(x), Br(x) (r = 1, 2, 3) such that n s=1(Ps(x))2 = (A1(x))2 + (B1(x))2 = (A2(x))2 + x(B2(x))2 = (A3(x))2 −x(B3(x))2.
IMO 1969 LL GBR26 Origin: GBR Problem A smooth solid consists of a right circular cylinder of height h and base-radius r, surmounted by a hemisphere of radius r and center O. The solid stands on a horizontal table. One end of a string is attached to a point on the base. The string is stretched (initially being kept in the vertical plane) over the highest point of the solid...
IMO 1985 LL SWE76 Origin: SWE Problem Are there integers m and n such that 5m2 −6mn + 7n2 = 1985?
IMO 1982 LL POL42 Origin: POL Problem Let F be the family of all k-element subsets of the set {1, 2, . . ., 2k + 1}. Prove that there exists a bijective function f : F \toF such that for every A \inF, the sets A and f(A) are disjoint.
IMO 1988 LL HKG33 Origin: HKG Problem Find a necessary and sufficient condition on the natural num- ber n for the equation xn + (2 + x)n + (2 −x)n = 0 to have a real root.
IMO 1979 LL FRA22 Origin: FRA Problem Consider two quadrilaterals ABCD and A′B′C′D′ in an affine Euclidian plane such that AB = A′B′, BC = B′C′, CD = C′D′, and DA = D′A′. Prove that the following two statements are true: (a) If the diagonals BD and AC are mutually perpendicular, then the diagonals B′D′ and A′C′ are also mutually perpendicular. (b) If the perpendicular bisector of BD intersects AC...
IMO 1969 LL SWE60 Origin: SWE Problem Find the natural number n with the following properties: (1) Let S = {p1, p2, . . . } be an arbitrary finite set of points in the plane, and rj the distance from Pj to the origin O. We assign to each Pj the closed disk Dj with center Pj and radius rj. Then some n of these disks contain all points...
IMO 1982 LL BEL5 Origin: BEL Problem Among all triangles with a given perimeter, find the one with the maximal radius of its incircle.
IMO 1967 LL USS55 Origin: USS Problem Find all x for which for all n, sin x + sin 2x + sin 3x + \cdot \cdot \cdot + sin nx \leq \sqrt 2 . Solution It is enough to find all x from (0, 2\pi] such that the given inequality holds for all n. Suppose 0 < x < 2\pi/3. If n is the maximum integer for which nx \leq...
IMO 1966 LL USS8 Origin: USS Problem We are given a bag of sugar, a two-pan balance, and a weight of 1 gram. How do we obtain 1 kilogram of sugar in the smallest possible number of weighings?
IMO 1988 LL HUN37 Origin: HUN Problem Let n points be given on the surface of a sphere. Show that the surface can be divided into n congruent regions such that each of them contains exactly one of the given points.
IMO 1986 LL GBR34 Origin: GBR Problem For each nonnegative integer n, Fn(x) is a polynomial in x of degreee n. Prove that if the identity Fn(2x) = n r=0 (−1)n−r n r 2rFr(x) holds for each n, then Fn(tx) = n r=0 n r tr(1 −t)n−rFr(x) for each n and all t.
IMO 1988 LL KOR55 Origin: KOR Problem Find all positive integers x such that the product of all digits of x is given by x2 −10x −22.
IMO 1983 LL CAN17 Origin: CAN Problem In how many ways can 1, 2, . . . , 2n be arranged in a 2 \times n rectangular array a1 a2 \cdot \cdot \cdot an b1 b2 \cdot \cdot \cdot bn for which: (i) a1 < a2 < \cdot \cdot \cdot < an, (ii) b1 < b2 < \cdot \cdot \cdot < bn, (iii) a1 < b1, a2 <...
IMO 1992 LL POL56 Origin: POL Problem A directed graph (any two distinct vertices joined by at most one directed line) has the following property: If x, u, and v are three distinct vertices such that x \tou and x \tov, then u \tow and v \tow for some vertex w. Suppose that x \tou \toy \to\cdot \cdot \cdot \toz is a path of length n, that cannot be extended...
IMO 1986 LL LUX50 Origin: LUX Problem Let D be the point on the side BC of the triangle ABC such that AD is the bisector of \angleCAB. Let I be the incenter of \triangleABC. (a) Construct the points P and Q on the sides AB and AC, respectively, such that PQ is parallel to BC and the perimeter of the triangle APQ is equal to k \cdot BC, where...
IMO 1983 LL LUX45 Origin: LUX Problem Let two glasses, numbered 1 and 2, contain an equal quantity of liquid, milk in glass 1 and coffee in glass 2. One does the following: Take one spoon of mixture from glass 1 and pour it into glass 2, and then take the same spoon of the new mixture from glass 2 and pour it back into the first glass. What happens...
IMO 1969 LL YUG69 Origin: YUG Problem Suppose that positive real numbers x1, x2, x3 satisfy x1x2x3 > 1, x1 + x2 + x3 < 1 x1 1 x2 1 x3 . Prove that: (a) None of x1, x2, x3 equals 1. (b) Exactly one of these numbers is less than 1.
IMO 1982 LL BRA10 Origin: BRA Problem Let r1, . . . , rn be the radii of n spheres. Call S1, S2, . . . , Sn the areas of the set of points of each sphere from which one cannot see any point of any other sphere. Prove that S1 r2 S2 r2 \cdot \cdot \cdot + Sn r2n = 4\pi.
IMO 1988 LL NET67 Origin: NET Problem Given a set of 1988 points in the plane, no three points of the set collinear, the points of a subset with 1788 points are colored blue, and the remaining 200 are colored red. Prove that there exists a line in the plane such that each of the two parts into which the line divides the plane contains 894 blue points and 100...
IMO 1976 LL GBR17 Origin: GBR Problem Show that there exists a convex polyhedron with all its vertices on the surface of a sphere and with all its faces congruent isosceles triangles whose ratio of sides are \sqrt 3 : \sqrt 3 : 2.
IMO 1972 LL SWE40 Origin: SWE Problem Prove the inequalities u v \leqsin u sin v \leq\pi u v , for 0 \lequ < v \leq\pi 2 .
IMO 1992 LL FIN14 Origin: FIN Problem Integers a1, a2, . . . , an satisfy |ak| = 1 and n k=1 akak+1ak+2ak+3 = 2, where an+j = aj. Prove that n ̸= 1992.
IMO 1992 LL PRK61 Origin: PRK Problem There are a board with 2n\cdot2n (= 4n2) squares and 4n2−1 cards numbered with different natural numbers. These cards are put one by one on each of the squares. One square is empty. We can move a card to an empty square from one of the adjacent squares (two squares are adjacent if they have a common edge). Is it possible to exchange...
IMO 1992 LL POL54 Origin: POL Problem Suppose that n > m \geq1 are integers such that the string of digits 143 occurs somewhere in the decimal representation of the fraction m/n. Prove that n > 125
IMO 1982 LL BRA11 Origin: BRA Problem A rectangular pool table has a hole at each of three of its corners. The lengths of sides of the table are the real numbers a and b. A billiard ball is shot from the fourth corner along its angle bisector. The ball falls in one of the holes. What should the relation between a and b be for this to happen?
IMO 1967 LL MON33 Origin: MON Problem In what case does the system x + y + mz = a, x + my + z = b, mx + y + z = c, have a solution? Find the conditions under which the unique solution of the above system is an arithmetic progression. Solution If m ̸\in{−2, 1}, the system has the unique solution x = b + a −(1...
IMO 1979 LL ROM56 Origin: ROM Problem Show that for every natural number n, n \sqrt 2 −[n \sqrt 2] > 2n \sqrt and that for every \epsilon > 0 there exists a natural number n with n \sqrt 2 − [n \sqrt 2] < 2n \sqrt 2 + \epsilon.
IMO 1970 LL BEL6 Origin: BEL Problem Prove that the equation in x n i=1 bi x −ai = c, bi > 0, a1 < a2 < a3 < \cdot \cdot \cdot < an, has n −1 roots x1, x2, x3, . . . , xn−1 such that a1 < x1 < a2 < x2 < a3 < x3 < \cdot \cdot \cdot < xn−1 < an.
IMO 1986 LL ROM62 Origin: ROM Problem Determine all pairs of positive integers (x, y) satisfying the equation px −y3 = 1, where p is a given prime number.
IMO 1989 LL INA46 Origin: INA Problem Given two distinct numbers b1 and b2, their product can be formed in two ways: b1 \times b2 and b2 \times b1. Given three distinct numbers, b1, b2, b3, their product can be formed in twelve ways: b1 \times(b2 \timesb3); (b1 \times b2) \times b3; b1 \times (b3 \times b2); (b1 \times b3) \times b2; b2 \times (b1 \times b3); (b2 \times b1)...
IMO 1978 LL GDR28 Origin: GDR Problem Let c, s be real functions defined on R{0} that are nonconstant on any interval and satisfy c x y = c(x)c(y) −s(x)s(y) for any x ̸= 0, y ̸= 0. Prove that: (a) c(1/x) = c(x), s(1/x) = −s(x) for any x ̸= 0, and also c(1) = 1, s(1) = s(−1) = 0; (b) c and s are either both...
IMO 1971 LL GBR14 Origin: GBR Problem Note that 83 −73 = 169 = 132 and 13 = 22 + 32. Prove that if the difference between two consecutive cubes is a square, then it is the square of the sum of two consecutive squares.
IMO 1974 LL CZS11 Origin: CZS Problem Given a line p and a triangle \trianglein the plane, construct an equilateral triangle one of whose vertices lies on the line p, while the other two halve the perimeter of \triangle.
IMO 1986 LL CZS16 Origin: CZS Problem Given a positive integer k, find the least integer nk for which there exist five sets S1, S2, S3, S4, S5 with the following properties: |Sj| = k for j = 1, . . . , 5, j=1 Sj = nk; |Si \capSi+1| = 0 = |S5 \capS1|, for i = 1, . . . , 4.
IMO 1985 LL ITA48 Origin: ITA Problem In a given country, all inhabitants are knights or knaves. A knight never lies; a knave always lies. We meet three persons, A, B, and C. Person A says, “If C is a knight, B is a knave.” Person C says, “A and I are different; one is a knight and the other is a knave.” Who are the knights, and who are...
IMO 1966 LL USS57 Origin: USS Problem Is it possible to choose a set of 100 (or 200) points on the boundary of a cube such that this set is fixed under each isometry of the cube into itself? Justify your answer.
IMO 1967 LL SWE48 Origin: SWE Problem Determine all positive roots of the equation xx = 1/ \sqrt 2. Solution Put f(x) = x ln x. The given equation is equivalent to f(x) = f(1/2), which has the solutions x1 = 1/2 and x2 = 1/4. Since the function f is decreasing on (0, 1/e), and increasing on (1/e, +\infty), this equation has no other solutions.
IMO 1986 LL TUR71 Origin: TUR Problem Two straight lines perpendicular to each other meet each side of a triangle in points symmetric with respect to the midpoint of that side. Prove that these two lines intersect in a point on the nine-point circle.
IMO 1987 LL USA64 Origin: USA Problem Let r > 1 be a real number, and let n be the largest integer smaller than r. Consider an arbitrary real number x with 0 \leqx \leq n r−1. By a base-r expansion of x we mean a representation of x in the form x = a1 r + a2 r2 + a3 r3 + \cdot \cdot \cdot , where the ai...
IMO 1977 LL USA53 Origin: USA Problem Find all pairs of integers a and b for which 7a + 14b = 5a2 + 5ab + 5b2. Solution The discriminant of the given equation considered as a quadratic equation in b is 196−75a2. Thus 75a2 \leq196 and hence −1 \leqa \leq1. Now the integer solutions of the given equation are easily found: (−1, 3), (0, 0), (1, 2).
IMO 1986 LL CHN13 Origin: CHN Problem Let N = {1, 2, . . ., n}, n \geq3. To each pair i, j of elements of N, i ̸= j, there is assigned a number fij \in{0, 1} such that fij + fji = 1. Let r(i) = j̸=i fij and write M = maxi\inN r(i), m = mini\inN r(i). Prove that for any w \inN with r(w) =...
IMO 1969 LL CZS17 Origin: CZS Problem Let d and p be two real numbers. Find the first term of an arith- metic progression a1, a2, a3, . . . with difference d such that a1a2a3a4 = p. Find the number of solutions in terms of d and p.
IMO 1966 LL USS50 Origin: USS Problem Given a quadrangle of sides a, b, c, d and area S, show that S \leq a+c \cdot b+d 2 .
IMO 1978 LL VIE47 Origin: VIE Problem Given the expression Pn(x) = 1 2n x + x2 −1 n + x − x2 −1 n , prove: (a) Pn(x) satisfies the identity Pn(x) −xPn−1(x) + 1 4Pn−2(x) \equiv0. (b) Pn(x) is a polynomial in x of degree n.
IMO 1978 LL CUB5 Origin: CUB Problem Prove that for any triangle ABC there exists a point P in the plane of the triangle and three points A′, B′, and C′ on the lines BC, AC, and AB respectively such that AB \cdot PC′ = AC \cdot PB′ = BC \cdot PA′ = 0.3M 2, where M = max{AB, AC, BC}.
IMO 1978 LL NET31 Origin: NET Problem Let the polynomials P(x) = xn + an−1xn−1 + \cdot \cdot \cdot + a1x + a0, Q(x) = xm + bm−1xm−1 + \cdot \cdot \cdot + b1x + b0, be given satisfying the identity P(x)2 = (x2 −1)Q(x)2 + 1. Prove the identity P ′(x) = nQ(x).
IMO 1987 LL AUS1 Origin: AUS Problem Let x1, x2, . . . , xn be n integers. Let n = p + q, where p and q are positive integers. For i = 1, 2, . . ., n, put Si = xi + xi+1 + \cdot \cdot \cdot + xi+p−1 and Ti = xi+p + xi+p+1 + \cdot \cdot \cdot + xi+n−1 (it is assumed that xi+n =...
IMO 1987 LL NET47 Origin: NET Problem Through a point P within a triangle ABC the lines l, m, and n perpendicular respectively to AP, BP, CP are drawn. Prove that if l intersects the line BC in Q, m intersects AC in R, and n intersects AB in S, then the points Q, R, and S are collinear.
IMO 1986 LL TUR72 Origin: TUR Problem A one-person game with two possible outcomes is played as follows: After each play, the player receives either a or b points, where a and b are integers with 0 < b < a < 1986. The game is played as many times as one wishes and the total score of the game is defined as the sum of points received after successive...
IMO 1989 LL CUB13 Origin: CUB Problem Let n be a natural number not greater than 44. Prove that for any function f defined over N2 whose images are in the set {1, 2, . . ., n}, there are four ordered pairs (i, j), (i, k), (l, j), and (l, k) such that f(i, j) = f(i, k) = f(l, j) = f(l, k), where i, j, k, l...
IMO 1988 LL CUB5 Origin: CUB Problem Let k be a positive integer and Mk the set of all the integers that are between 2k2 + k and 2k2 + 3k, both included. Is it possible to partition Mk into two subsets A and B such that x\inA x2 = x\inB x2?
IMO 1976 LL GDR21 Origin: GDR Problem Find the largest positive real number p (if it exists) such that the inequality x2 1 + x2 2 + \cdot \cdot \cdot + x2 n \geqp(x1x2 + x2x3 + \cdot \cdot \cdot + xn−1xn) (1) is satisfied for all real numbers xi, and (a) n = 2; (b) n = 5. Find the largest positive real number p (if it exists) such...
IMO 1969 LL MON41 Origin: MON Problem Given two numbers x0 and x1, let \alpha and \beta be coefficients of the equation 1 −\alphay −\betay2 = 0. Under the given conditions, find an expression for the solution of the system xn+2 −\alphaxn+1 −\betaxn = 0, n = 0, 1, 2, . . . .
IMO 1976 LL GBR16 Origin: GBR Problem Prove that there is a positive integer n such that the decimal representation of 7n contains a block of at least m consecutive zeros, where m is any given positive integer.
IMO 1989 LL HUN38 Origin: HUN Problem Connecting the vertices of a regular n-gon we obtain a closed (not necessarily convex) n-gon. Show that if n is even, then there are two parallel segments among the connecting segments and if n is odd then there cannot be exactly two parallel segments.
IMO 1966 LL GDR10 Origin: GDR Problem How many real solutions are there to the equation x = 1964 sinx −189?
IMO 1978 LL FIN12 Origin: FIN Problem The equation x3 + ax2 + bx + c = 0 has three (not necessarily distinct) real roots t, u, v. For which a, b, c do the numbers t3, u3, v3 satisfy the equation x3 + a3x2 + b3x + c3 = 0?
IMO 1977 LL BUL1 Origin: BUL Problem A pentagon ABCDE inscribed in a circle for which BC < CD and AB < DE is the base of a pyramid with vertex S. If AS is the longest edge starting from S, prove that BS > CS. Solution Let P be the projection of S onto the plane ABCDE. Obviously BS > CS is equivalent to BP > CP. The conditions...
IMO 1983 LL KUW38 Origin: KUW Problem Let {un} be the sequence defined by its first two terms u0, u1 and the recursion formula un+2 = un −un+1. (a) Show that un can be written in the form un = \alphaan + \betabn, where a, b, \alpha, \beta are constants independent of n that have to be determined. (b) If Sn = u0 + u1 + \cdot \cdot \cdot +...
IMO 1988 LL SWE78 Origin: SWE Problem A two-person game is played with nine boxes arranged in a 3 \times 3 square, initially empty, and with white and black stones. At each move a player puts three stones, not necessarily of the same color, in three boxes in either a horizontal or a vertical row. No box can contain stones of different colors: If, for instance, a player puts a...
IMO 1992 LL USA80 Origin: USA Problem Given a graph with n vertices and a positive integer m that is less than n, prove that the graph contains a set of m + 1 vertices in which the difference between the largest degree of any vertex in the set and the smallest degree of any vertex in the set is at most m −1.
IMO 1979 LL VIE72 Origin: VIE Problem Let f(x) be a polynomial with integer coefficients. Prove that if f(x) equals 1979 for four different integer values of x, then f(x) cannot be equal to 2 \times 1979 for any integral value of x.
IMO 1988 LL GBR20 Origin: GBR Problem It is proposed to partition the set of positive integers into two disjoint subsets A and B subject to the following conditions: (i) 1 is in A; (ii) no two distinct members of A have a sum of the form 2k + 2 (k = 0, 1, 2, . . .); and (iii) no two distinct members of B have a sum of...
IMO 1992 LL CAN5 Origin: CAN Problem Let I, H, O be the incenter, centroid, and circumcenter of the nonisosceles triangle ABC. Prove that AI\parallelHO if and only if ∡BAC = 120◦.
IMO 1989 LL HKG35 Origin: HKG Problem Find all square numbers S1 and S2 such that S1 −S2 = 1989.
IMO 1979 LL USS70 Origin: USS Problem There are 1979 equilateral triangles: T1, T2, . . . , T1979. A side of triangle Tk is equal to 1/k, k = 1, 2, . . . , 1979. At what values of a number a can one place all these triangles into the equilateral triangle with side length a so that they don’t intersect (points of contact are allowed)?
IMO 1976 LL USA37 Origin: USA Problem From a square board 11 squares long and 11 squares wide, the central square is removed. Prove that the remaining 120 squares cannot be covered by 15 strips each 8 units long and one unit wide.
IMO 1976 LL FIN12 Origin: FIN Problem Five points lie on the surface of a ball of unit radius. Find the maximum of the smallest distance between any two of them.
IMO 1979 LL HUN41 Origin: HUN Problem Prove the following statement: There does not exist a pyramid with square base and congruent lateral faces for which the measures of all edges, total area, and volume are integers.
IMO 1976 LL VIE50 Origin: VIE Problem Find a function f(x) defined for all real values of x such that for all x, f(x + 2) −f(x) = x2 + 2x + 4, and if x \in[0, 2), then f(x) = x2.
IMO 1989 LL ROM93 Origin: ROM Problem For \Phi : N \toZ let us define M\Phi = {f : N \toZ; f(x) > F(\Phi(x)), \forallx \inN}. (a) Prove that if M\Phi1 = M\Phi2 ̸= \emptyset, then \Phi1 = \Phi2. (b) Does this property remain true if M\Phi = {f : N \toN; f(x) > F(\Phi(x)), \forallx \inN}?
IMO 1985 LL TUR81 Origin: TUR Problem Given the side a and the corresponding altitude ha of a triangle ABC, find a relation between a and ha such that it is possible to construct, with straightedge and compass, triangle ABC such that the altitudes of ABC form a right triangle admitting ha as hypotenuse.
IMO 1966 LL BUL33 Origin: BUL Problem Two circles touch each other from inside, and an equilateral triangle is inscribed in the larger circle. From the vertices of the triangle one draws segments tangent to the smaller circle. Prove that the length of one of these segments equals the sum of the lengths of the other two.
IMO 1972 LL NET28 Origin: NET Problem The lengths of the sides of a rectangle are given to be odd integers. Prove that there does not exist a point within that rectangle that has integer distances to each of its four vertices.
IMO 1967 LL GBR19 Origin: GBR Problem The n points P1, P2, . . . , Pn are placed inside or on the bound- ary of a disk of radius 1 in such a way that the minimum distance dn between any two of these points has its largest possible value Dn. Calcu- late Dn for n = 2 to 7 and justify your answer. Solution Suppose n \leq6. Let...
IMO 1979 LL FRA21 Origin: FRA Problem Let E be the set of all bijective mappings from R to R satisfying (\forallt \inR) f(t) + f −1(t) = 2t, where f −1 is the mapping inverse to f. Find all elements of E that are monotonic mappings.
IMO 1987 LL GDR28 Origin: GDR Problem In a chess tournament there are n \geq5 players, and they have already played n2 2 games (each pair have played each other at most once). (a) Prove that there are five players a, b, c, d, e for which the pairs ab, ac, bc, ad, ae, de have already played. (b) Is the statement also valid for the n2...
IMO 1967 LL HUN22 Origin: HUN Problem The distance between the centers of the circles k1 and k2 with radii r is equal to r. Points A and B are on the circle k1, symmetric with respect to the line connecting the centers of the circles. Point P is an arbitrary point on k2. Prove that PA2 + PB2 \geq2r2. When does equality hold? Solution Let O1 and O2 be...
IMO 1984 LL BUL7 Origin: BUL Problem Prove that for any natural number n, the number 2n n divides the least common multiple of the numbers 1, 2, . . . , 2n −1, 2n.
IMO 1982 LL FRA26 Origin: FRA Problem Let (an)n\geq0 and (bn)n\geq0 be two sequences of natural numbers. Determine whether there exists a pair (p, q) of natural numbers that satisfy p < q and ap \leqaq, bp \leqbq.
IMO 1971 LL BUL7 Origin: BUL Problem In a triangle ABC, let H be its orthocenter, O its circumcenter, and R its circumradius. Prove that: (a) |OH| = R\sqrt1 −8 cos \alpha cos \beta cos \gamma, where \alpha, \beta, \gamma are angles of the tri- angle ABC; (b) O \equivH if and only if ABC is equilateral.
IMO 1985 LL POL64 Origin: POL Problem Let p be a prime. For which k can the set {1, 2, . . ., k} be partitioned into p subsets with equal sums of elements?
IMO 1983 LL COL21 Origin: COL Problem Prove that there are infinitely many positive integers n for which it is possible for a knight, starting at one of the squares of an n \times n chessboard, to go through each of the squares exactly once.
IMO 1984 LL GDR30 Origin: GDR Problem Decide whether it is possible to color the 1984 natural numbers 1, 2, 3, . . ., 1984 using 15 colors so that no geometric sequence of length 3 of the same color exists.
IMO 1966 LL USS52 Origin: USS Problem A figure of area 1 is cut out from a sheet of paper and divided into 10 parts, each of which is colored in one of 10 colors. Then the figure is turned to the other side and again divided into 10 parts (not necessarily in the same way). Show that it is possible to color these parts in the 10 colors so...
IMO 1970 LL NET37 Origin: NET Problem Solve the set of simultaneous equations v2+ w2+ x2+ y2 = 6 −2u, u2+ w2+ x2+ y2 = 6 −2v, u2+ v2+ x2+ y2 = 6 −2w, u2+ v2+ w2+ y2 = 6 −2x, u2+ v2+ w2+ x2 = 6 −2y.
IMO 1967 LL HUN24 Origin: HUN Problem Father has left to his children several identical gold coins. According to his will, the oldest child receives one coin and one-seventh of the remaining coins, the next child receives two coins and one-seventh of the remaining coins, the third child receives three coins and one-seventh of the remaining coins, and so on through the youngest child. If every child inherits an integer...
