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Let a, b be natural numbers with 1 \leqa \leqb, and M =

Find a set A of positive integers such that for any infinite

Let P(z) and Q(z) be complex-variable polynomials, with degree

Let n \geq4 be a fixed positive integer. Given a set S =

Let … be real numbers such that

Find all positive integers n such that

For n an odd positive integer, the unit squares of an n imes n

Suppose that every integer has been given one of the colors

For which positive integers n does there exist a positive

Let … denote the set of nonnegative integers. Find all functions … such that

Several equal spherical planets are given in outer space. On the

Given a set M of 1985 positive integers, none of which

From a bag containing 5 pairs of socks, each pair a different

Let a, b, c > 0 and ab + bc + ca = 1. Prove the inequality

Let …, …, … be real numbers. Prove that the system of equations

Let A, B, and C be three points on the edge of a circular

Solve the system of equations

Let a, b, c be integers and p an odd prime number. Prove that

Let n be a positive integer that is not a perfect cube. Define

Find all pairs of positive integers (x, p) such that p is

Three distinct points A, B, C are fixed on a line in this order.

A sequence (an) is defined by means of the recursion

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

Let P be a polynomial of degree n satisfying

Forty-nine students solve a set of three problems. The score for

Find all bases of logarithms in which a real positive number

Let V be a finite subset of Euclidean space consisting of

For a given triangle ABC, let X be a variable point on

The positive integers … and … are such that the numbers

A5 (NET 2)IMO5 Let A1A2A3A4A5A6 be a regular hexagon. Each of its

Two identically oriented equilateral triangles, ABC with center

The quadrilateral ABCD has the following properties:

Let a, b, c be positive numbers with \sqrta+

The prolongation of the bisector AL (L \inBC) in the acute-

S4 (NZL) Suppose that x1, x2, x3, . . . are positive real numbers for which

In acute triangle ABC with circumcenter O and altitude

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

Let n \geq3 be an integer and t1, t2, . . . , tn positive real

Given a point … and lengths …, prove that there exists an equilateral triangle … for which …, …, …, if and only if …,…

The point M inside the convex quadrilateral ABCD is such

I 4 (USS 4) The sum of the squares of five real numbers a1, a2, a3, a4, a5

Find all positive integers x and y such that x+y2+z3 = xyz,

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

Let n \geq2 be a natural number. Find a way to assign nat-

Let n, k be positive integers such that n is not divisible by

In the plane are given a point … and a polygon … (not necessarily convex). Let … denote the perimeter of …, … the sum…

Given an integer n > 1, denote by Pn the product of all

Show that for any vectors a, b in Euclidean space,

A set of three nonnegative integers {x, y, z} with x < y < z

The expression contains the denominator

We seek all three-digit integers whose quotient upon division by $11$ equals the sum of the squares of their digits.

Define

We seek all real numbers $x$ satisfying

The previous construction failed because adding a small geometric perturbation to $a_k$ cannot repair arbitrarily bad ratios.

The soldier moves inside an equilateral triangle $ABC$ of side length $a$.

Let $G$ be a set of non-constant affine functions of the real variable $x$ of the form

Let M be a nonempty subset of Z+ such that for every element

Let a, b, c denote the lengths of the sides BC, CA, AB, respec-

Let (an)n\geq1 and (bn)n\geq1 be two sequences of natural numbers

Let A, B, C, D be points in space. If for every point M on the

(a) The polynomial x2k + 1 + (x+ 1)2k is not divisible by x2 +x+ 1. Find

From point P on arc BC of the circumcircle about triangle

A triangle ABC with ngleA = 30◦and ngleC = 54◦is given. On

Prove that the center of the sphere circumscribed around a

In a triangle ABC, let H be its orthocenter, O its circumcenter,

Consider h + 1 chessboards. Number the squares of each board

In a multiple choice test there were 4 questions and 3 possible

Let ABC be a triangle. Prove that there is a unique point U

If the inradius of a triangle is half of its circumradius, prove

Five points in the plane are given, no three of which are collinear.

M = (ai,j), i, j = 1, 2, 3, 4, is a square matrix of order four.

Show that if 994 integers are chosen from 1, 2, . . . , 1992 and

The sum of all the face angles about all of the vertices except

To every natural number k, k \geq2, there corresponds a sequence

Let f be a real-valued function defined on I = (0, +\infty) and

Let p(x, y) and q(x, y) be polynomials in two variables such

Let f1(x) = x3 +a1x2 +b1x+c1 = 0 be an equation with three

Find all the functions f : R+ oR satisfying the identity

We are given three equal rectangles with the same center in

Let f1 = (a1, a2, . . . , an), n > 2, be a sequence of integers.

A convex quadrilateral is inscribed in a circle of radius 1. Prove

Two circles touch each other from inside, and an equilateral

Integers a1, a2, . . . , an satisfy |ak| = 1 and

Determine all continuous functions f such that

A park has the shape of a convex pentagon of area 5

Find the integer solutions of the equation

Let E be the set of all bijective mappings from R to R satisfying

Let A, B, C, D be four arbitrary distinct points in space.

Let Ax, By be two noncoplanar rays with AB as a common per-

Let ABCD be a tetrahedron such that AB \perpCD,

In a plane, three pairwise intersecting circles C1, C2, C3 with

Let M be a set of points in a plane with at least two elements.

Find x such that

Numbers 1, 2, . . . , 16 are written in a 4 imes4 square matrix so that

Let p be a prime odd number. Is it possible to find p−1 natural

The n points P1, P2, . . . , Pn are placed inside or on the bound-

The lengths of the sides of a rectangle are given to be odd

Two mirror walls are placed to form an angle of measure lpha. There

Let ABC be an isosceles triangle, AB = AC, ngleA = 20◦. Let

Let p be a prime number. A rational number x, with 0 < x < 1,