brain

Let AX, BY, CZ be three cevians concurrent at an inte-

Suppose that {x1, x2, . . . , xn} are positive integers for which

Find all natural numbers n for which every natural number

In a triangle ABC, let D and E be the intersections of the bisec-

Let \Gamma1, \Gamma2, \Gamma3, \Gamma4 be distinct circles such that \Gamma1, \Gamma3 are

Around a circular table an even number of persons have a

Suppose that every integer has been given one of the colors

If an acute-angled triangle ABC is given, construct an equilat-

Let D be the interior of the circle C and let A \inC. Show

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

Prove that in the Euclidean plane every regular polygon having

Let n \geq2 be a positive integer and \lambda a positive real

C7 (CZS 3)

Does there exist a finite set M of points in space, not all in

Let ϕ : {1, 2, 3, . . .} … be injective. Prove that

What is the smallest positive integer t such that there exist

Is it possible to find 100 positive integers not exceeding

In the plane, let there be given a circle C, a line l tangent

Find all positive integer solutions x, y, z of the equation 3x +

Let x1 \geqx2 \geq\cdot \cdot \cdot \geqxn and y1 \geqy2 \geq\cdot \cdot \cdot \geqyn be two

An acute triangle ABC is given. Points A1 and A2 are taken

We call a positive integer alternate if its decimal digits

For each finite set U of nonzero vectors in the plane we define

Let S and F be two opposite vertices of a regular octagon.

Let dn be the last nonzero digit of the decimal representation

Let D be an internal point on the side BC of a triangle ABC.

Let Z denote the set of all integers. Prove that for any integers

Consider a sequence of circles K1, K2, K3, K4, . . . of radii

Cards numbered 1 to 9 are arranged at random in a row. In a

1a.(CZS 3) The positive integers x1, . . . , xn, n \geq3, satisfy x1 < x2 <

Let ABCDEFGH be a parallelepiped with AE\parallelBF\parallelCG\parallelDH.

Prove that the product of the radii of three circles exscribed to a given triangle does not exceed … times the product…

S6 (IND) Let N denote the set of all positive integers. Prove that there

Unit cubes are made into beads by drilling a hole through

Given nine points in space, no four of which are coplanar,

We are given a positive integer … and a rectangular board … with dimensions …, …. The rectangle is divided into a grid…

Determine all pairs (a, b) of positive real numbers with a ̸= 1

Let n be a positive integer and let (x1, . . . , xn), (y1, . . . , yn)

Let B be a point on a circle S1, and let A be a point distinct

Consider the polynomial p(x) = xn+nxn−1+a2xn−2 +\cdot \cdot \cdot+an

In the convex pentagon ABCDE, the sides BC, CD, DE have

Let m be a fixed integer greater than 1. The sequence

Let f(x) = x8 + 4x6 + 2x4 + 28x2 + 1. Let p > 3 be a prime

Let … be an equilateral triangle and let … be a point in its interior. Let the lines …, …, … meet the sides …, …, … in…

An international society has its members in 6 different

At a round table are 1994 girls, playing a game with a deck

B2 (POL 4) A convex, closed figure lies inside a given circle. The figure

A function f from the set of positive integers N into itself is

Natural numbers from 1 to 99 (not necessarily distinct) are

Show that if a, b, c are the lengths of the sides of a triangle

2a.(USA 3) Determine the radius of a sphere S that passes through the

Let R+ be the set of all positive real numbers. Find all

Let x1, x2, . . . , xn be arbitrary real numbers. Prove the

Show that in the plane there exists a convex polygon of 1992

For any set S of five points in the plane, no three of which

We are given 100 points in the plane, no three of which are

(a) Show that the set Q+ of all positive rational numbers can be par-

Prove that every positive rational number can be repre-

Find the highest degree k of 1991 for which 1991k divides the

Let triangle ABC be such that its circumradius R is equal to

A lattice point in the plane is a point both of whose coordinates

Let R1, R2, . . . be the family of finite sequences of positive inte-

For what values of n does there exist an n imes n array of entries

A circle S is said to cut a circle \Sigma diametrally if their common

There are n + 1 fixed positions in a row, labeled 0 to n in

Let ABCD be a convex quadrilateral and O the intersection of

The bisectors of angles A, B, C of a triangle ABC meet its cir-

On a circle, 2n −1 (n \geq3) different points are given. Find

Prove the inequality

In a triangle ABC, choose any points K \inBC, L \inAC,

Denote by S the set of all primes p such that the decimal

Find the minimum value of

A semicircle \Gamma is drawn on one side of a straight line l. C

Consider the triangle ABC, its circumcircle k with center O

Let ABCD be a convex quadrilateral with AB not parallel

Prove that from x + y = 1 (x, y \inR) it follows that

Let A, B be adjacent vertices of a regular n-gon in the

Find all pairs of functions f : R oR, g : R oR such that

The set M = {1, 2, . . ., 2n} is partitioned into k nonintersecting

N is an arbitrary point on the bisector of ngleBAC.

Let K denote the set {a, b, c, d, e}. F is a collection of 16 different

On an infinite square grid, two players alternately mark sym-

Let p \geq5 be a prime number. Prove that there exists an

Prove that for every natural number k (k \geq2) there exists an

Let f, g, and a be polynomials with real coefficients, f and g

Given any integer n \geq2, assume that the integers a1, a2, . . . , an

We are given 3n points A1, A2, . . . , A3n in the plane, no three

Find all functions f : R oR satisfying the equation

Prove or disprove: Given a finite set of points with integer

Let n > m \geq1 be natural numbers such that the groups of

Let p and q be integers. Show that there exists an interval I of

Let … be given, and define recursively

3a.(USS 3) Find a method by which one can compute the coefficients

Let S = {1, 2, 3, . . ., 280}. Find the minimal natural num-

An infinite square grid is colored in the chessboard pattern.

A polynomial … with integer coefficients is said to be divisible by an integer … if … is divisible by … for all…

Let P be a polynomial with real coefficients such that P(x) > 0

In acute triangle ABC with circumcenter O and altitude

Find all functions f : R oR such that

A soldier has to investigate whether there are mines in an