IMO 1979 LL VIE75 Origin: VIE Problem Given an equilateral triangle ABC, let M be an arbitrary point in space. (a) Prove that one can construct a triangle from the segments MA, MB, MC. (b) Suppose that P and Q are two points symmetric with respect to the center O of ABC. Prove that the two triangles constructed from the segments PA, PB, PC and QA, QB, QC are of...
IMO 1987 LL VIE76 Origin: VIE Problem Given two sequences of positive numbers {ak} and {bk} (k \inN) such that (i) ak < bk, (ii) cos akx + cos bkx \geq−1 k for all k \inN and x \inR, prove the existence of limk\to\inftyak bk and find this limit.
IMO 1969 LL NET46 Origin: NET Problem The vertices of an (n + 1)-gon are placed on the edges of a regular n-gon so that the perimeter of the n-gon is divided into equal parts. How does one choose these n + 1 points in order to obtain the (n + 1)gon with (a) maximal area; (b) minimal area?
IMO 1989 LL POL80 Origin: POL Problem We are given a finite collection of segments in the plane, of total length 1. Prove that there exists a line ℓsuch that the sum of the lengths of the projections of the given segments to the line ℓis less than 2/\pi.
IMO 1988 LL HKG32 Origin: HKG Problem Assuming that the roots of x3+px2+qx+r = 0 are all real and positive, find a relation between p, q, and r that gives a necessary condition for the roots to be exactly the cosines of three angles of a triangle.
IMO 1967 LL GDR14 Origin: GDR Problem Which fraction p/q, where p, q are positive integers less than 100, is closest to \sqrt 2? Find all digits after the decimal point in the decimal representation of this fraction that coincide with digits in the decimal representation of \sqrt 2 (without using any tables). Solution We have that p q − \sqrt = |p −q \sqrt 2| q = |p2 −2q2|...
IMO 1966 LL CZS26 Origin: CZS Problem (a) Prove that (a1 +a2 +\cdot \cdot \cdot+ak)2 \leqk(a2 1 +\cdot \cdot \cdot+a2 k), where k \geq1 is a natural number and a1, . . . , ak are arbitrary real numbers. (b) If real numbers a1, . . . , an satisfy a1 + a2 + \cdot \cdot \cdot + an \geq (n −1)(a2 1 + \cdot \cdot \cdot + a2n), show...
IMO 1976 LL GDR20 Origin: GDR Problem Let (an), n = 0, 1, . . ., be a sequence of real numbers such that a0 = 0 and a3 n+1 = 1 2a2 n −1, n = 0, 1, . . .. Prove that there exists a positive number q, q < 1, such that for all n = 1, 2, . . ., |an+1 −an| \leqq|an −an−1|, and give...
IMO 1989 LL HKG32 Origin: HKG Problem Let ABC be an equilateral triangle. Let D, E, F, M, N, and P bee the mid-points of BC, CA, AB, FD, FB, and DC respectively. (a) Show that the line segments AM, EN, and FP are concurrent. (b) Let O be the point of intersection of AM, EN, and FP. Find OM : OF : ON : OE : OP : OA.
IMO 1989 LL KOR64 Origin: KOR Problem Let a regular (2n + 1)-gon be inscribed in a circle of radius r. We consider all the triangles whose vertices are from those of the regular (2n + 1)-gon. (a) How many triangles among them contain the center of the circle in their interior? (b) Find the sum of the areas of all those triangles that contain the center of the circle...
IMO 1986 LL GRE41 Origin: GRE Problem Let M, N, P be the midpoints of the sides BC, CA, AB of a triangle ABC. The lines AM, BN, CP intersect the circumcircle of ABC at points A′, B′, C′, respectively. Show that if A′B′C′ is an equilateral triangle, then so is ABC.
IMO 1976 LL GBR15 Origin: GBR Problem Let ABC and A′B′C′ be any two coplanar triangles. Let L be a point such that AL\parallelBC, A′L\parallelB′C′, and M, N similarly defined. The line BC meets B′C′ at P, and similarly defined are Q and R. Prove that PL, QM, RN are concurrent.
IMO 1967 LL CZS7 Origin: CZS Problem Find all real solutions of the system of equations x1 + x2 + \cdot \cdot \cdot + xn = a, x2 1 + x2 2 + \cdot \cdot \cdot + x2 n = a2, . . . . . . . . . . . . . . . . . . xn 1 + xn 2 + \cdot \cdot \cdot + xn...
IMO 1988 LL SPA75 Origin: SPA Problem Let ABC be a triangle with inradius r and circumradius R. Show that sin A 2 sin B 2 + sin B 2 sin C 2 + sin C 2 sin A 2 \leq5 8 + r 4R.
IMO 1985 LL ISR42 Origin: ISR Problem Prove that the product of two sides of a triangle is always greater than the product of the diameters of the inscribed circle and the circumscribed circle.
IMO 1974 LL CZS9 Origin: CZS Problem Solve the following system of linear equations with unknown x1, . . . , xn (n \geq2) and parameters c1, . . . , cn: 2x1 −x2 c1; −x1 +2x2 −x3 c2; −x2 +2x3 −x4 c3; . . . . . . . . . . . . −xn−2 +2xn−1 −xn = cn−1; −xn−1 +2xn = cn.
IMO 1979 LL FIN19 Origin: FIN Problem For k = 1, 2, . . . consider the k-tuples (a1, a2, . . . , ak) of positive integers such that a1 + 2a2 + \cdot \cdot \cdot + kak = 1979. Show that there are as many such k-tuples with odd k as there are with even k.
IMO 1992 LL TUR74 Origin: TUR Problem Let S = \pin 1992m | n, m \inZ . Show that every real number x \geq0 is an accumulation point of S.
IMO 1967 LL ROM43 Origin: ROM Problem The equation x5 + 5\lambdax4 −x3 + (\lambda\alpha −4)x2 −(8\lambda + 3)x + \lambda\alpha −2 = 0 is given. (a) Determine \alpha such that the given equation has exactly one root inde- pendent of \lambda. (b) Determine \alpha such that the given equation has exactly two roots inde- pendent of \lambda. Solution We can write the given equation in the form x5 −x3...
IMO 1977 LL USS50 Origin: USS Problem Determine all positive integers n for which there exists a poly- nomial Pn(x) of degree n with integer coefficients that is equal to n at n different integer points and that equals zero at zero. Solution Suppose that Pn(x) = n for x \in{x1, x2, . . . , xn}. Then Pn(x) = (x −x1)(x −x2) \cdot \cdot \cdot (x −xn) + n....
IMO 1977 LL ROM37 Origin: ROM Problem Let A1, A2, . . . , An+1 be positive integers such that (Ai, An+1) = 1 for every i = 1, 2, . . ., n. Show that the equation xA1 xA2 \cdot \cdot \cdot + xAn n = xAn+1 n+1 has an infinite set of solutions (x1, x2, . . . , xn+1) in positive integers. Solution We look for a...
IMO 1967 LL CZS8 Origin: CZS Problem ABCD is a parallelogram; AB = a, AD = 1, \alpha is the size of \angleDAB, and the three angles of the triangle ABD are acute. Prove that the four circles KA, KB, KC, KD, each of radius 1, whose centers are the vertices A, B, C, D, cover the parallelogram if and only if a \leq cos \alpha + \sqrt 3 sin...
IMO 1979 LL VIE76 Origin: VIE Problem Suppose that a triangle whose sides are of integer lengths is inscribed in a circle of diameter 6.25. Find the sides of the triangle.
IMO 1984 LL ROM46 Origin: ROM Problem Let (an)n\geq1 and (bn)n\geq1 be two sequences of natural numbers such that an+1 = nan + 1, bn+1 = nbn −1 for every n \geq1. Show that these two sequences can have only a finite number of terms in common.
IMO 1978 LL SWE35 Origin: SWE Problem A sequence (an)N 0 of real numbers is called concave if 2an \geq an−1 + an+1 for all integers n, 1 \leqn \leqN −1. (a) Prove that there exists a constant C > 0 such that N n=0 an \geqC(N −1) N n=0 a2 n (1) for all concave positive sequences (an)N 0 . (b) Prove that (1) holds with C...
IMO 1985 LL SPA72 Origin: SPA Problem Construct a triangle ABC given the side AB and the distance OH from the circumcenter O to the orthocenter H, assuming that OH and AB are parallel.
IMO 1987 LL BEL8 Origin: BEL Problem Determine the least possible value of the natural number n such that n! ends in exactly 1987 zeros.
IMO 1992 LL ICE24 Origin: ICE Problem Let Q+ denote the set of nonnegative rational numbers. Show that there exists exactly one function f : Q+ \toQ+ satisfying the follow- ing conditions: (i) if 0 < q < 1 2, then f(q) = 1 + f q 1−2q ; (ii) if 1 < q \leq2, then f(q) = 1 + f(q + 1); (iii) f(q)f(1/q) = 1 for all...
IMO 1985 LL ROM68 Origin: ROM Problem Show that the sequence {an}n\geq1 defined by an = [n \sqrt 2] contains an infinite number of integer powers of 2. ([x] is the integer part of x.)
IMO 1979 LL GDR32 Origin: GDR Problem Let n, k \geq1 be natural numbers. Find the number A(n, k) of solutions in integers of the equation |x1| + |x2| + \cdot \cdot \cdot + |xk| = n.
IMO 1988 LL FRA9 Origin: FRA Problem If a0 is a positive real number, consider the sequence {an} defined by an+1 = a2 n −1 n + 1 for n \geq0. Show that there exists a real number a > 0 such that: (i) for all real a0 \geqa, the sequence {an} \to+\infty(n \to\infty); (ii) for all real a0 < a, the sequence {an} \to0.
IMO 1970 LL SWE53 Origin: SWE Problem A square ABCD is divided into (n −1)2 congruent squares, with sides parallel to the sides of the given square. Consider the grid of all n2 corners obtained in this manner. Determine all integers n for which it is possible to construct a nondegenerate parabola with its axis parallel to one side of the square and that passes through exactly n points of...
IMO 1966 LL USS56 Origin: USS Problem Let ABCD be a tetrahedron such that AB \perpCD, AC \perpBD, and AD \perpBC. Prove that the midpoints of the edges of the tetrahedron lie on a sphere.
IMO 1988 LL GDR24 Origin: GDR Problem Let Zm,n be the set of all ordered pairs (i, j) with i \in {1, . . . , m} and j \in{1, . . ., n}. Also let am,n be the number of all those subsets of Zm,n that contain no two ordered pairs (i1, j1), (i2, j2) with |i1 −i2| + |j1 −j2| = 1. Show that for all positive integers...
IMO 1966 LL BUL32 Origin: BUL Problem The sides a, b, c of a triangle ABC form an arithmetic progression; the sides of another triangle A1B1C1 also form an arithmetic progression. Suppose that \angleA = \angleA1. Prove that the triangles ABC and A1B1C1 are similar.
IMO 1969 LL FRA19 Origin: FRA Problem Let n be an integer that is not divisible by any square greater than 1. Denote by xm the last digit of the number xm in the number system with base n. For which integers x is it possible for xm to be 0? Prove that the sequence xm is periodic with period t independent of x. For which x do we have...
IMO 1979 LL USA63 Origin: USA Problem If a1, a2, . . . , an denote the lengths of the sides of an arbitrary n-gon, prove that 2 \geq a1 s −a1 + a2 s −a2 \cdot \cdot \cdot + an s −an \geq n n −1, where s = a1 + a2 + \cdot \cdot \cdot + an.
IMO 1989 LL LUX65 Origin: LUX Problem A regular n-gon A1A2A3 . . . Ak . . . An inscribed in a circle of radius R is given. If S is a point on the circle, calculate T = SA2 1 +SA2 2 + \cdot \cdot \cdot + SA2 n.
IMO 1984 LL BEL4 Origin: BEL Problem Given a triangle ABC, three equilateral triangles AEB, BFC, and CGA are constructed in the exterior of ABC. Prove that: (a) CE = AF = BG; (b) CE, AF, and BG have a common point.
IMO 1985 LL CAN15 Origin: CAN Problem Superchess is played on on a 12 \times 12 board, and it uses su- perknights, which move between opposite corner cells of any 3\times4 subboard. Is it possible for a superknight to visit every other cell of a superchessboard exactly once and return to its starting cell?
IMO 1985 LL CZS20 Origin: CZS Problem Let T be the set of all lattice points (i.e., all points with integer coordinates) in three-dimensional space. Two such points (x, y, z) and (u, v, w) are called neighbors if |x −u| + |y −v| + |z −w| = 1. Show that there exists a subset S of T such that for each p \inT , there is exactly one point...
IMO 1984 LL USS68 Origin: USS Problem In the Martian language every finite sequence of letters of the Latin alphabet letters is a word. The publisher “Martian Words” makes a collection of all words in many volumes. In the first volume there are only one-letter words, in the second, two-letter words, etc., and the numeration of the words in each of the volumes continues the numeration of the previous volume....
IMO 1992 LL MON49 Origin: MON Problem Given real numbers xi (i = 1, 2, . . . , 4x + 2) such that 4x+2 i=1 (−1)i+1xixi+1 = 4m (x1 = x4k+3), prove that it is possible to choose numbers xk1, . . . , xk6 such that 6 The problem in this formulation is senseless. The correct formulation could be, “Find . . . such that \infty m=1...
IMO 1979 LL YUG79 Origin: YUG Problem Let S be a unit circle and K a subset of S consisting of several closed arcs. Let K satisfy the following properties: (i) K contains three points A, B, C, that are the vertices of an acute- angled triangle; (ii) for every point A that belongs to K its diametrically opposite point A′ and all points B on an arc of length...
IMO 1969 LL BEL6 Origin: BEL Problem Evaluate (cos(\pi/4) + i sin(\pi/4))10 in two different ways and prove that 10 − 10 1 10 = 24.
IMO 1989 LL ICE41 Origin: ICE Problem Alice has two urns. Each urn contains four balls and on each ball a natural number is written. She draws one ball from each urn at random, notes the sum of the numbers written on them, and replaces the balls in the urns from which she took them. This she repeats a large number of times. Bill, on examining the numbers recorded, notices...
IMO 1992 LL ITA40 Origin: ITA Problem The colonizers of a spherical planet have decided to build N towns, each having area 1/1000 of the total area of the planet. They also decided that any two points belonging to different towns will have different latitude and different longitude. What is the maximal value of N?
IMO 1969 LL FRA18 Origin: FRA Problem Let a and b be two nonnegative integers. Denote by H(a, b) the set of numbers n of the form n = pa + qb, where p and q are positive integers. Determine H(a) = H(a, a). Prove that if a ̸= b, it is enough to know all the sets H(a, b) for coprime numbers a, b in order to know all...
IMO 1967 LL POL38 Origin: POL Problem Does there exist an integer such that its cube is equal to 3n2 + 3n + 7, where n is integer? Solution Suppose that there exist integers n and m such that m3 = 3n2 + 3n + 7. Then from m3 \equiv1 (mod 3) it follows that m = 3k + 1 for some k \inZ. Substituting into the initial equation we...
IMO 1977 LL USA55 Origin: USA Problem Through a point O on the diagonal BD of a parallelogram ABCD, segments MN parallel to AB, and PQ parallel to AD, are drawn, with M on AD, and Q on AB. Prove that diagonals AO, BP, DN (ex- tended if necessary) will be concurrent. Solution The statement is true without the assumption that O \inBD. Let BP \cap DN = {K}. If...
IMO 1982 LL TUN44 Origin: TUN Problem Let A and B be positions of two ships M and N, respectively, at the moment when N saw M moving with constant speed v following the line Ax. In search of help, N moves with speed kv (k < 1) along the line By in order to meet M as soon as possible. Denote by C the point of meeting of the...
IMO 1986 LL ROM61 Origin: ROM Problem Given a positive integer n, find the greatest integer p with the property that for any function f : P(X) \toC, where X and C are sets of cardinality n and p, respectively, there exist two distinct sets A, B \inP(X) such that f(A) = f(B) = f(A \cupB). (P(X) is the family of all subsets of X.)
IMO 1982 LL BUL13 Origin: BUL Problem A regular n-gonal truncated pyramid is circumscribed around a sphere. Denote the areas of the base and the lateral surfaces of the pyramid by S1, S2, and S, respectively. Let \sigma be the area of the polygon whose vertices are the tangential points of the sphere and the lateral faces of the pyramid. Prove that \sigmaS = 4S1S2 cos2 \pi n.
IMO 1970 LL NET36 Origin: NET Problem Let x, y, z be nonnegative real numbers satisfying x2 + y2 + z2 = 5 and yz + zx + xy = 2. Which values can the greatest of the numbers x2 −yz, y2 −xz, z2 −xy have?
IMO 1974 LL ROM27 Origin: ROM Problem Let C1 and C2 be circles in the same plane, P1 and P2 arbitrary points on C1 and C2 respectively, and Q the midpoint of segment P1P2. Find the locus of points Q as P1 and P2 go through all possible positions. Alternative version. Let C1, C2, C3 be three circles in the same plane. Find the locus of the centroid of triangle...
IMO 1967 LL MON35 Origin: MON Problem Prove the identity n k=0 n k tan x 2k 1 + 2k 1 −tan2(x/2) k = sec2n x 2 + secn x. Solution The given sum can be rewritten as n k=0 n k tan2 x k + n k=0 n k 2 tan2 x 1 −tan2 x k . Since 2...
IMO 1966 LL BUL34 Origin: BUL Problem Determine all pairs of positive integers (x, y) satisfying the equa- tion 2x = 3y + 5.
IMO 1977 LL GDR17 Origin: GDR Problem A ball K of radius r is touched from the outside by mutually equal balls of radius R. Two of these balls are tangent to each other. Moreover, for two balls K1 and K2 tangent to K and tangent to each other there exist two other balls tangent to K1, K2 and also to K. How many balls are tangent to K? For...
IMO 1989 LL THA98 Origin: THA Problem Let f : N \toN be such that (i) f is strictly increasing; (ii) f(mn) = f(m)f(n) \forallm, n \inN; and (iii) if m ̸= n and mn = nm, then f(m) = n or f(n) = m. Determine f(30).
IMO 1969 LL MON42 Origin: MON Problem Let Ak (1 \leqk \leqh) be n-element sets such that each two of them have a nonempty intersection. Let A be the union of all the sets Ak, and let B be a subset of A such that for each k (1 \leqk \leqh) the intersection of Ak and B consists of exactly two different elements ak and bk. Find all subsets X...
IMO 1972 LL SWE37 Origin: SWE Problem On a chessboard (8 \times 8 squares with sides of length 1) two diagonally opposite corner squares are taken away. Can the board now be covered with nonoverlapping rectangles with sides of lengths 1 and 2?
IMO 1970 LL POL38 Origin: POL Problem Find the greatest integer A for which in any permutation of the numbers 1, . . . , 100 there exist ten consecutive numbers whose sum is at least A.
IMO 1984 LL USA59 Origin: USA Problem Determine the smallest positive integer m such that 529n +m\cdot 132n is divisible by 262417 for all odd positive integers n.
IMO 1979 LL BEL5 Origin: BEL Problem Describe which natural numbers do not belong to the set E = {[n + \sqrtn + 1/2] | n \inN}.
IMO 1979 LL BEL3 Origin: BEL Problem Is it possible to partition 3-dimensional Euclidean space into 1979 mutually isometric subsets?
IMO 1966 LL POL4 Origin: POL Problem Five points in the plane are given, no three of which are collinear. Show that some four of them form a convex quadrilateral.
IMO 1978 LL TUR37 Origin: TUR Problem Simplify loga(abc) + logb(abc) + logc(abc), where a, b, c are positive real numbers.
IMO 1969 LL POL52 Origin: POL Problem Prove that a regular polygon with an odd number of edges cannot be partitioned into four pieces with equal areas by two lines that pass through the center of polygon.
IMO 1987 LL TUR62 Origin: TUR Problem Let l, l′ be two lines in 3-space and let A, B, C be three points taken on l with B as midpoint of the segment AC. If a, b, c are the distances of A, B, C from l′, respectively, show that b \leq a2+c2 , equality holding if l, l′ are parallel.
IMO 1992 LL IND27 Origin: IND Problem Let ABC be an arbitrary scalene triangle. Define \Sigma to be the set of all circles y that have the following properties: (i) y meets each side of \triangleABC in two (possibly coincident) points; (ii) if the points of intersection of y with the sides of the triangle are la- beled by P, Q, R, S, T , U, with the points occurring...
IMO 1983 LL GBR29 Origin: GBR Problem Let O be a point outside a given circle. Two lines OAB, OCD through O meet the circle at A, B, C, D, where A, C are the midpoints of OB, OD, respectively. Additionally, the acute angle \theta between the lines is equal to the acute angle at which each line cuts the circle. Find cos \theta and show that the tangents at...
IMO 1967 LL GBR18 Origin: GBR Problem If x is a positive rational number, show that x can be uniquely expressed in the form x = a1 + a2 2! + a3 3! + \cdot \cdot \cdot , where a1, a2, . . . are integers, 0 \leqan \leqn −1 for n > 1, and the series terminates. Show also that x can be expressed as the sum of reciprocals...
IMO 1971 LL HUN26 Origin: HUN Problem An infinite set of rectangles in the Cartesian coordinate plane is given. The vertices of each of these rectangles have coordinates (0, 0), (p, 0), (p, q), (0, q) for some positive integers p, q. Show that there must exist two among them one of which is entirely contained in the other.
IMO 1979 LL NET48 Origin: NET Problem In the plane a circle C of unit radius is given. For any line l a number s(l) is defined in the following way: If l and C intersect in two points, s(l) is their distance; otherwise, s(l) = 0. Let P be a point at distance r from the center of C. One defines M(r) to be the maximum value of the...
IMO 1989 LL ROM92 Origin: ROM Problem Find the set of all a \inR for which there is no infinite sequence (xn)n\geq0 \subsetR satisfying x0 = a, xn+1 = xn+\alpha \betaxn+1, n = 0, 1, . . . , where \alpha\beta > 0.
IMO 1971 LL USS51 Origin: USS Problem Suppose that the sides AB and DC of a convex quadrilateral ABCD are not parallel. On the sides BC and AD, pairs of points (M, N) and (K, L) are chosen such that BM = MN = NC and AK = KL = LD. Prove that the areas of triangles OKM and OLN are different, where O is the intersection point of AB...
IMO 1988 LL USS88 Origin: USS Problem There are six circles inside a fixed circle, each tangent to the fixed circle and tangent to the two adjacent smaller circles. If the points of contact between the six circles and the larger circle are, in order, A1, A2, A3, A4, A5, and A6, prove that A1A2 \cdot A3A4 \cdot A5A6 = A2A3 \cdot A4A5 \cdot A6A1.
IMO 1977 LL USA54 Origin: USA Problem If 0 \leqa \leqb \leqc \leqd, prove that abbccdda \geqbacbdcad. Solution We shall use the following lemma. Lemma. If a real function f is convex on the interval I and x, y, z \inI, x \leqy \leqz, then (y −z)f(x) + (z −x)f(y) + (x −y)f(z) \leq0. Proof. The inequality is obvious for x = y = z. If x < z, then...
IMO 1977 LL GDR18 Origin: GDR Problem Given an isosceles triangle ABC with a right angle at C, construct the center M and radius r of a circle cutting on segments AB, BC, CA the segments DE, FG, and HK, respectively, such that \angleDME + \angleFMG + \angleHMK = 180◦and DE : FG : HK = AB : BC : CA. Solution Let U be the midpoint of the segment...
IMO 1967 LL ITA28 Origin: ITA Problem Find values of the parameter u for which the expression y = tan(x −u) + tan x + tan(x + u) tan(x −u) tan x tan(x + u) does not depend on x. Solution The given expression can be transformed into y = 4 cos 2u + 2 cos 2u −cos 2x −3. It does not depend on x if and only if...
IMO 1979 LL GRE35 Origin: GRE Problem Given a sequence (an), with a1 = 4 and an+1 = a2 n−2 (\foralln \inN), prove that there is a triangle with side lengths an −1, an, an + 1, and that its area is equal to an integer.
IMO 1992 LL POR58 Origin: POR Problem Let ABC be a triangle. Denote by a, b, and c the lengths of the sides opposite to the angles A, B, and C, respectively. Prove that7 bc a + b + c = sin A + sin B + sin C cos(A/2) sin(B/2) sin(C/2).
IMO 1979 LL VIE74 Origin: VIE Problem Given an equilateral triangle ABC of side a in a plane, let M be a point on the circumcircle of the triangle. Prove that the sum s = MA4 + MB4 + MC4 is independent of the position of the point M on the circle, and determine that constant value as a function of a.
IMO 1983 LL SWE62 Origin: SWE Problem A circle \gamma is drawn and let AB be a diameter. The point C on \gamma is the midpoint of the line segment BD. The line segments AC and DO, where O is the center of \gamma, intersect at P. Prove that there is a point E on AB such that P is on the circle with diameter AE.
