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Find all functions f from the reals to the reals such that

Let ABC be a triangle, H its orthocenter, O its circumcenter,

C6 (FRA 2) Let O be a point of three-dimensional space and let l1, l2, l3

A1 (GBR 3)IMO1 The function f(n) is defined for all positive integers

Let A1A2 . . . An be a regular n-gon. The points B1, . . . , Bn−1

For a polynomial P of degree 2000 with distinct real co-

For a triangle ABC, let k be its circumcircle with radius r. The

Let … be three positive integers with ….

Let ABC be a triangle with ∡BAC = 60◦. Let AP bisect

Let n and k be positive integers. There are given n circles

Prove that the product of five consecutive positive integers

Let Q be the center of the inscribed circle of a triangle ABC.

Determine all the triples (a, b, c) of positive real numbers such

Two congruent equilateral triangles ABC and A′B′C′ in the

II 3 (CUB 3) Let x, y, z be real numbers each of whose absolute value

Show that any two points lying inside a regular n-gon E can

For every integer n \geq2 determine the minimum value that the

For n \geq3 and a1 \leqa2 \leq\cdot \cdot \cdot \leqan given real numbers we

Prove for each triangle ABC the inequality

(a) Do there exist functions f : R oR and g : R oR such that

For each positive integer n, denote by s(n) the greatest

On a semicircle with unit radius four consecutive chords AB, BC,

On a 5 imes 5 board, two players alternately mark numbers on

6b.(CAN 5) Let x1, x2, . . . , xn be positive numbers. Prove that

Prove or disprove: From the interval [1, . . . , 30000] one

An … square is divided into … unit squares in the usual manner. Each of the … vertices of these squares is to be…

A nonempty set A of real numbers is called a B3-set if the

Given a tetrahedron ABCD, let x = AB \cdot CD, y = AC \cdot BD,

For a positive integer n define a sequence of zeros and ones

We consider three distinct half-lines Ox, Oy, Oz in a plane.

Prove that the volume of a tetrahedron inscribed in a right

Let n \inN, n \geq2, and A0 = (a01, a02, . . . , a0n) be any n-tuple

Let n \geq2 be an integer. Find the maximal cardinality of a set

A convex quadrilateral ABCD has perpendicular diagonals.

Let a tetrahedron ABCD be inscribed in a sphere S. Find the

Prove:

Let a be a positive integer and let {an} be defined by a0 = 0

Prove that the intersection of a plane and a regular tetrahedron

Does there exist a second-degree polynomial p(x, y) in two

Consider pairs of sequences of positive real numbers a1 \geq

Let n be a positive integer having at least two different prime

II 1 (POL 2) Let ai, bi be coprime positive integers for i = 1, 2, . . . , k,

For three points A, B, C in the plane we define m(ABC)

For an n imes n matrix A, let Xi be the set of entries in row

Let M be an interior point of the tetrahedron ABCD. Prove

A natural number n is said to have the property P if whenever

In an acute-angled triangle ABC, let AD, BE be altitudes and

Let x, y, z be nonnegative real numbers with x+y +z = 1.

Given a point M on the side AB of the triangle ABC, let

Let p be a prime number and let A be a set of positive integers

Let f(x) = x2+1

Let K be one of the two intersection points of the circles W1

Let real numbers x1, x2, . . . , xn satisfy 0 < x1 < x2 < \cdot \cdot \cdot <

Let T be the set of ordered triples (x, y, z), where x, y, z are

A circle of radius 1 is located in a right-angled trihedron and

Does there exist an integer n > 1 that satisfies the following

Let ABCD be a cyclic quadrilateral. Let P, Q, R be the

A positive integer is written in each square of an m imesn board.

Consider two concentric circles of radii R and r (R > r)

A2 (YUG 1) Let K be a convex polygon in the plane and suppose that

For an acute triangle ABC, M is the midpoint of the segment

Determine all positive integers n \geq2 that satisfy the following

We consider permutations (x1, . . . , x2n) of the set {1, . . . ,

Find all functions f defined on the positive real numbers

For points A1, . . . , A5 on the sphere of radius 1, what is the

A6 (VIE 1)IMO6 Let S be a square with sides of length 100 and let L be

A broken line A1A2 . . . An is drawn in a 50 imes50 square, so that

Determine all pairs (x, y) of positive integers such that x2y+

A, B, C, D are four points in the plane, with C, D on the

O is a point inside a convex quadrilateral ABCD of area

Let A1A2A3A4 be a tetrahedron, G its centroid, and

For every integer d \geq1, let Md be the set of all positive

Let A, B, C be fixed points in the plane. A man starts

Let f(n, r) be the arithmetic mean of the minima of all r-

Let ABC be a triangle with bisectors AA1, BB1, CC1 (A1 \in

A circle O with center O on base BC of an isosceles triangle

Determine the smallest natural number n having the following

Let x, y, and z be positive real numbers such that xyz = 1. Prove

Let n be a positive integer. How many integer solutions

In the coordinate plane a rectangle with vertices (0, 0), (m, 0),

Let m and n be positive integers. The set A = {a1, a2, . . . ,

Find all positive integers … and … for which

Let aij, i = 1, 2, 3, j = 1, 2, 3, be real numbers such that aij

Let O be the circumcenter of an acute-angled triangle ABC

Let D1, . . . , Dn be closed disks in the plane. (A closed disk

Let ABC be a triangle and M an interior point. Prove that

Find the largest number obtainable as the product of pos-

Given five real numbers u0, u1, u2, u3, u4, prove that it is always

Let A be the sum of the digits of the number 1616 and B

Let m positive integers a1, . . . , am be given. Prove that there

(FIN 2‘) Let E be a finite set of points such that E is not contained in

Let r \geq2 be a fixed positive integer, and let F be an infinite

Consider the set of all strictly decreasing sequences of n natural

(POL 1b) Let I = (0, 1] be the unit interval of the real line. For a given

Is there a 1990-gon with the following properties (i) and

Let … be a convex hexagon such that …, …, and …. Let …, …, … be the circumradii of triangles …, …, … respectively, and…

Let n be an integer, n \geq3. Let x1, x2, . . . , xn be real numbers

Prove that for all positive real numbers a, b, c,

Find the number of all …-digit numbers for which some fixed digit stands only in the …th … place and the last … digits…

Let a and b be two positive integers such that ab+1 divides