IMO 1978 LL GDR29 Origin: GDR Problem (Variant of GDR 4) Given a nonconstant function f : R+ \toR such that f(xy) = f(x)f(y) for any x, y > 0, find functions c, s : R+ \toR that satisfy c(x/y) = c(x)c(y)−s(x)s(y) for all x, y > 0 and c(x)+s(x) = f(x) for all x > 0.
IMO 1988 LL USS86 Origin: USS Problem Let a, b, c be integers different from zero. It is known that the equation ax2 + by2 + cz2 = 0 has a solution (x, y, z) in integers different from the solution x = y = z = 0. Prove that the equation ax2+by2+cz2 = 1 has a solution in rational numbers.
IMO 1969 LL CZS15 Origin: CZS Problem Let K1, . . . , Kn be nonnegative integers. Prove that K1!K2! \cdot \cdot \cdot Kn! \geq[K/n]!n, where K = K1 + \cdot \cdot \cdot + Kn.
IMO 1989 LL COL9 Origin: COL Problem Let m be a positive integer and define f(m) to be the number of factors of 2 in m! (that is, the greatest positive integer k such that 2k | m!). Prove that there are infinitely many positive integers m such that m −f(m) = 1989.
IMO 1983 LL ISR36 Origin: ISR Problem The set X has 1983 members. There exists a family of subsets {S1, S2, . . . , Sk} such that: (i) the union of any three of these subsets is the entire set X, while (ii) the union of any two of them contains at most 1979 members. What is the largest possible value of k?
IMO 1977 LL GBR21 Origin: GBR Problem Given that x1+x2+x3 = y1+y2+y3 = x1y1+x2y2+x3y3 = 0, prove that x2 x2 1 + x2 2 + x2 + y2 y2 1 + y2 2 + y2 = 2 3. Solution Let us consider the vectors v1 = (x1, x2, x3), v2 = (y1, y2, y3), v3 = (1, 1, 1) in space. The given equalities express the condition that these three...
IMO 1983 LL ISR37 Origin: ISR Problem The points A1, A2, . . . , A1983 are set on the circumference of a circle and each is given one of the values \pm1. Show that if the number of points with the value +1 is greater than 1789, then at least 1207 of the points will have the property that the partial sums that can be formed by taking the...
IMO 1970 LL ROM43 Origin: ROM Problem Prove that the equation x3 −3 tan \pi 12x2 −3x + tan \pi 12 = 0 has one root x1 = tan \pi 36, and find the other roots.
IMO 1972 LL USS43 Origin: USS Problem A fixed point A inside a circle is given. Consider all chords XY of the circle such that \angleXAY is a right angle, and for all such chords construct the point M symmetric to A with respect to XY . Find the locus of points M.
IMO 1986 LL GDR37 Origin: GDR Problem Prove that the set {1, 2, . . ., 1986} can be partitioned into 27 disjoint sets so that no one of these sets contains an arithmetic triple (i.e., three distinct numbers in an arithmetic progression).
IMO 1970 LL BEL7 Origin: BEL Problem Let ABCD be any quadrilateral. A square is constructed on each side of the quadrilateral, all in the same manner (i.e., outward or inward). Denote the centers of the squares by M1, M2, M3, and M4. Prove: (a) M1M3 = M2M4; (b) M1M3 is perpendicular to M2M4.
IMO 1978 LL TUR40 Origin: TUR Problem If Cp n = n! p!(n−p)! (p \geq1), prove the identity Cp n = Cp−1 n−1 + Cp−1 n−2 + \cdot \cdot \cdot + Cp−1 p Cp−1 p−1 and then evaluate the sum S = 1 \cdot 2 \cdot 3 + 2 \cdot 3 \cdot 4 + \cdot \cdot \cdot + 97 \cdot 98 \cdot 99.
IMO 1971 LL YUG54 Origin: YUG Problem A set M is formed of 2n n men, n = 1, 2, . . .. Prove that we can choose a subset P of the set M consisting of n + 1 men such that one of the following conditions is satisfied: (1) every member of the set P knows every other member of the set P; (2) no member of...
IMO 1974 LL CUB6 Origin: CUB Problem Prove that the product of two natural numbers with their sum cannot be the third power of a natural number.
IMO 1979 LL BRA9 Origin: BRA Problem The real numbers \alpha1, \alpha2, \alpha3, . . . , \alphan are positive. Let us denote by h = n 1/\alpha1+1/\alpha2+\cdot\cdot\cdot+1/\alphan the harmonic mean, g = n\sqrt\alpha1\alpha2 \cdot \cdot \cdot \alphan the geometric mean, a = \alpha1+\alpha2+\cdot\cdot\cdot+\alphan n the arithmetic mean. Prove that h \leq g \leqa, and that each of the equalities implies the other one.
IMO 1986 LL MON53 Origin: MON Problem For given positive integers r, v, n let S(r, v, n) denote the num- ber of n-tuples of nonnegative integers (x1, . . . , xn) satisfying the equation x1 + \cdot \cdot \cdot + xn = r and such that xi \leqv for i = 1, . . . , n. Prove that S(r, v, n) = m k=0 (−1)k n...
IMO 1971 LL USS49 Origin: USS Problem Diagonals of a convex quadrilateral ABCD intersect at a point O. Find all angles of this quadrilateral if ∡OBA = 30◦, ∡OCB = 45◦, ∡ODC = 45◦, and ∡OAD = 30◦.
IMO 1976 LL USA39 Origin: USA Problem In \triangleABC, the inscribed circle is tangent to side BC at X. Segment AX is drawn. Prove that the line joining the midpoint of segment AX to the midpoint of side BC passes through the center I of the inscribed circle.
IMO 1969 LL MON44 Origin: MON Problem Find the radius of the circle circumscribed about the isosceles triangle whose sides are the solutions of the equation x2 −ax + b = 0.
IMO 1970 LL BUL14 Origin: BUL Problem Let \alpha + \beta + \gamma = \pi. Prove that sin 2\alpha + sin 2\beta + sin 2\gamma = 2(sin \alpha + sin \beta + sin \gamma)(cos \alpha + cos \beta + cos \gamma) −2(sin \alpha + sin \beta + sin \gamma).
IMO 1969 LL GBR24 Origin: GBR Problem The polynomial P(x) = a0xk + a1xk−1 + \cdot \cdot \cdot + ak, where a0, . . . , ak are integers, is said to be divisible by an integer m if P(x) is a multiple of m for every integral value of x. Show that if P(x) is divisible by m, then a0 \cdotk! is a multiple of m. Also prove that...
IMO 1985 LL ROM69 Origin: ROM Problem Let A and B be two finite disjoint sets of points in the plane such that any three distinct points in A\cupB are not collinear. Assume that at least one of the sets A, B contains at least five points. Show that there exists a triangle all of whose vertices are contained in A or in B that does not contain in its...
IMO 1984 LL USS64 Origin: USS Problem For a matrix (pij) of the format m \times n with real entries, set ai = n j=1 pij for i = 1, . . . , m and bj = m i=1 pij for j = 1, . . . , n. (1) By integering a real number we mean replacing the number with the in- teger closest to it....
IMO 1983 LL VIE74 Origin: VIE Problem In a plane we are given two distinct points A, B and two lines a, b passing through B and A respectively (a ∋B, b ∋A) such that the line AB is equally inclined to a and b. Find the locus of points M in the plane such that the product of distances from M to A and a equals the product of...
IMO 1977 LL HUN25 Origin: HUN Problem Prove the identity (z + a)n = zn + a n k=1 n k (a −kb)k−1(z + kb)n−k. Solution Let fn(z) = zn + a n k=1 n k (a −kb)k−1(z + kb)n−k. We shall prove by induction on n that fn(z) = (z + a)n. This is trivial for n = 1. Suppose that the statement is true...
IMO 1986 LL MON51 Origin: MON Problem Let a, b, c, d be the lengths of the sides of a quadrilateral circumscribed about a circle and let S be its area. Prove that S \leq \sqrt abcd and find conditions for equality.
IMO 1982 LL USS52 Origin: USS Problem We are given 2n natural numbers 1, 1, 2, 2, 3, 3, . . ., n −1, n −1, n, n. Find all n for which these numbers can be arranged in a row such that for each k \leqn, there are exactly k numbers between the two numbers k.
IMO 1989 LL INA47 Origin: INA Problem Let log2 2 x −4 log2 x −m2 −2m −13 = 0 be an equation in x. Prove: (a) For any real value of m the equation has has two distinct solutions. (b) The product of the solutions of the equation does not depend on m. (c) One of the solutions of the equation is less than 1, while the other solution is...
IMO 1974 LL ROM29 Origin: ROM Problem Let A, B, C, D be points in space. If for every point M on the segment AB the sum area(AMC)+area(CMD)+area(DMB) is constant show that the points A, B, C, D lie in the same plane.
IMO 1970 LL BEL10 Origin: BEL Problem Let A, B, C be angles of a triangle. Prove that 1 < cos A + cos B + cos C \leq3 2.
IMO 1986 LL NET60 Origin: NET Problem Prove the inequality (−a+b+c)2(a−b+c)2(a+b−c)2 \geq(−a2+b2+c2)(a2−b2+c2)(a2+b2−c2) for all real numbers a, b, c.
IMO 1969 LL HUN37 Origin: HUN Problem If a1, a2, . . . , an are real constants, and if y = cos(a1 + x) + 2 cos(a2 + x) + \cdot \cdot \cdot + n cos(an + x) has two zeros x1 and x2 whose difference is not a multiple of \pi, prove that y \equiv0.
IMO 1976 LL USA40 Origin: USA Problem Let g(x) be a fixed polynomial and define f(x) by f(x) = x2 + xg(x3). Show that f(x) is not divisible by x2 −x + 1.
IMO 1984 LL USA62 Origin: USA Problem From a point P exterior to a circle K, two rays are drawn intersecting K in the respective pairs of points A, A′ and B, B′. For any other pair of points C, C′ on K, let D be the point of intersection of the circumcircles of triangles PAC and PB′C′ other than point P. Similarly, let D′ be the point of intersection...
IMO 1982 LL USA47 Origin: USA Problem Evaluate sec′′ \pi 4 +sec′′ 3\pi 4 +sec′′ 5\pi 4 +sec′′ 7\pi 4 . (Here sec′′ means the second derivative of sec.)
IMO 1985 LL ITA47 Origin: ITA Problem Let F be the correspondence associating with every point P = (x, y) the point P ′ = (x′, y′) such that x′ = ax + b, y′ = ay + 2b. (1) Show that if a ̸= 1, all lines PP ′ are concurrent. Find the equation of the set of points corresponding to P = (1, 1) for b = a2....
IMO 1966 LL CZS1 Origin: CZS Problem We are given n > 3 points in the plane, no three of which lie on a line. Does there necessarily exist a circle that passes through at least three of the given points and contains none of the other given points in its interior?
IMO 1967 LL POL40 Origin: POL Problem Exactly one side of a tetrahedron is of length greater than Solution Suppose CD is the longest edge of the tetrahedron ABCD, AB = a, CK and DL are the altitudes of the triangles ABC and ABD respectively, and DM is the altitude of the tetrahedron ABCD. Then CK2 \leq1 −a2/4, since CK is a leg of the right triangle whose other leg...
IMO 1971 LL SWE42 Origin: SWE Problem Let Li, i = 1, 2, 3, be line segments on the sides of an equilateral triangle, one segment on each side, with lengths li, i = 1, 2, 3. By L∗ i we denote the segment of length li with its midpoint on the midpoint of the corresponding side of the triangle. Let M(L) be the set of points in the plane...
IMO 1972 LL ROM34 Origin: ROM Problem If p is a prime number greater than 2 and a, b, c integers not divisible by p, prove that the equation ax2 + by2 = pz + c has an integer solution.
IMO 1969 LL POL54 Origin: POL Problem Given a polynomial f(x) with integer coefficients whose value is divisible by 3 for three integers k, k + 1, and k + 2, prove that f(m) is divisible by 3 for all integers m.
IMO 1987 LL USS71 Origin: USS Problem To every natural number k, k \geq2, there corresponds a sequence an(k) according to the following rule: a0 = k, an = \tau(an−1) for n \geq1, in which \tau(a) is the number of different divisors of a. Find all k for which the sequence an(k) does not contain the square of an integer.
IMO 1987 LL USA65 Origin: USA Problem The runs of a decimal number are its increasing or decreasing blocks of digits. Thus 024379 has three runs: 024, 43, and 379. Determine the average number of runs for a decimal number in the set {d1d2 . . . dn | dk ̸= dk+1, k = 1, 2, . . ., n −1}, where n \geq2.
IMO 1976 LL NET27 Origin: NET Problem In a plane three points P, Q, R, not on a line, are given. Let k, l, m be positive numbers. Construct a triangle ABC whose sides pass through P, Q, and R such that P divides the segment AB in the ratio 1 : k, Q divides the segment BC in the ratio 1 : l, and R divides the segment CA...
IMO 1988 LL HUN38 Origin: HUN Problem In a multiple choice test there were 4 questions and 3 possible answers for each question. A group of students was tested and it turned out that for any 3 of them there was a question that the three students answered differently. What is the maximal possible number of students tested?
IMO 1978 LL BUL3 Origin: BUL Problem Find all numbers \alpha for which the equation x2 −2x[x] + x −\alpha = 0 has two nonnegative roots. ([x] denotes the largest integer less than or equal to x.)
IMO 1977 LL FRG12 Origin: FRG Problem Let z be an integer > 1 and let M be the set of all numbers of the form zk = 1 + z + \cdot \cdot \cdot + zk, k = 0, 1, . . . . Determine the set T of divisors of at least one of the numbers zk from M. Solution According to part (a) of the previous problem...
IMO 1986 LL CHN12 Origin: CHN Problem Let O be an interior point of a tetrahedron A1A2A3A4. Let S1, S2, S3, S4 be spheres with centers A1, A2, A3, A4, respectively, and let U, V be spheres with centers at O. Suppose that for i, j = 1, 2, 3, 4, i ̸= j, the spheres Si and Sj are tangent to each other at a point Bij lying on...
IMO 1987 LL YUG77 Origin: YUG Problem Find the least natural number k such that for any n \in[0, 1] and any natural number n, ak(1 −a)n < (n + 1)3 .
IMO 1986 LL BEL4 Origin: BEL Problem Find the last eight digits of the binary development of 271986.
IMO 1969 LL BUL10 Origin: BUL Problem Let M be the point inside the right-angled triangle ABC (\angleC = 90◦) such that \angleMAB = \angleMBC = \angleMCA = ϕ. Let \psi be the acute angle between the medians of AC and BC. Prove that sin(ϕ+\psi) sin(ϕ−\psi) = 5.
IMO 1970 LL ROM44 Origin: ROM Problem If a, b, c are side lengths of a triangle, prove that (a + b)(b + c)(c + a) \geq8(a + b −c)(b + c −a)(c + a −b).
IMO 1972 LL BUL4 Origin: BUL Problem Given a triangle, prove that the points of intersection of three pairs of trisectors of the inner angles at the sides lying closest to those sides are vertices of an equilateral triangle.
IMO 1989 LL ISR59 Origin: ISR Problem Let v1, v2, . . . , v1989 be a set of coplanar vectors with |vr| \leq1 for 1 \leqr \leq1989. Show that it is possible to find ϵr (1 \leqr \leq1989), each equal to \pm1, such that 1989 r=1 ϵrvr \leq \sqrt 3.
IMO 1984 LL SPA52 Origin: SPA Problem Construct a scalene triangle such that a(tan B −tan C) = b(tan A −tan C).
IMO 1984 LL SPA51 Origin: SPA Problem Two cyclists leave simultaneously a point P in a circular run- way with constant velocities v1, v2 (v1 > v2) and in the same sense. A pedestrian leaves P at the same time, moving with velocity v3 = v1+v2 . If the pedestrian and the cyclists move in opposite directions, the pedes- trian meets the second cyclist 91 seconds after he meets the...
IMO 1984 LL NET42 Origin: NET Problem Triangle ABC is given for which BC = AC + 1 2AB. The point P divides AB such that RP : PA = 1 : 3. Prove that \angleCAP = 2\angleCPA.
IMO 1985 LL TUR80 Origin: TUR Problem Let E = {1, 2, . . ., 16} and let M be the collection of all 4 \times 4 matrices whose entries are distinct members of E. If a matrix A = (aij)4\times4 is chosen randomly from M, compute the probability p(k) of maxi minj aij = k for k \inE. Furthermore, determine l \inE such that p(l) = max{p(k) | k...
IMO 1985 LL USS93 Origin: USS Problem The sphere inscribed in tetrahedron ABCD touches the sides ABD and DBC at points K and M, respectively. Prove that ∡AKB = ∡DMC.
IMO 1977 LL ROM38 Origin: ROM Problem Let mj > 0 for j = 1, 2, . . ., n and a1 \leq\cdot \cdot \cdot \leqan < b1 \leq\cdot \cdot \cdot \leq bn < c1 \leq\cdot \cdot \cdot \leqcn be real numbers. Prove: ⎡ ⎣ n j=1 mj(aj + bj + cj) ⎤ ⎦ 3 ⎛ ⎝ n j=1 mj ⎞ ⎠ ⎡ ⎣ n j=1 mj(ajbj...
IMO 1982 LL GDR34 Origin: GDR Problem Let M be the set of all functions f with the following proper- ties: (i) f is defined for all real numbers and takes only real values. (ii) For all x, y \inR the following equality holds: f(x)f(y) = f(x + y) + f(x −y). (iii) f(0) ̸= 0. Determine all functions f \inM such that (a) f(1) = 5/2; (b) f(1) =...
IMO 1992 LL HKG23 Origin: HKG Problem An Egyptian number is a positive integer that can be expressed as a sum of positive integers, not necessarily distinct, such that the sum of their reciprocals is 1. For example, 32 = 2 + 3 + 9 + 18 is Egyptian because 1 2 + 1 3 + 1 9 + 18 = 1. Prove that all integers greater than 23 are...
IMO 1971 LL SWE45 Origin: SWE Problem Let m and n denote integers greater than 1, and let \nu(n) be the number of primes less than or equal to n. Show that if the equation n \nu(n) = m has a solution, then so does the equation n \nu(n) = m −1.
IMO 1986 LL AUS2 Origin: AUS Problem Let ABCD be a convex quadrilateral. DA and CB meet at F and AB and DC meet at E. The bisectors of the angles DFC and AED are perpendicular. Prove that these angle bisectors are parallel to the bisectors of the angles between the lines AC and BD.
IMO 1985 LL MOR55 Origin: MOR Problem The points A, B, C are in this order on line D, and AB = 4BC. Let M be a variable point on the perpendicular to D through C. Let MT1 and MT2 be tangents to the circle with center A and radius AB. Determine the locus of the orthocenter of the triangle MT1T2.
IMO 1986 LL CHN15 Origin: CHN Problem Let N = B1 \cup\cdot \cdot \cdot\cupBq be a partition of the set N of all positive integers and let an integer l \inN be given. Prove that there exist a set X \subsetN of cardinality l, an infinite set T \subsetN, and an integer k with 1 \leqk \leqq such that for any t \inT and any finite set Y \subsetX, the...
IMO 1972 LL BUL3 Origin: BUL Problem On a line a set of segments is given of total length less than n. Prove that every set of n points of the line can be translated in some direction along the line for a distance smaller than n/2 so that none of the points remain on the segments.
IMO 1977 LL GDR15 Origin: GDR Problem Let n be an integer greater than 1. In the Cartesian coordinate system we consider all squares with integer vertices (x, y) such that 1 \leq x, y \leqn. Denote by pk (k = 0, 1, 2, . . . ) the number of pairs of points that are vertices of exactly k such squares. Prove that k(k −1)pk = 0. Solution...
IMO 1969 LL SWE59 Origin: SWE Problem For each \lambda (0 < \lambda < 1 and \lambda ̸= 1/n for all n = 1, 2, 3, . . .) construct a continuous function f such that there do not exist x, y with 0 < \lambda < y = x + \lambda \leq1 for which f(x) = f(y).
IMO 1969 LL BEL5 Origin: BEL Problem Let G be the centroid of the triangle OAB. (a) Prove that all conics passing through the points O, A, B, G are hyper- bolas. (b) Find the locus of the centers of these hyperbolas.
IMO 1988 LL KOR56 Origin: KOR Problem The Fibonacci sequence is defined by an+1 = an + an−1 (n \geq1), a0 = 0, a1 = a2 = 1. Find the greatest common divisor of the 1960th and 1988th terms of the Fibonacci sequence.
IMO 1969 LL BUL9 Origin: BUL Problem One hundred convex polygons are placed on a square with edge of length 38 cm. The area of each of the polygons is smaller than \pi cm2, and the perimeter of each of the polygons is smaller than 2\pi cm. Prove that there exists a disk with radius 1 in the square that does not intersect any of the polygons.
IMO 1982 LL FIN23 Origin: FIN Problem Determine the sum of all positive integers whose digits (in base ten) form either a strictly increasing or a strictly decreasing sequence.
IMO 1972 LL SWE42 Origin: SWE Problem The decimal number 13101 is given. It is instead written as a ternary number. What are the two last digits of this ternary number?
IMO 1992 LL MON48 Origin: MON Problem Find all the functions f : R+ \toR satisfying the identity f(x)f(y) = y\alpha \cdot f x x\beta \cdot f y , x, y \inR+, where \alpha, \beta are given real numbers.
IMO 1972 LL USS45 Origin: USS Problem Let ABCD be a convex quadrilateral whose diagonals AC and BD intersect at point O. Let a line through O intersect segment AB at M and segment CD at N. Prove that the segment MN is not longer than at least one of the segments AC and BD.
IMO 1986 LL FRG27 Origin: FRG Problem In an urn there are n balls numbered 1, 2, . . . , n. They are drawn at random one by one one without replacement and the numbers are recorded. What is the probability that the resulting random permutation has only one local maximum? A term in a sequence is a local maximum if it is greater than all its neighbors.
IMO 1987 LL SPA58 Origin: SPA Problem Find, with argument, the integer solutions of the equation 3z2 = 2x3 + 385x2 + 256x −58195.
IMO 1984 LL ROM48 Origin: ROM Problem Let ABC be a triangle with interior angle bisectors AA1, BB1, CC1 and incenter I. If \sigma[IA1B] + \sigma[IB1C] + \sigma[IC1A] = 1 2\sigma[ABC], where \sigma[ABC] denotes the area of ABC, show that ABC is isosceles.
IMO 1979 LL ROM55 Origin: ROM Problem Let a, b be coprime integers. Show that the equation ax2 + by2 = z3 has an infinite set of solutions (x, y, z) with x, y, z \inZ and x, y mutually coprime (in each solution).
IMO 1983 LL VIE75 Origin: VIE Problem Find the sum of the fiftieth powers of all sides and diagonals of a regular 100-gon inscribed in a circle of radius R.
IMO 1974 LL FIN14 Origin: FIN Problem Let n and k be natural numbers and a1, a2, . . . , an positive real numbers satisfying a1 + a2 + \cdot \cdot \cdot + an = 1. Prove that a−k a−k \cdot \cdot \cdot + a−k n \geqnk+1.
IMO 1977 LL USA52 Origin: USA Problem Two perpendicular chords are drawn through a given interior point P of a circle with radius R. Determine, with proof, the maximum and the minimum of the sum of the lengths of these two chords if the distance from P to the center of the circle is kR. Solution The maximum and minimum are 2R \sqrt 4 −2k2 and 2R 1 +...
IMO 1972 LL MON24 Origin: MON Problem The diagonals of a convex 18-gon are colored in 5 different colors, each color appearing on an equal number of diagonals. The diag- onals of one color are numbered 1, 2, . . .. One randomly chooses one-fifth of all the diagonals. Find the number of possibilities for which among the chosen diagonals there exist exactly n pairs of diagonals of the same...
IMO 1967 LL SWE50 Origin: SWE Problem The function ϕ(x, y, z), defined for all triples (x, y, z) of real numbers, is such that there are two functions f and g defined for all pairs of real numbers such that ϕ(x, y, z) = f(x + y, z) = g(x, y + z) for all real x, y, and z. Show that there is a function h of one...
IMO 1986 LL HUN42 Origin: HUN Problem The integers 1, 2, . . ., n2 are placed on the fields of an n \times n chessboard (n > 2) in such a way that any two fields that have a common edge or a vertex are assigned numbers differing by at most n + 1. What is the total number of such placements?
IMO 1987 LL TUR60 Origin: TUR Problem It is given that x = −2272, y = 103 +102c+10b+a, and z = 1 satisfy the equation ax + by + cz = 1, where a, b, c are positive integers with a < b < c. Find y.
IMO 1977 LL POL31 Origin: POL Problem Let f be a function defined on the set of pairs of nonzero rational numbers whose values are positive real numbers. Suppose that f satisfies the following conditions: (1) f(ab, c) = f(a, c)f(b, c), f(c, ab) = f(c, a)f(c, b); (2) f(a, 1 −a) = 1. Prove that f(a, a) = f(a, −a) = 1, f(a, b)f(b, a) = 1. Solution We...
IMO 1982 LL FRA28 Origin: FRA Problem Let (u1, . . . , un) be an ordered ntuple. For each k, 1 \leqk \leqn, define vk = k\sqrtu1u2 \cdot \cdot \cdot uk. Prove that n k=1 vk \leqe \cdot n k=1 uk. (e is the base of the natural logarithm).
IMO 1989 LL GBR25 Origin: GBR Problem Let ABC be a triangle. Prove that there is a unique point U in the plane of ABC such that there exist real numbers \lambda, µ, \nu, \kappa, not all zero, such that \lambdaPL2 + µPM 2 + \nuPN 2 −\kappaUP 2 is constant for all points P of the plane, where L, M, N are the feet of the perpendiculars from P...
IMO 1967 LL GDR15 Origin: GDR Problem Suppose tan \alpha = p/q, where p and q are integers and q ̸= 0. Prove that the number tan \beta for which tan 2\beta = tan 3\alpha is rational only when p2 + q2 is the square of an integer. Solution Given that tan \alpha \inQ, we have that tan \beta is rational if and only if tan \gamma is rational, where...
IMO 1989 LL AUS1 Origin: AUS Problem In the set Sn = {1, 2, . . ., n} a new multiplication a∗b is defined with the following properties: (i) c = a ∗b is in Sn for any a \inSn, b \inSn. (ii) If the ordinary product a\cdotb is less than or equal to n, then a∗b = a\cdotb. (iii) The ordinary rules of multiplication hold for ∗, i.e., (1)...
IMO 1972 LL BUL6 Origin: BUL Problem Prove the inequality (n + 1) cos \pi n + 1 −n cos \pi n > 1 for all natural numbers n \geq2.
IMO 1992 LL FRG20 Origin: FRG Problem Let X and Y be two sets of points in the plane and M be a set of segments connecting points from X and Y . Let k be a natural number. Prove that the segments from M can be painted using k colors in such a way that for any point x \inX \cupY and two colors \alpha and \beta (\alpha ̸=...
IMO 1992 LL ITA39 Origin: ITA Problem Let n \geq2 be an integer. Find the minimum k for which there exists a partition of {1, 2, . . ., k} into n subsets X1, X2, . . . , Xn such that the following condition holds: for any i, j, 1 \leqi < j \leqn, there exist x1 \inX1, x2 \inX2 such that |xi −xj| = 1.
IMO 1987 LL ICE36 Origin: ICE Problem A game consists in pushing a flat stone along a sequence of squares S0, S1, S2, . . . that are arranged in linear order. The stone is ini- tially placed on square S0. When the stone stops on a square Sk it is pushed again in the same direction and so on until it reaches S1987 or goes beyond it; then the...
IMO 1992 LL FIN16 Origin: FIN Problem Find all triples (x, y, z) of integers such that x2 + 2 y2 + 3 z2 = 2 3.
IMO 1992 LL VIE82 Origin: VIE Problem Let f(x) = xm + a1xm−1 + \cdot \cdot \cdot + am−1x + am and g(x) = xn + b1xn−1 + \cdot \cdot \cdot + bn−1 + bn be two polynomials with real coefficients such that for each real number x, f(x) is the square of an integer if and only if so is g(x). Prove that if n + m > 0,...
IMO 1989 LL POR86 Origin: POR Problem Given two natural numbers w and n, the tower of n w’s is the natural number Tn(w) defined by Tn(w) = ww...w , with n w’s on the right side. More precisely, T1(w) = w and Tn+1(w) = wTn(w). For example, T3(2) = 222 = 16, T4(2) = 216 = 65536, and T2(3) = 33 = 27. Find the smallest tower of 3’s...
IMO 1989 LL KOR61 Origin: KOR Problem Let A be a set of positive integers such that no positive integer greater than 1 divides all the elements of A. Prove that any sufficiently large positive integer can be written as a sum of elements of A. (Elements may occur several times in the sum.)
IMO 1967 LL BUL2 Origin: BUL Problem Prove that 1 3n2 + 1 2n + 1 6 \geq(n!)2/n (n is a positive integer) and that equality is possible only in the case n = 1. Solution (n!)2/n = ((1 \cdot 2 \cdot \cdot \cdot n)1/n)2 \leq 1+2+\cdot\cdot\cdot+n n 2 = n+1 2 \leq1 3n2 + 1 2n + 1 6.
IMO 1966 LL ROM17 Origin: ROM Problem Suppose ABCD and A′B′C′D′ are two parallelograms arbi- trarily arranged in space, and let points M, N, P, Q divide the segments AA′, BB′, CC′, DD′ respectively in equal ratios. (a) Show that MNPQ is a parallelogram; (b) Find the locus of MNPQ as M varies along the segment AA′.
IMO 1974 LL BUL4 Origin: BUL Problem Let Ka, Kb, Kc with centers Oa, Ob, Oc be the excircles of a triangle ABC, touching the interiors of the sides BC, CA, AB at points Ta, Tb, Tc respectively. Prove that the lines OaTa, ObTb, OcTc are concurrent in a point P for which POa = POb = POc = 2R holds, where R denotes the circumradius of ABC. Also prove...
IMO 1987 LL TUR61 Origin: TUR Problem Let PQ be a line segment of constant length \lambda taken on the side BC of a triangle ABC with the order B, P, Q, C, and let the lines through P and Q parallel to the lateral sides meet AC at P1 and Q1 and AB at P2 and Q2 respectively. Prove that the sum of the areas of the trapezoids PQQ1P1...
IMO 1977 LL BUL3 Origin: BUL Problem In a company of n persons, each person has no more than d acquaintances, and in that company there exists a group of k persons, k \geqd, who are not acquainted with each other. Prove that the number of acquainted pairs is not greater than [n2/4]. Solution Let v1, v2, . . . , vk be k persons who are not acquainted with...
IMO 1970 LL BUL12 Origin: BUL Problem Let x1, x2, x3, x4, x5, x6 be given integers, not divisible by 7. Prove that at least one of the expressions of the form \pmx1 \pm x2 \pm x3 \pm x4 \pm x5 \pm x6 is divisible by 7, where the signs are selected in all possible ways. (Gener- alize the statement to every prime number!)
IMO 1969 LL YUG71 Origin: YUG Problem Let four points Ai (i = 1, 2, 3, 4) in the plane determine four triangles. In each of these triangles we choose the smallest angle. The sum of these angles is denoted by S. What is the exact placement of the points Ai if S = 180◦?
IMO 1966 LL USS51 Origin: USS Problem In a school, n children numbered 1 to n are initially arranged in the order 1, 2, . . . , n. At a command, every child can either exchange its position with any other child or not move. Can they rearrange into the order n, 1, 2, . . . , n −1 after two commands?
IMO 1989 LL POR85 Origin: POR Problem Let P(x) be a polynomial with integer coefficients such that P(m1) = P(m2) = P(m3) = P(m4) = 7 for given distinct integers m1, m2, m3, and m4. Show that there is no integer m such that P(m) = 14.
IMO 1992 LL CAN4 Origin: CAN Problem Let p, q, and r be the angles of a triangle, and let a = sin 2p, b = sin 2q, and c = sin 2r. If s = (a + b + c)/2, show that s(s −a)(s −b)(s −c) \geq0. When does equality hold?
IMO 1989 LL TUR102 Origin: TUR Problem If in a convex quadrilateral ABCD, E and F are the midpoints of the sides BC and DA respectively. Show that the sum of the areas of the triangles EDA and FBC is equal to the area of the quadrangle.
IMO 1985 LL VIE96 Origin: VIE Problem Determine all functions f : R \toR satisfying the following two conditions: (a) f(x + y) + f(x −y) = 2f(x)f(y) for all x, y \inR, (b) limx\to\inftyf(x) = 0.
IMO 1967 LL POL39 Origin: POL Problem Show that the triangle whose angles satisfy the equality sin2 A + sin2 B + sin2 C cos2 A + cos2 B + cos2 C = 2 is a right-angled triangle. Solution Since sin2 A + sin2 B + sin2 C + cos2 A + cos2 B + cos2 C = 3, the given equality is equivalent to cos2 A+cos2 B+cos2 C =...
IMO 1992 LL COL12 Origin: COL Problem Given a triangle ABC such that the circumcenter is in the interior of the incircle, prove that the triangle ABC is acute-angled.
IMO 1988 LL HKG35 Origin: HKG Problem In the triangle ABC, let D, E, and F be the midpoints of the three sides, X, Y , and Z the feet of the three altitudes, H the orthocenter, and P, Q, and R the midpoints of the line segments joining H to the three vertices. Show that the nine points D, E, F, P, Q, R, X, Y, Z lie on...
IMO 1992 LL CAN7 Origin: CAN Problem Let X be a bounded, nonempty set of points in the Cartesian plane. Let f(X) be the set of all points that are at a distance of at most 1 from some point in X. Let f n(X) = f(f(. . . (f(X)) . . . )) (n times). Show that f n(X) becomes “more circular” as n gets larger. In other words,...
IMO 1974 LL USA37 Origin: USA Problem Let a, b, and c denote the three sides of a billiard table in the shape of an equilateral triangle. A ball is placed at the midpoint of side a and then propelled toward side b with direction defined by the angle \theta. For what values of \theta will the ball strike the sides b, c, a in that order?
IMO 1969 LL GDR30 Origin: GDR Problem Prove that there exist infinitely many natural numbers a with the following property: The number z = n4 + a is not prime for any natural number n.
IMO 1979 LL YUG81 Origin: YUG Problem Let P be the set of rectangular parallelepipeds that have at least one edge of integer length. If a rectangular parallelepiped P0 can be decomposed into parallelepipeds P1, P2, . . . , Pn \inP, prove that P0 \inP.
IMO 1970 LL GDR31 Origin: GDR Problem Prove that for any triangle with sides a, b, c and area P the following inequality holds: P \leq \sqrt 4 (abc)2/3. Find all triangles for which equality holds.
IMO 1971 LL HUN22 Origin: HUN Problem We are given an n \times n board, where n is an odd number. In each cell of the board either +1 or −1 is written. Let ak and bk denote the products of numbers in the kth row and in the kth column respectively. Prove that the sum a1 + a2 + \cdot \cdot \cdot + an + b1 + b2 +...
IMO 1974 LL USA38 Origin: USA Problem Consider the binomial coefficients n k n! k!(n−k)! (k = 1, 2, . . . , n−1). Determine all positive integers n for which n , n , . . . , n n−1 are all even numbers.
IMO 1977 LL VIE60 Origin: VIE Problem Suppose x0, x1, . . . , xn are integers and x0 > x1 > \cdot \cdot \cdot > xn. Prove that at least one of the numbers |F(x0)|, |F(x1)|, |F(x2)|, . . . , |F(xn)|, where F(x) = xn + a1xn−1 + \cdot \cdot \cdot + an, ai \inR, i = 1, . . . , n, is greater than n! 2n...
IMO 1989 LL VIE108 Origin: VIE Problem For every sequence (x1, x2, . . . , xn) of the numbers {1, 2, . . ., n} arranged in any order, denote by f(s) the sum of absolute values of the differences between two consecutive members of s. Find the maximum value of f(s) (where s runs through the set of all such sequences).
IMO 1987 LL VIE72 Origin: VIE Problem Is it possible to cover a rectangle of dimensions m \times n with bricks that have the trimino angular shape (an arrangement of three unit squares forming the letter L) if: (a) m \times n = 1985 \times 1987; (b) m \times n = 1987 \times 1989?
IMO 1983 LL CUB22 Origin: CUB Problem Does there exist an infinite number of sets C consisting of 1983 consecutive natural numbers such that each of the numbers is divisible by some number of the form a1983, with a \inN, a ̸= 1?
IMO 1987 LL POL50 Origin: POL Problem Let P, Q, R be polynomials with real coefficients, satisfying P 4+Q4 = R2. Prove that there exist real numbers p, q, r and a polynomial S such that P = pS, Q = qS and R = rS2. Variants: (1) P 4 + Q4 = R4; (2) gcd(P, Q) = 1; (3) \pmP 4 + Q4 = R2 or R4.
IMO 1985 LL CUB17 Origin: CUB Problem Set An = n k=1 k6 2k . Find limn\to\inftyAn.
IMO 1972 LL BUL5 Origin: BUL Problem Given a pyramid whose base is an n-gon inscribable in a circle, let H be the projection of the top vertex of the pyramid to its base. Prove that the projections of H to the lateral edges of the pyramid lie on a circle.
IMO 1986 LL ROM64 Origin: ROM Problem Let (an)n\inN be the sequence of integers defined recursively by a1 = a2 = 1, an+2 = 7an+1 −an −2 for n \geq1. Prove that an is a perfect square for every n.
IMO 1988 LL ISR51 Origin: ISR Problem Let A1, A2, . . . , A29 be 29 different sequences of positive integers. For 1 \leqi < j \leq29 and any natural number x, we define Ni(x) to be the number of elements of the sequence Ai that are less than or equal to x, and Nij(x) to be the number of elements of the intersection Ai \capAj that are less...
IMO 1987 LL MON42 Origin: MON Problem Find the integer solutions of the equation \sqrt 2 m (2 + \sqrt 2)n .
IMO 1972 LL ROM33 Origin: ROM Problem A rectangle ABCD is given whose sides have lengths 3 and 2n, where n is a natural number. Denote by U(n) the number of ways in which one can cut the rectangle into rectangles of side lengths 1 and 2. (a) Prove that U(n + 1) + U(n −1) = 4U(n); (b) Prove that U(n) = \sqrt 3[( \sqrt 3 + 1)(2 +...
IMO 1967 LL GBR17 Origin: GBR Problem Let k, m, and n be positive integers such that m+k + 1 is a prime number greater than n + 1. Write cs for s(s + 1). Prove that the product (cm+1 −ck)(cm+2 −ck) \cdot \cdot \cdot (cm+n −ck) is divisible by the product c1c2 \cdot \cdot \cdot cn. Solution Using cr −cs = (r −s)(r + s + 1) we can...
IMO 1979 LL BUL13 Origin: BUL Problem The plane is divided into equal squares by parallel lines; i.e., a square net is given. Let M be an arbitrary set of n squares of this net. Prove that it is possible to choose no fewer than n/4 squares of M in such a way that no two of them have a common point.
IMO 1971 LL NET29 Origin: NET Problem A rhombus with its incircle is given. At each vertex of the rhombus a circle is constructed that touches the incircle and two edges of the rhombus. These circles have radii r1, r2, while the incircle has radius r. Given that r1 and r2 are natural numbers and that r1r2 = r, find r1, r2, and r.
IMO 1989 LL GRE30 Origin: GRE Problem In a triangle ABC for which 6(a + b + c)r2 = abc, we consider a point M on the inscribed circle and the projections D, E, F of M on the sides BC, AC, and AB respectively. Let S, S1 denote the areas of the triangles ABC and DEF respectively. Find the maximum and minimum values of the quotient S S1 (here...
IMO 1983 LL CAN15 Origin: CAN Problem Find all possible finite sequences {n0, n1, n2, . . . , nk} of integers such that for each i, i appears in the sequence ni times (0 \leqi \leqk).
IMO 1979 LL GDR30 Origin: GDR Problem Let M be a set of points in a plane with at least two elements. Prove that if M has two axes of symmetry g1 and g2 intersecting at an angle \alpha = q\pi, where q is irrational, then M must be infinite.
IMO 1972 LL GBR16 Origin: GBR Problem Consider the set S of all the different odd positive integers that are not multiples of 5 and that are less than 30m, m being a positive integer. What is the smallest integer k such that in any subset of k integers from S there must be two integers one of which divides the other? Prove your result.
IMO 1983 LL FRG25 Origin: FRG Problem How many permutations a1, a2, . . . , an of {1, 2, . . ., n} are sorted into increasing order by at most three repetitions of the following operation: Move from left to right and interchange ai and ai+1 whenever ai > ai+1 for i running from 1 up to n −1?
IMO 1989 LL INA48 Origin: INA Problem Let S be the point of intersection of the two lines l1 : 7x−5y + 8 = 0 and l2 : 3x + 4y −13 = 0. Let P = (3, 7), Q = (11, 13), and let A and B be points on the line PQ such that P is between A and Q, and B is between P and Q, and...
IMO 1972 LL CZS10 Origin: CZS Problem Given five points in the plane, no three of which are collinear, prove that there can be found at least two obtuse-angled triangles with vertices at the given points. Construct an example in which there are exactly two such triangles.
IMO 1978 LL CZS10 Origin: CZS Problem Show that for any natural number n there exist two prime numbers p and q, p ̸= q, such that n divides their difference.
IMO 1984 LL SWE58 Origin: SWE Problem Let (an)\infty 1 be a sequence such that an \leqan+m \leqan + am for all positive integers n and m. Prove that an n has a limit as n approaches infinity.
IMO 1979 LL HUN39 Origin: HUN Problem A desert expedition camps at the border of the desert, and has to provide one liter of drinking water for another member of the expedition, residing on the distance of n days of walking from the camp, under the following conditions: (i) Each member of the expedition can pick up at most 3 liters of water. (ii) Each member must drink one liter...
IMO 1971 LL BUL4 Origin: BUL Problem Let xn = 22n + 1 and let m be the least common multiple of x2, x3, . . . , x1971. Find the last digit of m.
IMO 1969 LL SWE62 Origin: SWE Problem Which natural numbers can be expressed as the difference of squares of two integers?
IMO 1989 LL POL77 Origin: POL Problem Given that cos x + cos y + cos z cos(x + y + z) = sin x + sin y + sin z sin(x + y + z) = a, show that cos(y + z) + cos(z + x) + cos(x + y) = a.
IMO 1966 LL USS54 Origin: USS Problem Find the last two digits of a sum of eighth powers of 100 consecutive integers.
IMO 1974 LL CZS12 Origin: CZS Problem A circle K with radius r, a point D on K, and a convex angle with vertex S and rays a and b are given in the plane. Construct a parallelogram ABCD such that A and B lie on a and b respectively, SA + SB = r, and C lies on K.
IMO 1976 LL BUL4 Origin: BUL Problem Find all pairs of natural numbers (m, n) for which 2m \cdot 3n + 1 is the square of some integer.
IMO 1989 LL MON67 Origin: MON Problem A family of sets A1, A2, . . . , An has the following properties: (i) Each Ai contains 30 elements. (ii) Ai \capAj contains exactly one element for all i, j, 1 \leqi < j \leq30. Find the largest possible n if the intersection of all these sets is empty.
IMO 1979 LL POL51 Origin: POL Problem Let ABC be an arbitrary triangle and let S1, S2, . . . , S7 be circles satisfying the following conditions: S1 is tangent to CA and AB, S2 is tangent to S1, AB, and BC, S3 is tangent to S2, BC, and CA, \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot S7...
IMO 1976 LL USS43 Origin: USS Problem Prove that if for a polynomial P(x, y) we have P(x −1, y −2x + 1) = P(x, y), then there exists a polynomial \Phi(x) with P(x, y) = \Phi(y −x2).
IMO 1966 LL POL35 Origin: POL Problem If a, b, c, d are integers such that ad is odd and bc is even, prove that at least one root of the polynomial ax3 + bx2 + cx + d is irrational.
IMO 1977 LL BUL4 Origin: BUL Problem We are given n points in space. Some pairs of these points are connected by line segments so that the number of segments equals [n2/4], and a connected triangle exists. Prove that any point from which the maximal number of segments starts is a vertex of a connected triangle. Solution Consider any vertex vn from which the maximal number d of seg- ments...
IMO 1978 LL CZS8 Origin: CZS Problem For two given triangles A1A2A3 and B1B2B3 with areas ∆A and ∆B, respectively, AiAk \geqBiBk, i, k = 1, 2, 3. Prove that ∆A \geq∆B if the triangle A1A2A3 is not obtuse-angled.
IMO 1974 LL VIE47 Origin: VIE Problem Given two points A, B outside of a given plane P, find the positions of points M in the plane P for which the ratio MA MB takes a minimum or maximum.
IMO 1988 LL USA82 Origin: USA Problem The triangle ABC has a right angle at C. The point P is located on segment AC such that triangles PBA and PBC have congruent inscribed circles. Express the length x = PC in terms of a = BC, b = CA, and c = AB.
IMO 1987 LL MON41 Origin: MON Problem Let n points be given arbitrarily in the plane, no three of them collinear. Let us draw segments between pairs of these points. What is the minimum number of segments that can be colored red in such a way that among any four points, three of them are connected by segments that form a red triangle?
IMO 1982 LL BRA9 Origin: BRA Problem Let n be a natural number, n \geq2, and let \varphi be Euler’s function; i.e., \varphi(n) is the number of positive integers not exceeding n and coprime to n. Given any two real numbers \alpha and \beta, 0 \leq\alpha < \beta \leq1, prove that there exists a natural number m such that \alpha < \varphi(m) m < \beta.
IMO 1972 LL BUL1 Origin: BUL Problem Find all integer solutions of the equation 1 + x + x2 + x3 + x4 = y4.
IMO 1966 LL BUL12 Origin: BUL Problem Find digits x, y, z such that the equality \sqrtxx \cdot \cdot \cdot x 2n −yy \cdot \cdot \cdot y n = zz \cdot \cdot \cdot z n holds for at least two values of n \inN, and in that case find all n for which this equality is true.
IMO 1989 LL THA100 Origin: THA Problem Let A be an n\timesn matrix whose elements are nonnegative real numbers. Assume that A is a nonsingular matrix and all elements of A−1 are nonnegative real numbers. Prove that every row and every column of A has exactly one nonzero element.
IMO 1967 LL HUN25 Origin: HUN Problem Three disks of diameter d are touching a sphere at their centers. Moreover, each disk touches the other two disks. How do we choose the radius R of the sphere so that the axis of the whole figure makes an angle 1 The statement so formulated is false. It would be trivially true under the addi- tional assumption that the polygonal line is...
IMO 1967 LL CZS9 Origin: CZS Problem The circle k and its diameter AB are given. Find the locus of the centers of circles inscribed in the triangles having one vertex on AB and two other vertices on k. Solution The incenter of any such triangle lies inside the circle k. We shall show that every point S interior to the circle S is the incenter of one such triangle....
IMO 1976 LL USS46 Origin: USS Problem For a \geq0, b \geq0, c \geq0, d \geq0, prove the inequality a4 + b4 + c4 + d4 + 2abcd \geqa2b2 + a2c2 + a2d2 + b2c2 + b2d2 + c2d2.
IMO 1988 LL MON62 Origin: MON Problem The positive integer n has the property that in any set of n integers chosen from the integers 1, 2, . . . , 1988, twenty-nine of them form an arithmetic progression. Prove that n > 1788.
IMO 1986 LL FRG29 Origin: FRG Problem We define a binary operation ⋆in the plane as follows: Given two points A and B in the plane, C = A ⋆B is the third vertex of the equilateral triangle ABC oriented positively. What is the relative position of three points I, M, O in the plane if I ⋆(M ⋆O) = (O ⋆I) ⋆M holds?
IMO 1989 LL ROM90 Origin: ROM Problem Prove that the sequence (an)n\geq0, an = [n \sqrt 2], contains an infinite number of perfect squares.
IMO 1970 LL GDR29 Origin: GDR Problem Prove that the equation 4x+6x = 9x has no rational solutions.
IMO 1979 LL POL52 Origin: POL Problem Let a real number \lambda > 1 be given and a sequence (nk) of positive integers such that nk+1 nk \lambda for k = 1, 2, . . . . Prove that there exists a positive integer c such that no positive integer n can be represented in more than c ways in the form n = nk + nj or n =...
IMO 1988 LL KOR57 Origin: KOR Problem Let C be a cube with edges of length 2. Construct a solid with fourteen faces by cutting offall eight corners of C, keeping the new faces perpendicular to the diagonals of the cube and keeping the newly formed faces identical. If at the conclusion of this process the fourteen faces so formed have the same area, find the area of each face...
IMO 1967 LL POL37 Origin: POL Problem Prove that for arbitrary positive numbers the following in- equality holds: a + 1 b + 1 c \leqa8 + b8 + c8 a3b3c3 . Solution Using the A–G mean inequality we obtain 8a2b3c3 \leq2a8 + 3b8 + 3c8, 8a3b2c3 \leq3a8 + 2b8 + 3c8, 8a3b3c2 \leq3a8 + 3b8 + 2c8. By adding these inequalities and dividing by 3a3b3c3 we obtain the desired...
IMO 1986 LL FIN19 Origin: FIN Problem Let f : [0, 1] \to[0, 1] satisfy f(0) = 0, f(1) = 1 and f(x + y) −f(x) = f(x) −f(x −y) for all x, y \geq0 with x −y, x + y \in[0, 1]. Prove that f(x) = x for all x \in[0, 1].
IMO 1989 LL THA97 Origin: THA Problem Let n be a positive integer, X = {1, 2, . . ., n}, and k a positive integer such that n/2 \leqk \leqn. Determine, with proof, the number of all functions f : X \toX that satisfy the following conditions: (i) f 2 = f; (ii) the number of elements in the image of f is k; (iii) for each y in...
IMO 1969 LL CZS12 Origin: CZS Problem Given a unit cube, find the locus of the centroids of all tetra- hedra whose vertices lie on the sides of the cube.
IMO 1966 LL BUL21 Origin: BUL Problem Prove that the volume V and the lateral area S of a right circular cone satisfy the inequality 6V \pi 2 \leq 2S \pi \sqrt . When does equality occur?
IMO 1985 LL MOR56 Origin: MOR Problem Let ABCD be a rhombus with angle \angleA = 60◦. Let E be a point, different from D, on the line AD. The lines CE and AB intersect at F. The lines DF and BE intersect at M. Determine the angle ∡BMD as a function of the position of E on AD.
IMO 1988 LL USA81 Origin: USA Problem There are n \geq3 job openings at a factory, ranked 1 to n in order of increasing pay. There are n job applicants, ranked 1 to n in order of increasing ability. Applicant i is qualified for job j if and only if i \geqj. The applicants arrive one at a time in random order. Each in turn is hired to the highest-ranking...
IMO 1992 LL SPA68 Origin: SPA Problem Show that the numbers tan(r\pi/15), where r is a positive integer less than 15 and relatively prime to 15, satisfy x8 −92x6 + 134x4 −28x2 + 1 = 0.
IMO 1989 LL VIE110 Origin: VIE Problem Do there exist two sequences of real numbers {ai}, {bi}, i \in N = {1, 2, 3, . . .}, satisfying the following conditions: 3\pi 2 \leqai \leqbi, cos aix + cos bix \geq−1 i for all i \inN and all x, 0 < x < 1?
IMO 1986 LL AUS3 Origin: AUS Problem A line parallel to the side BC of a triangle ABC meets AB in F and AC in E. Prove that the circles on BE and CF as diameters intersect in a point lying on the altitude of the triangle ABC dropped from A to BC.
IMO 1988 LL POL69 Origin: POL Problem For a convex polygon P in the plane let P ′ denote the convex polygon with vertices at the midpoints of the sides of P. Given an integer n \geq3, determine sharp bounds for the ratio area(P ′) area(P) over all convex n-gons P.
IMO 1979 LL GRE36 Origin: GRE Problem A regular tetrahedron A1B1C1D1 is inscribed in a regular tetrahedron ABCD, where A1 lies in the plane BCD, B1 in the plane ACD, etc. Prove that A1B1 \geqAB/3.
IMO 1976 LL BUL5 Origin: BUL Problem Let ABCDS be a pyramid with four faces and with ABCD as a base, and let a plane \alpha through the vertex A meet its edges SB and SD at points M and N, respectively. Prove that if the intersection of the plane \alpha with the pyramid ABCDS is a parallelogram, then SM \cdot SN > BM \cdot DN.
IMO 1966 LL CZS41 Origin: CZS Problem If A1A2 . . . An is a regular n-gon (n \geq3), how many different obtuse triangles AiAjAk exist?
IMO 1971 LL HUN24 Origin: HUN Problem Let A, B, and C denote the angles of a triangle. If sin2 A + sin2 B + sin2 C = 2, prove that the triangle is right-angled.
IMO 1987 LL VIE75 Origin: VIE Problem Let ak be positive numbers such that a1 \geq1 and ak+1 −ak \geq1 (k = 1, 2, . . . ). Prove that for every n \inN, n k=1 ak+1 1987\sqrtak < 1987.
IMO 1989 LL BUL6 Origin: BUL Problem The circles c1 and c2 are tangent at the point A. A straight line l through A intersects c1 and c2 at points C1 and C2 respectively. A circle c, which contains C1 and C2, meets c1 and c2 at points B1 and B2 respectively. Let \kappa be the circle circumscribed around triangle AB1B2. The circle k tangent to \kappa at the point...
IMO 1983 LL USA68 Origin: USA Problem Three of the roots of the equation x4 −px3 + qx2 −rx + s = 0 are tan A, tan B, and tan C, where A, B, and C are angles of a triangle. Determine the fourth root as a function only of p, q, r, and s.
IMO 1969 LL POL57 Origin: POL Problem On the sides AB and AC of triangle ABC two points K and L are given such that KB AK + LC AL = 1. Prove that KL passes through the centroid of ABC.
IMO 1977 LL USS49 Origin: USS Problem Find all pairs of integers (p, q) for which all roots of the trino- mials x2 + px + q and x2 + qx + p are integers. Solution If one of p, q, say p, is zero, then −q is a perfect square. Conversely, (p, q) = (0, −t2) and (p, q) = (−t2, 0) satisfy the conditions for t \inZ. We...
IMO 1977 LL NET30 Origin: NET Problem A triangle ABC with \angleA = 30◦and \angleC = 54◦is given. On BC a point D is chosen such that \angleCAD = 12◦. On AB a point E is chosen such that \angleACE = 6◦. Let S be the point of intersection of AD and CE. Prove that BS = BC. Solution Suppose \angleSBA = x. By the trigonometric form of Ceva’s theorem...
IMO 1972 LL NET25 Origin: NET Problem We consider n real variables xi (1 \leqi \leqn), where n is an integer and n \geq2. The product of these variables will be denoted by p, their sum by s, and the sum of their squares by S. Furthermore, let \alpha be a positive constant. We now study the inequality ps \leqS\alpha. Prove that it holds for every n-tuple (xi) if and...
IMO 1977 LL USA56 Origin: USA Problem The four circumcircles of the four faces of a tetrahedron have equal radii. Prove that the four faces of the tetrahedron are congruent triangles. Solution See the solution to (LL67-38).
IMO 1976 LL POL30 Origin: POL Problem Prove that if P(x) = (x−a)kQ(x), where k is a positive integer, a is a nonzero real number, Q(x) is a nonzero polynomial, then P(x) has at least k + 1 nonzero coefficients.
IMO 1985 LL SPA74 Origin: SPA Problem Find the triples of positive integers x, y, z satisfying x + 1 y + 1 z = 4 5.
IMO 1979 LL USA64 Origin: USA Problem From point P on arc BC of the circumcircle about triangle ABC, PX is constructed perpendicular to BC, PY is perpendicular to AC, and PZ perpendicular to AB (all extended if necessary). Prove that BC PX = AC PY + AB PZ .
IMO 1989 LL MOR72 Origin: MOR Problem Let ABCD be a quadrilateral inscribed in a circle with diam- eter AB such that BC = a, CD = 2a, DA = 3 \sqrt 5−1 a. For each point M on the semicircle AB not containing C and D, denote by h1, h2, h3 the distances from M to the sides BC, CD, and DA. Find the maximum of h1 + h2...
IMO 1969 LL HUN34 Origin: HUN Problem Let a and b be arbitrary integers. Prove that if k is an integer not divisible by 3, then (a + b)2k + a2k + b2k is divisible by a2 + ab + b2.
IMO 1985 LL USS90 Origin: USS Problem Decompose the number 51985−1 into a product of three integers, each of which is larger than 5100.
IMO 1986 LL SWE66 Origin: SWE Problem One hundred red points and one hundred blue points are chosen in the plane, no three of them lying on a line. Show that these points can be connected pairwise, red ones with blue ones, by disjoint line segments.
IMO 1974 LL NET20 Origin: NET Problem For which natural numbers n do there exist n natural numbers ai (1 \leqi \leqn) such that n i=1 a−2 i = 1?
IMO 1978 LL VIE49 Origin: VIE Problem Let A, B, C, D be four arbitrary distinct points in space. (a) Prove that using the segments AB + CD, AC + BD and AD + BC it is always possible to construct a triangle T that is nondegenerate and has no obtuse angle. (b) What should these four points satisfy in order for the triangle T to be right-angled?
IMO 1985 LL CZS21 Origin: CZS Problem Let A be a set of positive integers such that for any two elements x, y of A, |x−y| \geqxy 25 . Prove that A contains at most nine elements. Give an example of such a set of nine elements.
IMO 1982 LL AUS3 Origin: AUS Problem Given n points X1, X2, . . . , Xn in the interval 0 \leqXi \leq1, i = 1, 2, . . . , n, show that there is a point y, 0 \leqy \leq1, such that n n i=1 |y −Xi| = 1 2.
IMO 1976 LL NET24 Origin: NET Problem Let 0 \leqx1 \leqx2 \leq\cdot \cdot \cdot \leqxn \leq1. Prove that for all A \geq1 there exists an interval I of length 2 n\sqrt A such that for all x \inI, |(x −x1)(x −x2) \cdot \cdot \cdot (x −xn)| \leqA.
IMO 1987 LL USS67 Origin: USS Problem If a, b, c, d are real numbers such that a2 + b2 + c2 + d2 \leq1, find the maximum of the expression (a + b)4 + (a + c)4 + (a + d)4 + (b + c)4 + (b + d)4 + (c + d)4.
IMO 1985 LL BEL6 Origin: BEL Problem On a one-way street, an unending sequence of cars of width a, length b passes with velocity v. The cars are separated by the distance c. A pedestrian crosses the street perpendicularly with velocity w, without paying attention to the cars. (a) What is the probability that the pedestrian crosses the street unin- jured? (b) Can he improve this probability by crossing the...
IMO 1978 LL GBR20 Origin: GBR Problem Let O be the center of a circle. Let OU, OV be perpendicular radii of the circle. The chord PQ passes through the midpoint M of UV . Let W be a point such that PM = PW, where U, V, M, W are collinear. Let R be a point such that PR = MQ, where R lies on the line PW. Prove...
IMO 1966 LL POL36 Origin: POL Problem Let ABCD be a cyclic quadrilateral. Show that the centroids of the triangles ABC, CDA, BCD, DAB lie on a circle.
IMO 1984 LL AUS1 Origin: AUS Problem The fraction 10 can be written as the sum of two positive fractions with numerator 1 as follows: 10 = 1 5 + 1 10 and also 10 = 1 4 + 1 20. There are the only two ways in which this can be done. In how many ways can 1984 be written as the sum of two positive fractions with numerator...
IMO 1972 LL CZS13 Origin: CZS Problem Given a sphere K, determine the set of all points A that are vertices of some parallelograms ABCD that satisfy AC \leqBD and whose entire diagonal BD is contained in K.
IMO 1986 LL FRA23 Origin: FRA Problem Let I and J be the centers of the incircle and the excircle in the angle BAC of the triangle ABC. For any point M in the plane of the triangle, not on the line BC, denote by IM and JM the centers of the incircle and the excircle (touching BC) of the triangle BCM. Find the locus of points M for which...
IMO 1986 LL MOR57 Origin: MOR Problem In a triangle ABC, the incircle touches the sides BC, CA, AB in the points A′, B′, C′, respectively; the excircle in the angle A touches the lines containing these sides in A1, B1, C1, and similarly, the excircles in the angles B and C touch these lines in A2, B2, C2 and A3, B3, C3. Prove that the triangle ABC is right-angled...
IMO 1992 LL POL57 Origin: POL Problem For positive numbers a, b, c define A = (a + b + c)/3, G = (abc)1/3, H = 3/(a−1 + b−1 + c−1). Prove that A G 3 \geq1 4 + 3 4 \cdot A H , for every a, b, c > 0.
IMO 1979 LL YUG78 Origin: YUG Problem By \omega(n), where n is an integer greater than 1, let us denote the number of different prime divisors of the number n. Prove that there exist infinitely many numbers n for which \omega(n) < \omega(n + 1) < \omega(n + 2) holds.
IMO 1985 LL POL65 Origin: POL Problem Define the functions f, F : N \toN, by f(n) = 3 − \sqrt n , F(k) = min{n \inN|f k(n) > 0}, where f k = f ◦\cdot \cdot \cdot ◦f is f iterated n times. Prove that F(k + 2) = 3F(k + 1) −F(k) for all k \inN.
IMO 1977 LL FIN44 Origin: FIN Problem Let E be a finite set of points in space such that E is not contained in a plane and no three points of E are collinear. Show that E contains the vertices of a tetrahedron T = ABCD such that T \capE = {A, B, C, D} (including interior points of T ) and such that the projection of A onto the...
IMO 1969 LL HUN39 Origin: HUN Problem Find the positions of three points A, B, C on the boundary of a unit cube such that min{AB, AC, BC} is the greatest possible.
IMO 1979 LL BRA8 Origin: BRA Problem The sequence (an) of real numbers is defined as follows: a1 = 1, a2 = 2 and an = 3an−1 −an−2, n \geq3. Prove that for n \geq3, an = a2 n−1 an−2 1, where [x] denotes the integer p such that p \leqx < p + 1.
IMO 1988 LL MON66 Origin: MON Problem Suppose \alphai > 0, \betai > 0 for 1 \leqi \leqn (n > 1) and that n i=1 \alphai = n i=1 \betai = \pi. Prove that n i=1 cos \betai sin \alphai \leq n i=1 cot \alphai.
IMO 1976 LL NET23 Origin: NET Problem Prove that in a Euclidean plane there are infinitely many concentric circles C such that all triangles inscribed in C have at least one irrational side.
IMO 1974 LL POL25 Origin: POL Problem Let f : R \toR be of the form f(x) = x + \epsilon sin x, where 0 < |\epsilon| \leq1. Define for any x \inR, xn = f ◦\cdot \cdot \cdot ◦f n times (x). Show that for every x \inR there exists an integer k such that limn\to\inftyxn = k\pi.
IMO 1970 LL USS56 Origin: USS Problem A square hole of depth h whose base is of length a is given. A dog is tied to the center of the square at the bottom of the hole by a rope of length L > \sqrt 2a2 + h2, and walks on the ground around the hole. The edges of the hole are smooth, so that the rope can freely slide...
IMO 1988 LL KOR58 Origin: KOR Problem For each pair of positive integers k and n, let Sk(n) be the base-k digit sum of n. Prove that there are at most two primes p less than 20,000 for which S31(p) is a composite number.
IMO 1966 LL CZS16 Origin: CZS Problem We are given a circle K with center S and radius 1 and a square Q with center M and side 2. Let XY be the hypotenuse of an isosceles right triangle XY Z. Describe the locus of points Z as X varies along K and Y varies along the boundary of Q.
IMO 1977 LL CZS7 Origin: CZS Problem Prove the following assertion: If c1, c2, . . . , cn (n \geq2) are real numbers such that (n −1)(c2 1 + c2 2 + \cdot \cdot \cdot + c2 n) = (c1 + c2 + \cdot \cdot \cdot + cn)2, then either all these numbers are nonnegative or all these numbers are nonpositive. Solution Let us suppose that c1 \leqc2 \leq\cdot...
IMO 1972 LL USS46 Origin: USS Problem Numbers 1, 2, . . . , 16 are written in a 4\times4 square matrix so that the sum of the numbers in every row, every column, and every diagonal is the same and furthermore that the numbers 1 and 16 lie in opposite corners. Prove that the sum of any two numbers symmetric with respect to the center of the square equals...
IMO 1974 LL USA41 Origin: USA Problem Through the circumcenter O of an arbitrary acute-angled trian- gle, chords A1A2, B1B2, C1C2 are drawn parallel to the sides BC, CA, AB of the triangle respectively. If R is the radius of the circumcircle, prove that A1O \cdot OA2 + B1O \cdot OB2 + C1O \cdot OC2 = R2.
IMO 1987 LL GRE30 Origin: GRE Problem Consider the regular 1987-gon A1A2 . . . A1987 with center O. Show that the sum of vectors belonging to any proper subset of M = {OAj | j = 1, 2, . . . , 1987} is nonzero.
IMO 1982 LL VIE56 Origin: VIE Problem Let f(x) = ax2 + bx + c and g(x) = cx2 + bx + a. If |f(0)| \leq1, |f(1)| \leq1, |f(−1)| \leq1, prove that for |x| \leq1, (a) |f(x)| \leq5/4, (b) |g(x)| \leq2.
IMO 1985 LL CZS23 Origin: CZS Problem Let N = {1, 2, 3, . . .}. For real x, y, set S(x, y) = {s | s = [nx + y], n \inN}. Prove that if r > 1 is a rational number, there exist real numbers u and v such that S(r, 0) \capS(u, v) = \emptyset, S(r, 0) \cupS(u, v) = N.
IMO 1976 LL CZS7 Origin: CZS Problem Let P be a fixed point and T a given triangle that contains the point P. Translate the triangle T by a given vector v and denote by T ′ this new triangle. Let r, R, respectively, be the radii of the smallest disks centered at P that contain the triangles T , T ′, respectively. Prove that r + |v| \leq3R and...
IMO 1979 LL USA65 Origin: USA Problem Given f(x) \leqx for all real x and f(x + y) \leqf(x) + f(y) for all real x, y, prove that f(x) = x for all x.
IMO 1985 LL MON53 Origin: MON Problem For each P inside the triangle ABC, let A(P), B(P), and C(P) be the points of intersection of the lines AP, BP, and CP with the sides opposite to A, B, and C, respectively. Determine P in such a way that the area of the triangle A(P)B(P)C(P) is as large as possible.
IMO 1983 LL GBR28 Origin: GBR Problem Show that if the sides a, b, c of a triangle satisfy the equation 2(ab2 + bc2 + ca2) = a2b + b2c + c2a + 3abc, then the triangle is equilateral. Show also that the equation can be satisfied by positive real numbers that are not the sides of a triangle.
IMO 1984 LL BEL6 Origin: BEL Problem Let P, Q, R be the polynomials with real or complex coefficients such that at least one of them is not constant. If P n +Qn +Rn = 0, prove that n < 3.
IMO 1988 LL IRE50 Origin: IRE Problem Let g(n) be defined as follows: g(1) = 0, g(2) = 1, g(n + 2) = g(n) + g(n + 1) + 1 (n \geq1). Prove that if n > 5 is a prime, then n divides g(n)(g(n) + 1).
IMO 1985 LL CAN13 Origin: CAN Problem Find the average of the quantity (a1 −a2)2 + (a2 −a3)2 + \cdot \cdot \cdot + (an−1 −an)2 taken over all permutations (a1, a2, . . . , an) of (1, 2, . . . , n).
IMO 1970 LL ROM46 Origin: ROM Problem Given a triangle ABC and a plane \pi having no common points with the triangle, find a point M such that the triangle determined by the points of intersection of the lines MA, MB, MC with \pi is congruent to the triangle ABC.
IMO 1977 LL ROM39 Origin: ROM Problem Consider 37 distinct points in space, all with integer coordi- nates. Prove that we may find among them three distinct points such that their barycenter has integers coordinates. Solution By the pigeonhole principle, we can find 5 distinct points among the given 37 such that their x-coordinates are congruent and their y-coordinates are congruent modulo 3. Now among these 5 points either there...
IMO 1992 LL KOR46 Origin: KOR Problem Prove that the sequence 5, 12, 19, 26, 33, . . . contains no term of the form 2n −1.
IMO 1979 LL BUL11 Origin: BUL Problem Prove that a pyramid A1A2 . . . A2k+1S with equal lateral edges and equal space angles between adjacent lateral walls is regular. Variant. Prove that a pyramid A1 . . . A2k+1S with equal space angles between adjacent lateral walls is regular if there exists a sphere tangent to all its edges.
IMO 1967 LL ROM47 Origin: ROM Problem Prove the inequality x1x2 \cdot \cdot \cdot xk xn−1 xn−1 \cdot \cdot \cdot + xn−1 k \leqxn+k−1 +xn+k−1 +\cdot \cdot \cdot+xn+k−1 k , where xi > 0 (i = 1, 2, . . ., k), k \inN, n \inN. Solution Using the A–G mean inequality we get (n + k −1)xn 1x2 \cdot \cdot \cdot xk \leqnxn+k−1 xn+k−1 \cdot \cdot \cdot...
IMO 1987 LL MON43 Origin: MON Problem Let 2n + 3 points be given in the plane in such a way that no three lie on a line and no four lie on a circle. Prove that the number of circles that pass through three of these points and contain exactly n interior points is not less than 1 2n+3 .
IMO 1978 LL GDR27 Origin: GDR Problem Determine the sixth number after the decimal point in the number \sqrt 1978 + /\sqrt 1978 020.
IMO 1970 LL BEL11 Origin: BEL Problem Let ABCD and A′B′C′D′ be two squares in the same plane and oriented in the same direction. Let A′′, B′′, C′′, and D′′ be the midpoints of AA′, BB′, CC′, and DD′. Prove that A′′B′′C′′D′′ is also a square.
IMO 1987 LL ROM54 Origin: ROM Problem Let n be a natural number. Solve in integers the equation xn + yn = (x −y)n+1.
IMO 1986 LL USS78 Origin: USS Problem If T and T1 are two triangles with angles x, y, z and x1, y1, z1, respectively, prove the inequality cos x1 sin x + cos y1 sin y + cos z1 sin z \leqcot x + cot y + cot z.
IMO 1972 LL SWE41 Origin: SWE Problem The ternary expansion x = 0.10101010 . . . is given. Give the binary expansion of x. Alternatively, transform the binary expansion y = 0.110110110 . . . into a ternary expansion.
IMO 1966 LL YUG45 Origin: YUG Problem An alphabet consists of n letters. What is the maximal length of a word, if (i) two neighboring letters in a word are always different, and (ii) no word abab (a ̸= b) can be obtained by omitting letters from the given word?
IMO 1966 LL ROM47 Origin: ROM Problem Find the number of lines dividing a given triangle into two parts of equal area which determine the segment of minimum possible length inside the triangle. Compute this minimum length in terms of the sides a, b, c of the triangle.
IMO 1976 LL USS45 Origin: USS Problem We are given n (n \geq5) circles in a plane. Suppose that every three of them have a common point. Prove that all n circles have a common point.
IMO 1986 LL GBR35 Origin: GBR Problem Establish the maximum and minimum values that the sum |a| + |b| + |c| can have if a, b, c are real numbers such that the maximum value of |ax2 + bx + c| is 1 for −1 \leqx \leq1.
IMO 1974 LL USS46 Origin: USS Problem Outside an arbitrary triangle ABC, triangles ADB and BCE are constructed such that \angleADB = \angleBEC = 90◦and \angleDAB = \angleEBC = 30◦. On the segment AC the point F with AF = 3FC is chosen. Prove that \angleDFE = 90◦ and \angleFDE = 30◦.
IMO 1984 LL SWE57 Origin: SWE Problem Let a, b, c, d be a permutation of the numbers 1, 9, 8, 4 and let n = (10a + b)10c+d. Find the probability that 1984! is divisible by n.
IMO 1979 LL POL53 Origin: POL Problem An infinite increasing sequence of positive integers nj (j = 1, 2, . . . ) has the property that for a certain c, N nj\leqN nj \leqc, for every N > 0 Prove that there exist finitely many sequences m(i) j (i = 1, 2, . . . , k) such that {n1, n2, . . . } = %k i=1{m(i)...
IMO 1978 LL NET32 Origin: NET Problem Let C be the circumcircle of the square with vertices (0, 0), (0, 1978), (1978, 0), (1978, 1978) in the Cartesian plane. Prove that C con- tains no other point for which both coordinates are integers.
IMO 1969 LL BEL2 Origin: BEL Problem (a) Find the equations of regular hyperbolas passing through the points A(\alpha, 0), B(\beta, 0), and C(0, \gamma). (b) Prove that all such hyperbolas pass through the orthocenter H of the triangle ABC. (c) Find the locus of the centers of these hyperbolas. (d) Check whether this locus coincides with the nine-point circle of the triangle ABC.
IMO 1984 LL SWE56 Origin: SWE Problem Let a, b, c be nonnegative integers such that a \leqb \leqc, 2b ̸= a + c and a+b+c is an integer. Is it possible to find three nonnegative integers d, e, and f such that d \leqe \leqf, f ̸= c, and such that a2+b2+c2 = d2 + e2 + f 2?
IMO 1988 LL HKG31 Origin: HKG Problem The circle x2 + y2 = r2 meets the coordinate axes at A = (r, 0), B = (−r, 0), C = (0, r), and D = (0, −r). Let P = (u, v) and Q = (−u, v) be two points on the circumference of the circle. Let N be the point of intersection of PQ and the y-axis, and let M...
IMO 1983 LL BUL14 Origin: BUL Problem Let l be tangent to the circle k at B. Let A be a point on k and P the foot of perpendicular from A to l. Let M be symmetric to P with respect to AB. Find the set of all such points M.
IMO 1966 LL ROM31 Origin: ROM Problem Solve the equation |x2 −1| + |x2 −4| = mx as a function of the parameter m. Which pairs (x, m) of integers satisfy this equation?
IMO 1974 LL CZS10 Origin: CZS Problem A regular octagon P is given whose incircle k has diameter 1. About k is circumscribed a regular 16-gon, which is also inscribed in P, cutting from P eight isosceles triangles. To the octagon P, three of these triangles are added so that exactly two of them are adjacent and no two of them are opposite to each other. Every 11-gon so obtained...
IMO 1989 LL IND51 Origin: IND Problem Let t(n), for n = 3, 4, 5, . . ., represent the number of distinct, incongruent, integer-sided triangles whose perimeter is n; e.g., t(3) = 1. Prove that t(2n −1) −t(2n) = n or n 6 + 1 .
IMO 1992 LL AUS1 Origin: AUS Problem Points D and E are chosen on the sides AB and AC of the triangle ABC in such a way that if F is the intersection point of BE and CD, then AE + EF = AD + DF. Prove that AC + CF = AB + BF.
IMO 1987 LL LUX39 Origin: LUX Problem Let A be a set of polynomials with real coefficients and let them satisfy the following conditions: (i) if f \inA and deg f \leq1, then f(x) = x −1; (ii) if f \inA and deg f \geq2, then either there exists g \inA such that f(x) = x2+deg g + xg(x) −1 or there exist g, h \inA such that f(x) =...
IMO 1979 LL HUN40 Origin: HUN Problem A polynomial P(x) has degree at most 2k, where k = 0, 1, 2, . . . . Given that for an integer i, the inequality −k \leqi \leqk implies |P(i)| \leq1, prove that for all real numbers x, with −k \leqx \leqk, the following inequality holds: |P(x)| < (2k + 1) 2k k .
IMO 1984 LL BUL10 Origin: BUL Problem Assume that the bisecting plane of the dihedral angle at edge AB of the tetrahedron ABCD meets the edge CD at point E. Denote by S1, S2, S3, respectively the areas of the triangles ABC, ABE, and ABD. Prove that no tetrahedron exists for which S1, S2, S3 (in this order) form an arithmetic or geometric progression.
IMO 1988 LL FRA12 Origin: FRA Problem Show that there do not exist more than 27 half-lines (or rays) emanating from the origin in 3-dimensional space such that the angle between each pair of rays is greater than of equal to \pi/4.
IMO 1974 LL YUG51 Origin: YUG Problem There are n points on a flat piece of paper, any two of them at a distance of at least 2 from each other. An inattentive pupil spills ink on a part of the paper such that the total area of the damaged part equals 3/2. Prove that there exist two vectors of equal length less than 1 and with their sum having...
IMO 1988 LL SPA76 Origin: SPA Problem The quadrilateral A1A2A3A4 is cyclic and its sides are a1 = A1A2, a2 = A2A3, a3 = A3A4, and a4 = A4A1. The respective circles with centers Ii and radii \rhoi are tangent externally to each side ai and to the sides ai+1 and ai−1 extended (a0 = a4). Show that i=1 ai \rhoi = 4(csc A1 + csc A2)2.
IMO 1985 LL GDR35 Origin: GDR Problem We call a coloring f of the elements in the set M = {(x, y) | x = 0, 1, . . . , kn −1; y = 0, 1, . . ., ln −1} with n colors allowable if every color appears exactly k and l times in each row and column and there are no rectangles with sides parallel to the...
IMO 1970 LL SWE51 Origin: SWE Problem Let p be a prime number. A rational number x, with 0 < x < 1, is written in lowest terms. The rational number obtained from x by adding p to both the numerator and the denominator differs from x by 1/p2. Determine all rational numbers x with this property.
IMO 1969 LL NET50 Origin: NET Problem The bisectors of the exterior angles of a pentagon B1B2B3B4B5 form another pentagon A1A2A3A4A5. Construct B1B2B3B4B5 from the given pentagon A1A2A3A4A5.
IMO 1966 LL YUG46 Origin: YUG Problem Let f(a, b, c) = |b −a| |ab| b + a ab −2 c |b −a| |ab| b + a ab 2 c . Prove that f(a, b, c) = 4 max{1/a, 1/b, 1/c}.
IMO 1988 LL GRE29 Origin: GRE Problem Find positive integers x1, x2, . . . , x29, at least one of which is greater than 1988, such that x2 1 + x2 2 + \cdot \cdot \cdot + x2 29 = 29x1x2 . . . x29.
IMO 1984 LL BEL5 Origin: BEL Problem For a real number x, let [x] denote the greatest integer not exceeding x. If m \geq3, prove that m(m + 1) 2(2m −1) m + 1 .
IMO 1970 LL AUT1 Origin: AUT Problem Prove that bc b + c + ca c + a + ab a + b \leq1 2(a + b + c) (a, b, c > 0).
IMO 1967 LL CZS10 Origin: CZS Problem The square ABCD is to be decomposed into n triangles (nonoverlapping) all of whose angles are acute. Find the smallest inte- ger n for which there exists a solution to this problem and construct at least one decomposition for this n. Answer whether it is possible to ask additionally that (at least) one of these triangles has a perimeter less than an arbitrarily...
IMO 1988 LL POL70 Origin: POL Problem In 3-dimensional space a point O is given and a finite set A of segments with the sum of the lengths equal to 1988. Prove that there exists a plane disjoint from A such that the distance from it to O does not exceed 574.
IMO 1969 LL NET49 Origin: NET Problem A boy has a set of trains and pieces of railroad track. Each piece is a quarter of circle, and by concatenating these pieces, the boy obtained a closed railway. The railway does not intersect itself. In passing through this railway, the train sometimes goes in the clockwise direction, and sometimes in the opposite direction. Prove that the train passes an even number...
IMO 1992 LL THA71 Origin: THA Problem Let P1(x, y) and P2(x, y) be two relatively prime polynomials with complex coefficients. Let Q(x, y) and R(x, y) be polynomials with complex coefficients and each of degree not exceeding d. Prove that there exist two integers A1, A2 not simultaneously zero with |Ai| \leqd + 1 (i = 1, 2) and such that the polynomial A1P1(x, y) + A2P2(x, y) is...
IMO 1977 LL ROM36 Origin: ROM Problem Consider a sequence of numbers (a1, a2, . . . , a2n). Define the operation S((a1, a2, . . . , a2n)) = (a1a2, a2a3, . . . , a2n−1a2n, a2na1). Prove that whatever the sequence (a1, a2, . . . , a2n) is, with ai \in{−1, 1} for i = 1, 2, . . . , 2n, after finitely many applications of...
IMO 1967 LL HUN23 Origin: HUN Problem Prove that for an arbitrary pair of vectors f and g in the plane, the inequality af 2 + bfg + cg2 \geq0 holds if and only if the following conditions are fulfilled: a \geq0, c \geq0, 4ac \geqb2. Solution Suppose that a \geq0, c \geq0, 4ac \geqb2. If a = 0, then b = 0, and the inequality reduces to the obvious...
IMO 1987 LL GRE31 Origin: GRE Problem Construct a triangle ABC given its side a = BC, its circum- radius R (2R \geqa), and the difference 1/k = 1/c−1/b, where c = AB and b = AC.
IMO 1979 LL BEL6 Origin: BEL Problem Prove that 1 4 sin2 36◦−1 = cos 72◦.
IMO 1978 LL USA45 Origin: USA Problem If r > s > 0 and a > b > c, prove that arbs + brcs + cras \geqasbr + bscr + csar.
IMO 1983 LL FIN24 Origin: FIN Problem Every x, 0 \leqx \leq1, admits a unique representation x = \infty j=0 aj2−j, where all the aj belong to {0, 1} and infinitely many of them are 0. If b(0) = 1+c 2+c, b(1) = 2+c, c > 0, and f(x) = a0 + \infty j=0 b(a0) \cdot \cdot \cdot b(aj)aj+1, show that 0 < f(x) −x < c for every...
IMO 1988 LL HKG34 Origin: HKG Problem Express the number 1988 as the sum of some positive integers in such a way that the product of these positive integers is maximal.
IMO 1972 LL CZS9 Origin: CZS Problem Given natural numbers k and n, k \leqn, n \geq3, find the set of all values in the interval (0, \pi) that the kth-largest among the interior angles of a convex ngon can take.
IMO 1989 LL IRE56 Origin: IRE Problem Let n = 2k −1, where k \geq6 is an integer. Let T be the set of all n-tuples (x1, x2, . . . , xn) where xi is 0 or 1 (i = 1, 2, . . ., n). For x = (x1, . . . , xn) and y = (y1, . . . , yn) in T , let d(x,...
IMO 1972 LL SWE39 Origin: SWE Problem How many tangents to the curve y = x3 −3x (y = x3 + px) can be drawn from different points in the plane?
IMO 1967 LL BUL3 Origin: BUL Problem Prove the trigonometric inequality cos x < 1 −x2 2 + x4 16, where x \in(0, \pi/2). Solution Consider the function f : [0, \pi/2] \toR defined by f(x) = 1 −x2/2 + x4/16 −cos x. It is easy to calculate that f ′(0) = f ′′(0) = f ′′′(0) = 0 and f ′′′′(x) = 3/2 −cos x. Since f ′′′′(x) >...
IMO 1970 LL FRA25 Origin: FRA Problem Suppose that f is a real function defined for 0 \leqx \leq1 having the first derivative f ′ for 0 \leqx \leq1 and the second derivative f ′′ for 0 < x < 1. Prove that if f(0) = f ′(0) = f ′(1) = f(1) −1 = 0, there exists a number 0 < y < 1 such that |f ′′(y)| \geq4.
IMO 1966 LL USS53 Origin: USS Problem Prove that in every convex hexagon of area S one can draw a diagonal that cuts offa triangle of area not exceeding 1 6S.
IMO 1967 LL CZS11 Origin: CZS Problem Let n be a positive integer. Find the maximal number of non- congruent triangles whose side lengths are integers less than or equal to n. Solution We have to find the number pn of triples of positive integers (a, b, c) satisfying a \leqb \leqc \leqn and a + b > c. Let us denote by pn(k) the number of such triples with...
IMO 1985 LL ITA45 Origin: ITA Problem Two persons, X and Y , play with a die. X wins a game if the outcome is 1 or 2; Y wins in the other cases. A player wins a match if he wins two consecutive games. For each player determine the probability of winning a match within 5 games. Determine the probabilities of winning in an unlimited number of games. If...
IMO 1969 LL USS67 Origin: USS Problem Under the conditions x1, x2 > 0, x1y1 > z2 1, and x2y2 > z2 2, prove the inequality (x1 + x2)(y1 + y2) −(z1 + z2)2 \leq x1y1 −z2 + x2y2 −z2 .
IMO 1971 LL BUL6 Origin: BUL Problem Let squares be constructed on the sides BC, CA, AB of a trian- gle ABC, all to the outside of the triangle, and let A1, B1, C1 be their cen- ters. Starting from the triangle A1B1C1 one analogously obtains a triangle A2B2C2. If S, S1, S2 denote the areas of triangles ABC, A1B1C1, A2B2C2, respectively, prove that S = 8S1 −4S2.
IMO 1971 LL GBR15 Origin: GBR Problem Let ABCD be a convex quadrilateral whose diagonals intersect at O at an angle \theta. Let us set OA = a, OB = b, OC = c, and OD = d, c > a > 0, and d > b > 0. Show that if there exists a right circular cone with vertex V , with the properties: (1) its axis passes through...
IMO 1988 LL MON64 Origin: MON Problem Given n points A1, A2, . . . , An, no three collinear, show that the n-gon A1A2 . . . An can be inscribed in a circle if and only if A1A2 \cdot A3An \cdot \cdot \cdot An−1An + A2A3 \cdot A4An \cdot \cdot \cdot An−1An \cdot A1An + \cdot \cdot \cdot +An−1An−2 \cdot A1An \cdot \cdot \cdot An−3An = A1An−1 \cdot...
IMO 1985 LL SWE79 Origin: SWE Problem Let a, b, and c be real numbers such that bc −a2 + ca −b2 + ab −c2 = 0. Prove that a (bc −a2)2 + b (ca −b2)2 + c (ab −c2)2 = 0.
IMO 1985 LL NOR62 Origin: NOR Problem A “large” circular disk is attached to a vertical wall. It rotates clockwise with one revolution per minute. An insect lands on the disk and immediately starts to climb vertically upward with constant speed \pi 3 cm per second (relative to the disk). Describe the path of the insect (a) relative to the disk; (b) relative to the wall.
IMO 1989 LL ICE40 Origin: ICE Problem A sequence of real numbers x0, x1, x2, . . . is defined as follows: x0 = 1989 and for each n \geq1 xn = −1989 n n−1 k=0 xk. Calculate the value of 1989 n=0 2nxn.
IMO 1977 LL HUN24 Origin: HUN Problem Determine all real functions f(x) that are defined and contin- uous on the interval (−1, 1) and that satisfy the functional equation f(x + y) = f(x) + f(y) 1 −f(x)f(y) (x, y, x + y \in(−1, 1)). Solution Setting x = y = 0 gives us f(0) = 0. Let us put g(x) = arctanf(x). The given functional equation becomes tan g(x...
IMO 1986 LL USS80 Origin: USS Problem Let ABCD be a tetrahedron and O its incenter, and let the line OD be perpendicular to AD. Find the angle between the planes DOB and DOC.
IMO 1978 LL VIE50 Origin: VIE Problem A variable tetrahedron ABCD has the following properties: Its edge lengths can change as well as its vertices, but the opposite edges remain equal (BC = DA, CA = DB, AB = DC); and the vertices A, B, C lie respectively on three fixed spheres with the same center P and radii 3, 4, 12. What is the maximal length of PD?
IMO 1984 LL MON36 Origin: MON Problem The set {1, 2, . . ., 49} is divided into three subsets. Prove that at least one of these subsets contains three different numbers a, b, c such that a + b = c.
IMO 1979 LL FRG26 Origin: FRG Problem Let n be a natural number. If 4n + 2n + 1 is a prime, prove that n is a power of three.
IMO 1983 LL ROM55 Origin: ROM Problem For every a \inN denote by M(a) the number of elements of the set {b \inN | a + b is a divisor of ab}. Find maxa\leq1983 M(a).
IMO 1967 LL USS56 Origin: USS Problem In a group of interpreters each one speaks one or several foreign languages; 24 of them speak Japanese, 24 Malay, 24 Farsi. Prove that it is possible to select a subgroup in which exactly 12 interpreters speak Japanese, exactly 12 speak Malay, and exactly 12 speak Farsi. Solution We shall prove by induction on n the following statement: If in some group of...
IMO 1992 LL USA81 Origin: USA Problem Suppose that points X, Y, Z are located on sides BC, CA, and AB, respectively, of \triangleABC in such a way that \triangleXY Z is similar to \triangleABC. Prove that the orthocenter of \triangleXY Z is the circumcenter of \triangleABC.
IMO 1984 LL NET41 Origin: NET Problem Determine positive integers p, q, and r such that the diagonal of a block consisting of p\timesq \timesr unit cubes passes through exactly 1984 of the unit cubes, while its length is minimal. (The diagonal is said to pass through a unit cube if it has more than one point in common with the unit cube.)
IMO 1985 LL BUL11 Origin: BUL Problem Let a and b be integers and n a positive integer. Prove that bn−1a(a + b)(a + 2b) \cdot \cdot \cdot (a + (n −1)b) n! is an integer.
IMO 1992 LL MON50 Origin: MON Problem Let N be a point inside the triangle ABC. Through the mid- points of the segments AN, BN, and CN the lines parallel to the opposite sides of \triangleABC are constructed. Let AN, BN, and CN be the intersection points of these lines. If N is the orthocenter of the triangle ABC, prove that the nine-point circles of \triangleABC and \triangleANBNCN coincide. Remark....
IMO 1969 LL SWE61 Origin: SWE Problem Let a0, a1, a2 be determined with a0 = 0, an+1 = 2an + 2n. Prove that if n is power of 2, then so is an.
IMO 1983 LL ROM53 Origin: ROM Problem Let a \inR and let z1, z2, . . . , zn be complex numbers of mod- ulus 1 satisfying the relation n k=1 z3 k = 4(a + (a −n)i) −3 n k=1 zk. Prove that a \in{0, 1, . . ., n} and zk \in{1, i} for all k.
IMO 1978 LL TUR38 Origin: TUR Problem Given a circle, construct a chord that is trisected by two given noncollinear radii.
IMO 1985 LL ROM71 Origin: ROM Problem For every integer r > 1 find the smallest integer h(r) > 1 having the following property: For any partition of the set {1, 2, . . ., h(r)} into r classes, there exist integers a \geq0, 1 \leqx \leqy such that the numbers a + x, a + y, a + x + y are contained in the same class of the...
IMO 1982 LL BEL6 Origin: BEL Problem On the three distinct lines a, b, and c three points A, B, and C are given, respectively. Construct three collinear points X, Y, Z on lines a, b, c, respectively, such that BY AX = 2 and CZ AX = 3.
IMO 1992 LL KOR45 Origin: KOR Problem Let n be a positive integer. Prove that the number of ways to express n as a sum of distinct positive integers (up to order) and the number of ways to express n as a sum of odd positive integers (up to order) are the same.
IMO 1983 LL USA67 Origin: USA Problem The altitude from a vertex of a given tetrahedron intersects the opposite face in its orthocenter. Prove that all four altitudes of the tetrahedron are concurrent.
IMO 1983 LL NET47 Origin: NET Problem In a plane, three pairwise intersecting circles C1, C2, C3 with centers M1, M2, M3 are given. For i = 1, 2, 3, let Ai be one of the points of intersection of Cj and Ck ({i, j, k} = {1, 2, 3}). Prove that if \angleM3A1M2 = \angleM1A2M3 = \angleM2A3M1 = \pi/3 (directed angles), then M1A1, M2A2, and M3A3 are concurrent.
IMO 1984 LL USS65 Origin: USS Problem A tetrahedron is inscribed in a sphere of radius 1 such that the center of the sphere is inside the tetrahedron. Prove that the sum of lengths of all edges of the tetrahedron is greater than 6.
IMO 1988 LL POL71 Origin: POL Problem Given integers a1, . . . , a10, prove that there exists a nonzero sequence (x1, . . . , x10) such that all xi belong to {−1, 0, 1} and the number 10 i=1 xiai is divisible by 1001.
IMO 1992 LL IRN37 Origin: IRN Problem Let the circles C1, C2, and C3 be orthogonal to the circle C and intersect each other inside C forming acute angles of measures A, B, and C. Show that A + B + C < \pi.
IMO 1985 LL ITA46 Origin: ITA Problem Let C be the curve determined by the equation y = x3 in the rectangular coordinate system. Let t be the tangent to C at a point P of C; t intersects C at another point Q. Find the equation of the set L of the midpoints M of PQ as P describes C. Is the correspondence associating P and M a bijection...
IMO 1986 LL MON55 Origin: MON Problem Given an integer n \geq2, determine all n-digit numbers M0 = a1a2 . . . an (ai ̸= 0, i = 1, 2, . . ., n) divisible by the numbers M1 = a2a3 . . . ana1, M2 = a3a4 . . . ana1a2, . . . , Mn−1 = ana1a2 . . . an−1.
IMO 1989 LL USA106 Origin: USA Problem Let n > 1 be a fixed integer. Define functions f0(x) = 0, f1(x) = 1 −cos x, and for k > 0, fk+1(x) = 2fk(x) cos x −fk−1(x). If F(x) = f1(x) + f2(x) + \cdot \cdot \cdot + fn(x), prove that (a) 0 < F(x) < 1 for 0 < x < \pi n+1, and (b) F(x) > 1 for \pi...
IMO 1985 LL NET58 Origin: NET Problem Prove that there are infinitely many pairs (k, N) of positive integers such that 1 + 2 + \cdot \cdot \cdot + k = (k + 1) + (k + 2) + \cdot \cdot \cdot + N.
IMO 1976 LL CZS6 Origin: CZS Problem For each point X of a given polytope, denote by f(X) the sum of the distances of the point X from all the planes of the faces of the polytope. Prove that if f attains its maximum at an interior point of the polytope, then f is constant.
IMO 1976 LL POL32 Origin: POL Problem We consider the infinite chessboard covering the whole plane. In every field of the chessboard there is a nonnegative real number. Every number is the arithmetic mean of the numbers in the four adjacent fields of the chessboard. Prove that the numbers occurring in the fields of the chessboard are all equal.
IMO 1971 LL CUB11 Origin: CUB Problem Prove that n! cannot be the square of any natural number.
IMO 1987 LL FRA15 Origin: FRA Problem Let a1, a2, a3, b1, b2, b3, c1, c2, c3 be nine strictly positive real numbers. We set S1 = a1b2c3, S2 = a2b3c1, S3 = a3b1c2; T1 = a1b3c2, T2 = a2b1c3, T3 = a3b2c1. Suppose that the set {S1, S2, S3, T1, T2, T3} has at most two elements. Prove that S1 + S2 + S3 = T1 + T2 +...
IMO 1969 LL MON43 Origin: MON Problem Let p and q be two prime numbers greater than 3. Prove that if their difference is 2n, then for any two integers m and n, the number S = p2m+1 + q2m+1 is divisible by 3.
IMO 1971 LL NET33 Origin: NET Problem A square 2n \times 2n grid is given. Let us consider all possible paths along grid lines, going from the center of the grid to the border, such that (1) no point of the grid is reached more than once, and (2) each of the squares homothetic to the grid having its center at the grid center is passed through only once. (a)...
IMO 1982 LL CZS20 Origin: CZS Problem Consider a cube C and two planes \sigma, \tau, which divide Euclidean space into several regions. Prove that the interior of at least one of these regions meets at least three faces of the cube.
IMO 1969 LL NET47 Origin: NET Problem Let A and B be points on the circle \gamma. A point C, different from A and B, is on the circle \gamma. Let D be the projection of the point C onto the line AB. Consider three other circles \gamma1, \gamma2, and \gamma3 with the common tangent AB: \gamma1 inscribed in the triangle ABC, and \gamma2 and \gamma3 tangent to both (the...
IMO 1983 LL LUX42 Origin: LUX Problem Consider the square ABCD in which a segment is drawn between each vertex and the midpoints of both opposite sides. Find the ratio of the area of the octagon determined by these segments and the area of the square ABCD.
IMO 1979 LL BRA7 Origin: BRA Problem M = (ai,j), i, j = 1, 2, 3, 4, is a square matrix of order four. Given that: (i) for each i = 1, 2, 3, 4 and for each k = 5, 6, 7, ai,k = ai,k−4; Pi = a1,i + a2,i+1 + a3,i+2 + a4,i+3; Si = a4,i + a3,i+1 + a2,i+2 + a1,i+3; Li = ai,1 + ai,2 +...
IMO 1974 LL CUB7 Origin: CUB Problem Let P be a prime number and n a natural number. Prove that the product N = pn2 2n−1 i=1; 2∤i ((p −1)i)! p2i pi is a natural number that is not divisible by p.
IMO 1977 LL FIN43 Origin: FIN Problem Evaluate S = n k=1 k(k + 1) \cdot \cdot \cdot (k + p), where n and p are positive integers. Solution Since k(k + 1) \cdot \cdot \cdot (k + p) = (p + 1)! k+p p+1 = (p + 1)! k+p+1 p+2 − k+p p+2 , it follows that n k=1 k(k + 1) \cdot \cdot...
IMO 1989 LL CUB10 Origin: CUB Problem Given the equation 4x3 + 4x2y −15xy2 −18y3 −12x2 + 6xy + 36y2 + 5x −10y = 0, find all positive integer solutions.
IMO 1992 LL IND25 Origin: IND Problem (a) Show that the set N of all natural numbers can be parti- tioned into three disjoint subsets A, B, and C satisfying the following conditions: A2 = A, B2 = C, C2 = B, AB = B, AC = C, BC = A, where HK stands for {hk | h \inH, k \inK} for any two subsets H, K of N, and...
IMO 1967 LL USS58 Origin: USS Problem A linear binomial l(z) = Az + B with complex coefficients A and B is given. It is known that the maximal value of |l(z)| on the segment −1 \leqx \leq1 (y = 0) of the real line in the complex plane (z = x + iy) is equal to M. Prove that for every z |l(z)| \leqM\rho, where \rho is the sum...
IMO 1983 LL GBR32 Origin: GBR Problem Let a, b, c be positive real numbers and let [x] denote the greatest integer that does not exceed the real number x. Suppose that f is a function defined on the set of nonnegative integers n and taking real values such that f(0) = 0 and f(n) \leqan + f([bn]) + f([cn]), for all n \geq1. Prove that if b + c...
IMO 1982 LL USA48 Origin: USA Problem Given a finite sequence of complex numbers c1, c2, . . . , cn, show that there exists an integer k (1 \leqk \leqn) such that for every finite sequence a1, a2, . . . , an of real numbers with 1 \geqa1 \geqa2 \geq\cdot \cdot \cdot \geqan \geq0, the following inequality holds: n m=1 amcmn \leq n m=1 cm .
IMO 1986 LL CAN10 Origin: CAN Problem A set of n standard dice are shaken and randomly placed in a straight line. If n < 2r and r < s, then the probability that there will be a string of at least r, but not more than s, consecutive 1’s can be written as P/6s+2. Find an explicit expression for P.
IMO 1986 LL IRE45 Origin: IRE Problem Given n real numbers a1 \leqa2 \leq\cdot \cdot \cdot \leqan, define M1 = 1 n n i=1 ai, M2 = n(n −1) 1\leqi<j\leqn aiaj, Q = M 2 1 −M2. Prove that a1 \leqM1 −Q \leqM1 + Q \leqan and that equality holds if and only if a1 = a2 = \cdot \cdot \cdot = an.
IMO 1989 LL POR87 Origin: POR Problem A balance has a left pan, a right pan, and a pointer that moves along a graduated ruler. Like many other grocer balances, this one works as follows: An object of weight L is placed in the left pan and another of weight R in the right pan, the pointer stops at the number R −L on the graduated ruler. There are n...
IMO 1969 LL BEL3 Origin: BEL Problem Construct the circle that is tangent to three given circles.
IMO 1985 LL AUS3 Origin: AUS Problem A function f has the following property: If k > 1, j > 1, and (k, j) = m, then f(kj) = f(m) (f(k/m) + f(j/m)). What values can f(1984) and f(1985) take?
IMO 1983 LL SPA59 Origin: SPA Problem Solve the equation tan2(2x) + 2 tan(2x) \cdot tan(3x) −1 = 0.
IMO 1977 LL SWE42 Origin: SWE Problem The sequence an,k, k = 1, 2, 3, . . ., 2n, n = 0, 1, 2, . . ., is defined by the following recurrence formula: a1 = 2, an,k = 2a3 n−1,k, an,k+2n−1 = 1 2a3 n−1,k for k = 1, 2, 3, . . ., 2n−1, n = 0, 1, 2, . . . . Prove that the numbers an,k...
IMO 1966 LL POL37 Origin: POL Problem Prove that the perpendiculars drawn from the midpoints of the sides of a cyclic quadrilateral to the opposite sides meet at one point.
IMO 1970 LL AUT5 Origin: AUT Problem Prove that n n+1 + n+1 + \cdot \cdot \cdot + n n+1 \geq1 for n \geq2.
IMO 1978 LL TUR36 Origin: TUR Problem The integers 1 through 1000 are located on the circumference of a circle in natural order. Starting with 1, every fifteenth number (i.e., 1, 16, 31, . . .) is marked. The marking is continued until an already marked number is reached. How many of the numbers will be left unmarked?
IMO 1979 LL YUG77 Origin: YUG Problem By h(n), where n is an integer greater than 1, let us denote the greatest prime divisor of the number n. Are there infinitely many numbers n for which h(n) < h(n + 1) < h(n + 2) holds?
IMO 1987 LL POL51 Origin: POL Problem The function F is a one-to-one transformation of the plane into itself that maps rectangles into rectangles (rectangles are closed; continu- ity is not assumed). Prove that F maps squares into squares.
IMO 1970 LL AUT4 Origin: AUT Problem Solve the system of equations x2 + xy = a2 + ab y2 + xy = a2 −ab, a, b real, a ̸= 0.
IMO 1985 LL BRA7 Origin: BRA Problem A convex quadrilateral is inscribed in a circle of radius 1. Prove that the difference between its perimeter and the sum of the lengths of its diagonals is greater than zero and less than 2.
IMO 1974 LL USA39 Origin: USA Problem Let n be a positive integer, n \geq2, and consider the polynomial equation xn −xn−2 −x + 2 = 0. For each n, determine all complex numbers x that satisfy the equation and have modulus |x| = 1.
IMO 1976 LL SWE33 Origin: SWE Problem A finite set of points P in the plane has the following prop- erty: Every line through two points in P contains at least one more point belonging to P. Prove that all points in P lie on a straight line.
IMO 1988 LL INA45 Origin: INA Problem (a) Consider a circle K with diameter AB, a circle L tangent to AB and to K, and a circle M tangent to circle K, circle L, and AB. Calculate the ratio of the area of circle K to the area of circle M. (b) In triangle ABC, AB = AC and ∡CAB = 80◦. If points D, E, and F lie on...
IMO 1987 LL GBR26 Origin: GBR Problem Prove that if x, y, z are real numbers such that x2+y2+z2 = 2, then x + y + z \leqxyz + 2.
IMO 1967 LL ITA29 Origin: ITA Problem The triangles A0B0C0 and A′B′C′ have all their angles acute. Describe how to construct one of the triangles ABC, similar to A′B′C′ and circumscribing A0B0C0 (so that A, B, C correspond to A′, B′, C′, and AB passes through C0, BC through A0, and CA through B0). Among these triangles ABC, describe, and prove, how to construct the triangle with the maximum area....
IMO 1985 LL USS91 Origin: USS Problem Thirty-four countries participated in a jury session of the IMO, each represented by the leader and the deputy leader of the team. Before the meeting, some participants exchanged handshakes, but no team leader shook hands with his deputy. After the meeting, the leader of the Illyrian team asked every other participant the number of people they had shaken hands with, and all the...
IMO 1966 LL ROM29 Origin: ROM Problem (a) Find the number of ways 500 can be represented as a sum of consecutive integers. (b) Find the number of such representations for N = 2\alpha3\beta5\gamma, \alpha, \beta, \gamma \inN. Which of these representations consist only of natural numbers? (c) Determine the number of such representations for an arbitrary natural number N.
IMO 1989 LL IRE54 Origin: IRE Problem Let f be a function from the real numbers to the real numbers such that f(1) = 1, f(a+b) = f(a)+f(b) for all a, b, and f(x)f(1/x) = 1 for all x ̸= 0. Prove that f(x) = x for all real numbers x.
IMO 1988 LL IRE49 Origin: IRE Problem Let −1 < x < 1. Show that k=0 1 −x2 1 −2x cos(2\pik/7) + x2 = 7(1 + x7) 1 −x7 . Deduce that csc2 \pi 7 + csc2 2\pi 7 + csc2 3\pi 7 = 8.
IMO 1967 LL MON31 Origin: MON Problem An urn contains balls of k different colors; there are ni balls of the ith color. Balls are drawn at random from the urn, one by one, without replacement. Find the smallest number of draws necessary for getting m balls of the same color. Solution Suppose that n1 \leqn2 \leq\cdot \cdot \cdot \leqnk. If nk < m, there is no solution. Otherwise, the...
IMO 1992 LL PRK59 Origin: PRK Problem Let a regular 7-gon A0A1A2A3A4A5A6 be inscribed in a circle. Prove that for any two points P, Q on the arc A0A6 the following equality holds: i=0 (−1)iPAi = i=0 (−1)iQAi.
IMO 1979 LL ISR43 Origin: ISR Problem Let a, b, c denote the lengths of the sides BC, CA, AB, respec- tively, of a triangle ABC. If P is any point on the circumference of the circle inscribed in the triangle, show that aPA2+bPB2+cPC2 is constant.
IMO 1972 LL CZS12 Origin: CZS Problem A circle k = (S, r) is given and a hexagon AA′BB′CC′ inscribed in it. The lengths of sides of the hexagon satisfy AA′ = A′B, BB′ = B′C, CC′ = C′A. Prove that the area P of triangle ABC is not greater than the area P ′ of triangle A′B′C′. When does P = P ′ hold?
IMO 1989 LL FRA19 Origin: FRA Problem Let a1, . . . , an be distinct positive integers that do not contain a 9 in their decimal representations. Prove that a1 \cdot \cdot \cdot + 1 an \leq30.
IMO 1978 LL YUG54 Origin: YUG Problem Let p, q and r be three lines in space such that there is no plane that is parallel to all three of them. Prove that there exist three planes \alpha, \beta, and \gamma, containing p, q, and r respectively, that are perpendicular to each other (\alpha \perp\beta, \beta \perp\gamma, \gamma \perp\alpha).
IMO 1985 LL IRE40 Origin: IRE Problem Each of the numbers x1, x2, . . . , xn equals 1 or −1 and x1x2x3x4 + x2x3x4x5 + \cdot \cdot \cdot + xn−3xn−2xn−1xn +xn−2xn−1xnx1 + xn−1xnx1x2 + xnx1x2x3 = 0. Prove that n is divisible by 4.
IMO 1989 LL GBR26 Origin: GBR Problem Let a, b, c, d be positive integers such that ab = cd and a + b = c −d. Prove that there exists a right-angled triangle the measures of whose sides (in some unit) are integers and whose area measure is ab square units.
IMO 1979 LL HUN42 Origin: HUN Problem Let a quadratic polynomial g(x) = ax2 + bx + c be given and an integer n \geq1. Prove that there exists at most one polynomial f(x) of nth degree such that f(g(x)) = g(f(x)).
IMO 1969 LL NET51 Origin: NET Problem A curve determined by y = x2 −10x + 52, 0 \leqx \leq100, is constructed in a rectangular grid. Determine the number of squares cut by the curve.
IMO 1983 LL LUX44 Origin: LUX Problem We are given twelve coins, one of which is a fake with a different mass from the other eleven. Determine that coin with three weighings and whether it is heavier or lighter than the others.
IMO 1987 LL ICE37 Origin: ICE Problem Five distinct numbers are drawn successively and at random from the set {1, . . . , n}. Show that the probability of a draw in which the first three numbers as well as all five numbers can be arranged to form an arithmetic progression is greater than (n−2)3 .
IMO 1969 LL CZS16 Origin: CZS Problem A convex quadrilateral ABCD with sides AB = a, BC = b, CD = c, DA = d and angles \alpha = \angleDAB, \beta = \angleABC, \gamma = \angleBCD, and \delta = \angleCDA is given. Let s = (a + b + c + d)/2 and P be the area of the quadrilateral. Prove that P 2 = (s −a)(s −b)(s −c)(s −d)...
IMO 1974 LL FIN13 Origin: FIN Problem Prove that 2147 −1 is divisible by 343.
IMO 1988 LL INA46 Origin: INA Problem (a) Calculate x = (11+6 \sqrt 2)\sqrt 11−6 \sqrt 2−(11−6 \sqrt 2)\sqrt 11+6 \sqrt (\sqrt\sqrt 5+2+\sqrt\sqrt 5−2)−(\sqrt\sqrt 5+1) . (b) For each positive number x, let k = (x+1/x)6−(x6+1/x6)−2 (x+1/x)3+(x3+1/x3) . Calculate the minimum value of k.
IMO 1977 LL FIN46 Origin: FIN Problem Let f be a strictly increasing function defined on the set of real numbers. For x real and t positive, set g(x, t) = f(x + t) −f(x) f(x) −f(x −t). Assume that the inequalities 2−1 < g(x, t) < 2 hold for all positive t if x = 0, and for all t \leq|x| otherwise. Show that 14−1 < g(x, t) <...
IMO 1984 LL GBR28 Origin: GBR Problem A “number triangle” (tnk) (0 \leqk \leqn) is defined by tn,0 = tn,n = 1 (n \geq0), tn+1,m = 2 − \sqrt m tn,m + 2 + \sqrt n−m+1 tn,m−1 (1 \leqm \leqn). Prove that all tn,m are integers.
IMO 1970 LL FRA23 Origin: FRA Problem Let E be a finite set, PE the family of its subsets, and f a mapping from PE to the set of nonnegative real numbers such that for any two disjoint subsets A, B of E, f(A \cupB) = f(A) + f(B). Prove that there exists a subset F of E such that if with each A \subsetE we associate a subset A′...
IMO 1985 LL USA89 Origin: USA Problem Given that n elements a1, a2, . . . , an are organized into n pairs P1, P2, . . . , Pn in such a way that two pairs Pi, Pj share exactly one el- ement when (ai, aj) is one of the pairs, prove that every element is in exactly two of the pairs.
IMO 1967 LL USS57 Origin: USS Problem Consider the sequence (cn): c1 = a1 + a2 + \cdot \cdot \cdot + a8, c2 = a2 1 + a2 2 + \cdot \cdot \cdot + a2 8, . . . . . . . . . . . . cn = an 1 + an 2 + \cdot \cdot \cdot + an 8, . . . . . . . ....
IMO 1969 LL MON40 Origin: MON Problem Find the number of five-digit numbers with the following properties: there are two pairs of digits such that digits from each pair are equal and are next to each other, digits from different pairs are different, and the remaining digit (which does not belong to any of the pairs) is different from the other digits.
IMO 1992 LL IRE33 Origin: IRE Problem Let a, b, c be positive real numbers and p, q, r complex numbers. Let S be the set of all solutions (x, y, z) in C of the system of simultaneous equations ax + by + cz = p, ax2 + by2 + cz2 = q, ax3 + bx3 + cx3 = r. Prove that S has at most six elements.
IMO 1966 LL USS7 Origin: USS Problem For which arrangements of two infinite circular cylinders does their intersection lie in a plane?
IMO 1967 LL POL36 Origin: POL Problem Prove that the center of the sphere circumscribed around a tetrahedron ABCD coincides with the center of a sphere inscribed in that tetrahedron if and only if AB = CD, AC = BD, and AD = BC. Solution Suppose that the skew edges of the tetrahedron ABCD are equal. Let K, L, M, P, Q, R be the midpoints of edges AB, AC,...
IMO 1970 LL POL40 Origin: POL Problem Let ABC be a triangle with angles \alpha, \beta, \gamma commensurable with \pi. Starting from a point P interior to the triangle, a ball reflects on the sides of ABC, respecting the law of reflection that the angle of incidence is equal to the angle of reflection. Prove that, supposing that the ball never reaches any of the vertices A, B, C, the...
IMO 1988 LL INA43 Origin: INA Problem (a) The polynomial x2k + 1 + (x+ 1)2k is not divisible by x2 +x+ 1. Find the value of k. (b) If p, q, and r are distinct roots of x3 −x2 + x −2 = 0, find the value of p3 + q3 + r3. (c) If r is the remainder when each of the numbers 1059, 1417, and 2312 is...
IMO 1974 LL SWE33 Origin: SWE Problem Let a be a real number such that 0 < a < 1, and let n be a positive integer. Define the sequence a0, a1, a2, . . . , an recursively by a0 = a; ak+1 = ak + 1 na2 k for k = 0, 1, . . ., n −1. Prove that there exists a real number A, depending on...
IMO 1988 LL KOR53 Origin: KOR Problem Let x = p, y = q, z = r, w = s be the unique solution of the system of linear equations x + aiy + a2 i z + a3 i w = a4 i , i = 1, 2, 3, 4. Express the solution of the following system in terms of p, q, r, and s: x + a2 i...
IMO 1978 LL USA44 Origin: USA Problem In \triangleABC with \angleC = 60o, prove that c a + c b \geq2.
IMO 1987 LL AUS4 Origin: AUS Problem Let a1, a2, a3, b1, b2, b3 be positive real numbers. Prove that (a1b2 + a2b1 + a1b3 + a3b1 + a2b3 + a3b2)2 \geq4(a1a2 + a2a3 + a3a1)(b1b2 + b2b3 + b3b1) and show that the two sides of the inequality are equal if and only if a1/b1 = a2/b2 = a3/b3.
IMO 1970 LL BEL9 Origin: BEL Problem If n is even, prove that 1 −1 2 + 1 3 −1 4 + \cdot \cdot \cdot −1 n = 2 n + 2 + n + 4 + n + 6 + \cdot \cdot \cdot + 1 2n .
IMO 1978 LL FIN13 Origin: FIN Problem The satellites A and B circle the Earth in the equatorial plane at altitude h. They are separated by distance 2r, where r is the radius of the Earth. For which h can they be seen in mutually perpendicular directions from some point on the equator?
IMO 1992 LL GBR21 Origin: GBR Problem Prove that if x, y, z > 1 and 1 x + 1 y + 1 z = 2, then \sqrtx + y + z \geq \sqrt x −1 + y −1 + \sqrt z −1.
IMO 1976 LL BUL2 Origin: BUL Problem Let P be a set of n points and S a set of l segments. It is known that: (i) No four points of P are coplanar. (ii) Any segment from S has its endpoints at P. (iii) There is a point, say g, in P that is the endpoint of a maximal number of segments from S and that is not a...
IMO 1992 LL AUS3 Origin: AUS Problem Let ABC be a triangle, O its circumcenter, S its centroid, and H its orthocenter. Denote by A1, B1, and C1 the centers of the circles circumscribed about the triangles CHB, CHA, and AHB, respectively. Prove that the triangle ABC is congruent to the triangle A1B1C1 and that the nine-point circle of \triangleABC is also the nine-point circle of \triangleA1B1C1.
IMO 1970 LL GDR28 Origin: GDR Problem A set G with elements u, v, w, . . . is a group if the following conditions are fulfilled: (1) There is a binary algebraic operation ◦defined on G such that for all u, v \inG there is a w \inG with u ◦v = w. (2) This operation is associative; i.e., for all u, v, w \inG, (u ◦v) ◦w =...
IMO 1971 LL CUB10 Origin: CUB Problem In how many different ways can three knights be placed on a chessboard so that the number of squares attacked would be maximal?
IMO 1983 LL USA65 Origin: USA Problem Let ABCD be a convex quadrilateral whose diagonals AC and BD intersect in a point P. Prove that AP PC = cot \angleBAC + cot \angleDAC cot \angleBCA + cot \angleDCA.
IMO 1992 LL SPA66 Origin: SPA Problem A circle of radius \rho is tangent to the sides AB and AC of the triangle ABC, and its center K is at a distance p from BC. (a) Prove that a(p −\rho) = 2s(r −\rho), where r is the inradius and 2s the perimeter of ABC. (b) Prove that if the circle intersect BC at D and E, then DE = 4...
IMO 1985 LL ROM67 Origin: ROM Problem Let k \geq2 and n1, n2, . . . , nk \geq1 natural numbers having the property n2 | 2n1 −1, n3 | 2n2 −1, . . . , nk | 2nk−1 −1, and n1 | 2nk −1. Show that n1 = n2 = \cdot \cdot \cdot = nk = 1.
IMO 1971 LL GDR19 Origin: GDR Problem In a triangle P1P2P3 let PiQi be the altitude from Pi for i = 1, 2, 3 (Qi being the foot of the altitude). The circle with diameter PiQi meets the two corresponding sides at two points different from Pi. Denote the length of the segment whose endpoints are these two points by li. Prove that l1 = l2 = l3.
IMO 1979 LL VIE73 Origin: VIE Problem In a plane a finite number of equal circles are given. These circles are mutually nonintersecting (they may be externally tangent). Prove that one can use at most four colors for coloring these circles so that two circles tangent to each other are of different colors. What is the smallest number of circles that requires four colors?
IMO 1988 LL BUL2 Origin: BUL Problem Let an = (n + 1)2 + n2 , n = 1, 2, . . . , where [x] denotes the integer part of x. Prove that (a) there are infinitely many positive integers m such that am+1−am > 1; (b) there are infinitely many positive integers m such that am+1−am = 1.
IMO 1966 LL ROM39 Origin: ROM Problem In a plane, a circle with center O and radius R and two points A, B are given. (a) Draw a chord CD parallel to AB so that AC and BD intersect at a point P on the circle. (b) Prove that there are two possible positions of point P, say P1, P2, and find the distance between them if OA = a,...
IMO 1977 LL ROM35 Origin: ROM Problem Find all numbers N = a1a2 . . . an for which 9 \times a1a2 . . . an = an . . . a2a1 such that at most one of the digits a1, a2, . . . , an is zero. Solution The solutions are 0 and Nk = 10 99 . . .9 k 89, where k =...
IMO 1985 LL SPA75 Origin: SPA Problem Let ABCD be a rectangle, AB = a, BC = b. Consider the family of parallel and equidistant straight lines (the distance between two consecutive lines being d) that are at an the angle \varphi, 0 \leq\varphi \leq90◦, with respect to AB. Let L be the sum of the lengths of all the segments intersecting the rectangle. Find: (a) how L varies, (b)...
IMO 1967 LL USS59 Origin: USS Problem On the circle with center O and radius 1 the point A0 is fixed and points A1, A2, . . . , A999, A1000 are distributed in such a way that \angleA0OAk = k (in radians). Cut the circle at points A0, A1, . . . , A1000. How many arcs with different lengths are obtained? Solution By the arc AB we shall...
IMO 1983 LL NET48 Origin: NET Problem Prove that in any parallelepiped the sum of the lengths of the edges is less than or equal to twice the sum of the lengths of the four diagonals.
IMO 1984 LL AUS2 Origin: AUS Problem Given a regular convex 2m-sided polygon P, show that there is a 2m-sided polygon \pi with the same vertices as P (but in different order) such that \pi has exactly one pair of parallel sides.
IMO 1971 LL POL38 Origin: POL Problem Let A, B, C be three points with integer coordinates in the plane and K a circle with radius R passing through A, B, C. Show that AB\cdotBC\cdotCA \geq2R, and if the center of K is in the origin of the coordinates, show that AB \cdot BC \cdot CA \geq4R.
IMO 1974 LL GBR19 Origin: GBR Problem (Alternative to GBR 2) Prove that there exists, for n \geq4, a set S of 3n equal circles in spacethat can be partitioned into three subsets s5, s4, and s3, each containing n circles, such that each circle in sr touches exactly r circles in S.
IMO 1969 LL HUN38 Origin: HUN Problem Let r and m (r \leqm) be natural numbers and Ak = 2k−1 2m \pi. Evaluate m2 m k=1 m l=1 sin(rAk) sin(rAl) cos(rAk −rAl).
IMO 1969 LL YUG70 Origin: YUG Problem A park has the shape of a convex pentagon of area 5 \sqrt 3 ha (= 50000 \sqrt 3 m2). A man standing at an interior point O of the park notices that he stands at a distance of at most 200 m from each vertex of the pentagon. Prove that he stands at a distance of at least 100 m from each...
IMO 1986 LL CZS17 Origin: CZS Problem We call a tetrahedron right-faced if each of its faces is a right- angled triangle. (a) Prove that every orthogonal parallelepiped can be partitioned into six right-faced tetrahedra. (b) Prove that a tetrahedron with vertices A1, A2, A3, A4 is fight-faced if and only if there exist four distinct real numbers c1, c2, c3, and c4 such that the edges AjAk have lengths...
IMO 1983 LL COL20 Origin: COL Problem Let f and g be functions from the set A to the same set A. We define f to be a functional nth root of g (n is a positive integer) if f n(x) = g(x), where f n(x) = f n−1(f(x)). (a) Prove that the function g : R \toR, g(x) = 1/x has an infinite number of nth functional roots for...
IMO 1971 LL HUN25 Origin: HUN Problem Let ABC, AA1A2, BB1B2, CC1C2 be four equilateral triangles in the plane satisfying only that they are all positively oriented (i.e., in the counterclockwise direction). Denote the midpoints of the segments A2B1, B2C1, C2A1 by P, Q, R in this order. Prove that the triangle PQR is equilateral.
IMO 1985 LL FRA24 Origin: FRA Problem Let d \geq1 be an integer that is not the square of an integer. Prove that for every integer n \geq1, (n \sqrt d + 1)| sin(n\pi \sqrt d)| \geq1.
IMO 1985 LL GBR31 Origin: GBR Problem Let E1, E2, and E3 be three mutually intersecting ellipses, all in the same plane. Their foci are respectively F2, F3; F3, F1; and F1, F2. The three foci are not on a straight line. Prove that the common chords of each pair of ellipses are concurrent.
IMO 1992 LL TUR73 Origin: TUR Problem Let {An | n = 1, 2, . . .} be a set of points in the plane such that for each n, the disk with center An and radius 2n contains no other point Aj. For any given positive real numbers a < b and R, show that there is a subset G of the plane satisfying: (i) the area of G...
IMO 1970 LL CZS19 Origin: CZS Problem Let n > 1 be a natural number, a \geq1 a real number, and x1, x2, . . . , xn numbers such that x1 = 1, xk+1 xk = a+\alphak for k = 1, 2, . . . , n− 1, where \alphak are real numbers with \alphak \leq k(k+1). Prove that n−1\sqrtxn < a + n −1.
IMO 1974 LL BUL2 Origin: BUL Problem Let {un} be the Fibonacci sequence, i.e., u0 = 0, u1 = 1, un = un−1 + un−2 for n > 1. Prove that there exist infinitely many prime numbers p that divide up−1.
IMO 1988 LL MON63 Origin: MON Problem Let ABCD be a quadrilateral. Let A′BCD′ be the reflection of ABCD in BC, while A′′B′CD′ is the reflection of A′BCD′ in CD′ and A′′B′′C′D′ is the reflection of A′′B′CD′ in D′A′′. Show that if the lines AA′′ and BB′′ are parallel, then ABCD is a cyclic quadrilateral.
IMO 1989 LL USA104 Origin: USA Problem For each nonzero complex number z, let arg z be the unique real number t such that −\pi < t \leq\pi and z = |z|(cos t + ı sin t). Given a real number c > 0 and a complex number z ̸= 0 with arg z ̸= \pi, define B(c, z) = {b \inR | |w −z| < b ⇒| arg w...
IMO 1985 LL ISR43 Origin: ISR Problem Suppose that 1985 points are given inside a unit cube. Show that one can always choose 32 of them in such a way that every (possibly degenerate) closed polygon with these points as vertices has a total length of less than 8 \sqrt 3.
IMO 1989 LL FIN17 Origin: FIN Problem Let a, 0 < a < 1, be a real number and f a continuous function on [0, 1] satisfying f(0) = 0, f(1) = 1, and f x + y = (1 −a)f(x) + af(y) for all x, y \in[0, 1] with x \leqy. Determine f(1/7).
IMO 1974 LL GBR16 Origin: GBR Problem A pack of 2n cards contains n different pairs of cards. Each pair consists of two identical cards, either of which is called the twin of the other. A game is played between two players A and B. A third person called the dealer shuffles the pack and deals the cards one by one face upward onto the table. One of the players,...
IMO 1967 LL ROM44 Origin: ROM Problem Suppose p and q are two different positive integers and x is a real number. Form the product (x + p)(x + q). (a) Find the sum S(x, n) = (x + p)(x + q), where p and q take values from 1 to n. (b) Do there exist integer values of x for which S(x, n) = 0? Solution (a) S(x, n)...
IMO 1988 LL VIE89 Origin: VIE Problem We match sets M of points in the coordinate plane to sets M∗ according to the rule that (x∗, y∗) belongs to M∗if and only if xx∗+yy∗\leq 1 whenever (x, y) \inM. Find all triangles Y such that Y∗is the reflection of Y at the origin.
IMO 1984 LL GBR27 Origin: GBR Problem The function f(n) is defined on the nonnegative integers n by: f(0) = 0, f(1) = 1, f(n) = f n −1 2m(m −1) −f 1 2m(m + 1) −n , for 1 2m(m −1) < n \leq1 2m(m + 1), m \geq2. Find the smallest integer n for which f(n) = 5.
IMO 1969 LL BUL7 Origin: BUL Problem Prove that the equation x3 + y3 + z3 = 1969 has no integral solutions.
IMO 1972 LL GBR17 Origin: GBR Problem A solid right circular cylinder with height h and base-radius r has a solid hemisphere of radius r resting upon it. The center of the hemisphere O is on the axis of the cylinder. Let P be any point on the surface of the hemisphere and Q the point on the base circle of the cylinder that is furthest from P (measuring along...
IMO 1992 LL ROM63 Origin: ROM Problem Let a and b be integers. Prove that 2a2−1 b2+2 is not an integer.
IMO 1992 LL TWN77 Origin: TWN Problem Show that if 994 integers are chosen from 1, 2, . . . , 1992 and one of the chosen integers is less than 64, then there exist two among the chosen integers such that one of them is a factor of the other.
IMO 1985 LL ROM70 Origin: ROM Problem Let C be a class of functions f : N \toN that contains the functions S(x) = x + 1 and E(x) = x −[\sqrtx]2 for every x \inN. ([x] is the integer part of x.) If C has the property that for every f, g \inC, f + g, fg, f ◦g \inC, show that the function max(f(x) −g(x), 0) is in...
IMO 1982 LL USA46 Origin: USA Problem Prove that if a diagonal is drawn in a quadrilateral inscribed in a circle, the sum of the radii of the circles inscribed in the two triangles thus formed is the same, no matter which diagonal is drawn.
IMO 1982 LL CAN15 Origin: CAN Problem Show that the set S of natural numbers n for which 3/n cannot be written as the sum of two reciprocals of natural numbers (S = {n | 3/n ̸= 1/p + 1/q for any p, q \inN}) is not the union of finitely many arithmetic progressions.
IMO 1992 LL CAN6 Origin: CAN Problem Suppose that n numbers x1, x2, . . . , xn are chosen randomly from the set {1, 2, 3, 4, 5}. Prove that the probability that x2 1 + x2 2 + \cdot \cdot \cdot + x2 n \equiv0 (mod 5) is at least 1/5.
IMO 1967 LL MON34 Origin: MON Problem The faces of a convex polyhedron are six squares and eight equilateral triangles, and each edge is a common side for one triangle and one square. All dihedral angles obtained from the triangle and square with a common edge are equal. Prove that it is possible to circumscribe a sphere around this polyhedron and compute the ratio of the squares of the volumes...
IMO 1969 LL GBR28 Origin: GBR Problem Let us define u0 = 0, u1 = 1 and for n \geq0, un+2 = aun+1+bun, a and b being positive integers. Express un as a polynomial in a and b. Prove the result. Given that b is prime, prove that b divides a(ub −1).
IMO 1969 LL GDR32 Origin: GDR Problem Find the maximal number of regions into which a sphere can be partitioned by n circles.
IMO 1977 LL VIE58 Origin: VIE Problem Prove that for every triangle the following inequality holds: ab + bc + ca 4S \geqcot \pi 6 , where a, b, c are lengths of the sides and S is the area of the triangle. Solution The following inequality (Finsler and Hadwiger, 1938) is sharper than the one we have to prove: 2ab + 2bc + 2ca −a2 −b2 −c2 \geq4S \sqrt...
IMO 1971 LL GDR20 Origin: GDR Problem Let M be the circumcenter of a triangle ABC. The line through M perpendicular to CM meets the lines CA and CB at Q and P respec- tively. Prove that CP CM CQ CM AB PQ = 2.
IMO 1987 LL AUS5 Origin: AUS Problem Let there be given three circles K1, K2, K3 with centers O1, O2, O3 respectively, which meet at a common point P. Also, let K1 \capK2 = {P, A}, K2 \capK3 = {P, B}, K3 \capK1 = {P, C}. Given an arbitrary point X on K1, join X to A to meet K2 again in Y , and join X to C to...
IMO 1969 LL POL53 Origin: POL Problem Given two segments AB and CD not in the same plane, find the locus of points M such that MA2 + MB2 = MC2 + MD2.
IMO 1976 LL USA38 Origin: USA Problem Let x = \sqrta + \sqrt b, where a and b are natural numbers, x is not an integer, and x < 1976. Prove that the fractional part of x exceeds 10−19.76.
IMO 1987 LL SPA59 Origin: SPA Problem It is given that a11, a22 are real numbers, that x1, x2, a12, b1, b2 are complex numbers, and that a11a22 = a12a12 (where a12 is the conju- gate of a12). We consider the following system in x1, x2: x1(a11x1 + a12x2) = b1, x2(a12x1 + a22x2) = b2. (a) Give one condition to make the system consistent. (b) Give one condition to...
IMO 1966 LL USS5 Origin: USS Problem Prove the inequality tan \pi sin x 4 sin \alpha + tan \pi cos x 4 cos \alpha > 1 for any x, \alpha with 0 \leqx \leq\pi/2 and \pi/6 < y < \pi/3.
IMO 1983 LL SWE61 Origin: SWE Problem Let a and b be integers. Is it possible to find integers p and q such that the integers p + na and q + nb have no common prime factor no matter how the integer n is chosen.
IMO 1978 LL USA42 Origin: USA Problem A, B, C, D, E are points on a circle O with radius equal to r. Chords AB and DE are parallel to each other and have length equal to x. Diagonals AC, AD, BE, CE are drawn. If segment XY on O meets AC at X and EC at Y , prove that lines BX and DY meet at Z on the...
IMO 1985 LL GBR32 Origin: GBR Problem A collection of 2n letters contains 2 each of n different letters. The collection is partitioned into n pairs, each pair containing 2 letters, which may be the same or different. Denote the number of distinct parti- tions by un. (Partitions differing in the order of the pairs in the partition or in the order of the two letters in the pairs are...
IMO 1969 LL GDR29 Origin: GDR Problem Find all real numbers \lambda such that the equation sin4 x −cos4 x = \lambda(tan4 x −cot4 x) (a) has no solution, (b) has exactly one solution, (c) has exactly two solutions, (d) has more than two solutions (in the interval (0, \pi/4)).
IMO 1982 LL POL40 Origin: POL Problem We consider a game on an infinite chessboard similar to that of solitaire: If two adjacent fields are occupied by pawns and the next field is empty (the three fields lie on a vertical or horizontal line), then we may remove these two pawns and put one of them on the third field. Prove that if in the initial position pawns fill a...
IMO 1978 LL FIN14 Origin: FIN Problem Let p(x, y) and q(x, y) be polynomials in two variables such that for x \geq0, y \geq0 the following conditions hold: (i) p(x, y) and q(x, y) are increasing functions of x for every fixed y. (ii) p(x, y) is an increasing and q(x) is a decreasing function of y for every fixed x. (iii) p(x, 0) = q(x, 0) for every...
IMO 1984 LL FRG23 Origin: FRG Problem A 2 \times 2 \times 12 box fixed in space is to be filled with twenty-four 1 \times 1 \times 2 bricks. In how many ways can this be done?
IMO 1970 LL BUL16 Origin: BUL Problem Show that the equation 2 −x2 + 3 −x3 = 0 has no real roots.
IMO 1989 LL BUL5 Origin: BUL Problem The sequences a0, a1, . . . and b0, b1, . . . are defined by the equal- ities a0 = \sqrt 2 , an+1 = \sqrt 1 − 1 −a2n, n = 0, 1, 2, . . . and b0 = 1, bn+1 = 1 + b2n −1 bn , n = 0, 1, 2, . . .. Prove the inequalities 2n+2an...
IMO 1979 LL NET49 Origin: NET Problem Let there be given two sequences of integers fi(1), fi(2), . . . (i = 1, 2) satisfying: (i) fi(nm) = fi(n)fi(m) if gcd(n, m) = 1; (ii) for every prime P and all k = 2, 3, 4, . . ., fi(P k) = fi(P)fi(P k−1) − P 2f(P k−2). Moreover, for every prime P: (iii) f1(P) = 2P, (iv) f2(P) <...
IMO 1986 LL USA75 Origin: USA Problem The incenter of a triangle is the midpoint of the line seg- ment of length 4 joining the centroid and the orthocenter of the triangle. Determine the maximum possible area of the triangle.
IMO 1971 LL SWE43 Origin: SWE Problem Show that for nonnegative real numbers a, b and integers n \geq2, an + bn \geq a + b n . When does equality hold?
IMO 1970 LL ROM45 Origin: ROM Problem Let M be an interior point of tetrahedron V ABC. Denote by A1, B1, C1 the points of intersection of lines MA, MB, MC with the planes V BC, V CA, V AB, and by A2, B2, C2 the points of intersection of lines V A1, V B1, V C1 with the sides BC, CA, AB. (a) Prove that the volume of the...
IMO 1983 LL FRG26 Origin: FRG Problem Let a, b, c be positive integers satisfying (a, b) = (b, c) = (c, a) =
IMO 1987 LL FRA16 Origin: FRA Problem Let ABC be a triangle. For every point M belonging to segment BC we denote by B′ and c′ the orthogonal projections of M on the straight lines AC and BC. Find points M for which the length of segment B′C′ is a minimum.
IMO 1989 LL CUB12 Origin: CUB Problem Let P(x) be a polynomial such that the following inequalities are satisfied: P(0) > 0; P(1) > P(0); P(2) > 2P(1) −P(0); P(3) > 3P(2) −3P(1) + P(0); and also for every natural number n, P(n + 4) > 4P(n + 3) −6P(n + 2) + 4P(n + 1) −P(n). Prove that for every positive natural number n, P(n) is positive.
IMO 1985 LL GBR30 Origin: GBR Problem A plane rectangular grid is given and a “rational point” is defined as a point (x, y) where x and y are both rational numbers. Let A, B, A′, B′ be four distinct rational points. Let P be a point such that A′B′ AB = B′P BC = PA′ PA . In other words, the triangles ABP, A′B′P are directly or oppositely similar....
IMO 1976 LL USS44 Origin: USS Problem A circle of radius 1 rolls around a circle of radius \sqrt
IMO 1986 LL BEL5 Origin: BEL Problem Let ABC and DEF be acute-angled triangles. Write d = EF, e = FD, f = DE. Show that there exists a point P in the interior of ABC for which the value of the expression d\cdotAP +e\cdotBP +f \cdotCP attains a minimum.
IMO 1978 LL GBR21 Origin: GBR Problem A circle touches the sides AB, BC, CD, DA of a square at points K, L, M, N respectively, and BU, KV are parallel lines such that U is on DM and V on DN. Prove that UV touches the circle.
IMO 1966 LL BUL3 Origin: BUL Problem A regular triangular prism has height h and a base of side length a. Both bases have small holes in the centers, and the inside of the three vertical walls has a mirror surface. Light enters through the small hole in the top base, strikes each vertical wall once and leaves through the hole in the bottom. Find the angle at which the...
IMO 1992 LL KOR43 Origin: KOR Problem Find the number of positive integers n satisfying \varphi(n) | n such that \infty m=1 n m −n −1 m = 1992. What is the largest number among them? As usual, \varphi(n) is the number of positive integers less than or equal to n and relatively prime to n.6
IMO 1985 LL VIE97 Origin: VIE Problem In a plane a circle with radius R and center w and a line \Lambda are given. The distance between w and \Lambda is d, d > R. The points M and N are chosen on \Lambda in such a way that the circle with diameter MN is externally tangent to the given circle. Show that there exists a point A in the...
IMO 1986 LL MON52 Origin: MON Problem Solve the system of equations tan x1 + cot x1 = 3 tan x2, tan x2 + cot x2 = 3 tan x3, \cdot \cdot \cdot \cdot \cdot \cdot tan xn + cot xn = 3 tan x1.
IMO 1971 LL AUT1 Origin: AUT Problem The points S(i, j) with integer Cartesian coordinates 0 < i \leqn, 0 < j \leqm, m \leqn, form a lattice. Find the number of: (a) rectangles with vertices on the lattice and sides parallel to the coor- dinate axes; (b) squares with vertices on the lattice and sides parallel to the coordinate axes; (c) squares in total, with vertices on the lattice.
IMO 1976 LL FIN10 Origin: FIN Problem Show that the reciprocal of any number of the form 2(m2 + m + 1), where m is a positive integer, can be represented as a sum of consecutive terms in the sequence (aj)\infty j=1, aj = j(j + 1)(j + 2).
IMO 1987 LL MON40 Origin: MON Problem The perpendicular line issued from the center of the circum- circle to the bisector of angle C in a triangle ABC divides the segment of the bisector inside ABC into two segments with ratio of lengths \lambda. Given b = AC and a = BC, find the length of side c.
IMO 1989 LL GBR27 Origin: GBR Problem Integers cm,n (m \geq0, n \geq0) are defined by cm,0 = 1 for all m \geq0, c0,n = 1 for all n \geq0, and cm,n = cm−1,n −ncm−1,n−1 for all m > 0, n > 0. Prove that cm,n = cn,m for all m \geq0, n \geq0.
IMO 1986 LL FRA21 Origin: FRA Problem Let AB be a segment of unit length and let C, D be variable points of this segment. Find the maximum value of the product of the lengths of the six distinct segments with endpoints in the set {A, B, C, D}.
IMO 1984 LL BUL9 Origin: BUL Problem The circle inscribed in the triangle A1A2A3 is tangent to its sides A1A2, A2A3, A3A1 at points T1, T2, T3, respectively. Denote by M1, M2, M3 the midpoints of the segments A2A3, A3A1, A1A2, respec- tively. Prove that the perpendiculars through the points M1, M2, M3 to the lines T2T3, T3T1, T1T2 meet at one point.
IMO 1986 LL AUS1 Origin: AUS Problem Let k be one of the integers 2, 3, 4 and let n = 2k −1. Prove the inequality 1 + bk + b2k + \cdot \cdot \cdot + bnk \geq(1 + bn)k for all real b \geq0.
IMO 1988 LL INA44 Origin: INA Problem (a) Let g(x) = x5 + x4 + x3 + x2 + x + 1. What is the remainder when the polynomial g(x12) is divided by the polynomial g(x)? (b) If k is a positive integer and f is a function such that for every positive number x, f(x2 +1) \sqrtx = k, find the value of f 9+y2 y2 \sqrt 12/y...
IMO 1982 LL BEL7 Origin: BEL Problem Find all solutions (x, y) \inZ2 of the equation x3 −y3 = 2xy + 8.
IMO 1983 LL LUX46 Origin: LUX Problem Let f be a real-valued function defined on I = (0, +\infty) and having no zeros on I. Suppose that lim x\to+\infty f ′(x) f(x) = +\infty. For the sequence un = ln f(n+1) f(n) , prove that un \to+\infty(n \to+\infty).
IMO 1974 LL USS44 Origin: USS Problem We are given n mass points of equal mass in space. We define a sequence of points O1, O2, O3, . . . as follows: O1 is an arbitrary point (within the unit distance of at least one of the n points); O2 is the center of gravity of all the n given points that are inside the unit sphere centered at O1;...
IMO 1972 LL ROM35 Origin: ROM Problem (a) Prove that for a, b, c, d \inR, m \in[1, +\infty) with am + b = −cm + d = m, (i) \sqrt a2 + b2 + \sqrt c2 + d2 + (a −c)2 + (b −d)2 \geq 4m2 1+m2 , and (ii) 2 \leq 4m2 1+m2 < 4. (b) Express a, b, c, d as functions of m so that there...
IMO 1977 LL CZS9 Origin: CZS Problem Let ABCD be a regular tetrahedron and Z an isometry map- ping A, B, C, D into B, C, D, A, respectively. Find the set M of all points X of the face ABC whose distance from Z(X) is equal to a given number t. Find necessary and sufficient conditions for the set M to be nonempty. Solution A necessary and sufficient condition...
IMO 1984 LL SPA54 Origin: SPA Problem Let P be a convex planar polygon with equal angles. Let l1, . . . , ln be its sides. Show that a necessary and sufficient condition for P to be regular is that the sum of the ratios li li+1 (i = 1, . . . , n; ln+1 = l1) equals the number of sides.
IMO 1989 LL THA99 Origin: THA Problem An arithmetic function is a real-valued function whose do- main is the set of positive integers. Define the convolution product of two arithmetic functions f and g to be the arithmetic function f ⋆g, where (f ⋆g)(n) = ij=n f(i)g(i), and f ⋆k = f ⋆f ⋆\cdot \cdot \cdot ⋆f (k times). We say that two arithmetic functions f and g are...
IMO 1969 LL FRA20 Origin: FRA Problem A polygon (not necessarily convex) with vertices in the lattice points of a rectangular grid is given. The area of the polygon is S. If I is the number of lattice points that are strictly in the interior of the polygon and B the number of lattice points on the border of the polygon, find the number T = 2S −B −2I +...
IMO 1987 LL FRA14 Origin: FRA Problem Given n real numbers 0 < t1 \leqt2 \leq\cdot \cdot \cdot \leqtn < 1, prove that (1 −t2 n) t1 (1 −t2 1)2 + t2 (1 −t3 2)2 + \cdot \cdot \cdot + tn n (1 −tn+1 n )2 < 1.
IMO 1989 LL IRE55 Origin: IRE Problem Let [x] denote the greatest integer less than or equal to x. Let \alpha be the positive root of the equation x2 −1989x −1 = 0. Prove that there exist infinitely many natural numbers n that satisfy the equation [\alphan + 1989\alpha[\alphan]] = 1989n + (19892 + 1)[\alphan].
IMO 1977 LL CZS6 Origin: CZS Problem Let x1, x2, . . . , xn (n \geq1) be real numbers such that 0 \leqxj \leq\pi, j = 1, 2, . . ., n. Prove that if n j=1(cos xj + 1) is an odd integer, then n j=1 sin xj \geq1. Solution Let ⟨y⟩denote the distance from y \inR to the closest even integer. We claim that ⟨1 + cos...
IMO 1985 LL CAN14 Origin: CAN Problem Let k be a positive integer. Define u0 = 0, u1 = 1, and un = kun−1 −un−2, n \geq2. Show that for each integer n, the number u3 1 + u3 2 + \cdot \cdot \cdot + u3 n is a multiple of u1 + u2 + \cdot \cdot \cdot + un.
IMO 1969 LL BEL4 Origin: BEL Problem Let O be a point on a nondegenerate conic. A right angle with vertex O intersects the conic at points A and B. Prove that the line AB passes through a fixed point located on the normal to the conic through the point O.
IMO 1985 LL MON52 Origin: MON Problem In the triangle ABC, let B1 be on AC, E on AB, G on BC, and let EG be parallel to AC. Furthermore, let EG be tangent to the inscribed circle of the triangle ABB1 and intersect BB1 at F. Let r, r1, and r2 be the inradii of the triangles ABC, ABB1, and BFG, respectively. Prove that r = r1 + r2.
IMO 1992 LL SAF65 Origin: SAF Problem If A, B, C, and D are four distinct points in space, prove that there is a plane P on which the orthogonal projections of A, B, C, and D form a parallelogram (possibly degenerate).
IMO 1976 LL GDR18 Origin: GDR Problem Prove that the number 191976 + 761976: (a) is divisible by the (Fermat) prime number F4 = 224 + 1; (b) is divisible by at least four distinct primes other than F4.
IMO 1986 LL CAN9 Origin: CAN Problem In a triangle ABC, \angleBAC = 100◦, AB = AC. A point D is chosen on the side AC such that \angleABD = \angleCBD. Prove that AD + DB = BC.
IMO 1983 LL GBR30 Origin: GBR Problem Prove the existence of a unique sequence {un} (n = 0, 1, 2 . . .) of positive integers such that u2 n = n r=0 n + r r un−r for all n \geq0, where m r is the usual binomial coefficient.
IMO 1985 LL FRG28 Origin: FRG Problem Let M be the set of the lengths of an octahedron whose sides are congruent quadrangles. Prove that M has at most three elements. (FRG 1a) Let an octahedron whose sides are congruent quadrangles be given. Prove that each of these quadrangles has two equal sides meeting at a common vertex.
IMO 1989 LL VIE109 Origin: VIE Problem Let Ax, By be two noncoplanar rays with AB as a common per- pendicular, and let M, N be two mobile points on Ax and By respectively such that AM + BN = MN. First version. Prove that there exist infinitely many lines coplanar with each of the lines MN. Second version. Prove that there exist infinitely many rotations around a fixed axis...
IMO 1977 LL USS48 Origin: USS Problem The intersection of a plane with a regular tetrahedron with edge a is a quadrilateral with perimeter P. Prove that 2a \leqP \leq3a. Solution Let a plane cut the edges AB, BC, CD, DA at points K, L, M, N respec- tively. Let D′, A′, B′ be distinct points in the plane ABC such that the triangles BCD′, CD′A′, D′A′B′ are equilateral, and...
IMO 1972 LL NET29 Origin: NET Problem Let A, B, C be points on the sides B1C1, C1A1, A1B1 of a triangle A1B1C1 such that A1A, B1B, C1C are the bisectors of angles of the triangle. We have that AC = BC and A1C1 ̸= B1C1. (a) Prove that C1 lies on the circumcircle of the triangle ABC. (b) Suppose that ∡BAC1 = \pi/6; find the form of triangle ABC.
IMO 1988 LL VIE91 Origin: VIE Problem A regular 14-gon with side length a is inscribed in a circle of radius one. Prove that 2 −a 2a 3 cos \pi 7 .
IMO 1989 LL PHI76 Origin: PHI Problem Let k and s be positive integers. For sets of real numbers {\alpha1, \alpha2, . . . , \alphas} and {\beta1, \beta2, . . . , \betas} that satisfy s i=1 \alphaj i = s i=1 \betaj i for each j = 1, 2, . . ., k, we write {\alpha1, \alpha2, . . . , \alphas} =k {\beta1, \beta2, . . ....
IMO 1966 LL YUG44 Origin: YUG Problem What is the greatest number of balls of radius 1/2 that can be placed within a rectangular box of size 10 \times 10 \times 1?
IMO 1969 LL GDR33 Origin: GDR Problem Given a ring G in the plane bounded by two concentric circles with radii R and R/2, prove that we can cover this region with 8 disks of radius 2R/5. (A region is covered if each of its points is inside or on the border of some disk.)