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Let P be a convex planar polygon with equal angles. Let

Let the numbers 1, 2, . . . , n2 be written in the cells of an n imes n

Given a sequence (an), with a1 = 4 and an+1 = a2

Let n > 1 be a fixed integer. Define functions f0(x) = 0,

Find all positive numbers p for which the equation x2+px+3p = 0

Given two segments AB and CD not in the same plane, find

A curve determined by

Suppose medians ma and mb of a triangle are orthogonal.

Let m and n be natural numbers with m > n. Prove that

Suppose that 1985 points are given inside a unit cube. Show

Show that the triangle whose angles satisfy the equality

Determine all positive integers n for which there exists a poly-

Five points lie on the surface of a ball of unit radius. Find the

The satellites A and B circle the Earth in the equatorial plane

Suppose that f is a real function defined for 0 \leqx \leq1 having

Let S be a subset of the real numbers with the following

We are given twelve coins, one of which is a fake with a different

Prove that there are infinitely many pairs (k, N) of positive

If lpha is the real root of the equation

Let CD be a diameter of circle K. Let AB be a chord that is

Let f : R oR be a continuous function. Suppose that the

There are a board with 2n\cdot2n (= 4n2) squares and 4n2−1 cards

Consider the regular 1987-gon A1A2 . . . A1987 with center O.

Two persons, X and Y , play with a die. X wins a game if the

Prove that there are infinitely many positive integers n for

Given two sequences of positive numbers {ak} and {bk} (k \inN)

From a point P exterior to a circle K, two rays are drawn

Three of the roots of the equation x4 −px3 + qx2 −rx + s = 0

Under the conditions x1, x2 > 0, x1y1 > z2

A rectangular pool table has a hole at each of three of its

(1) Start with a white balls and b black balls.

Let a, b, c be integers different from zero. It is known that the

Let a and b be two natural numbers that have an equal number

If

If in a convex quadrilateral ABCD, E and F are the midpoints

Consider infinite diagrams

Prove that for any triangle with sides a, b, c and area P the

Let E be the set of all triangles whose only points with integer

Let a and b be two positive real numbers. If x is a real solution

Find all numbers N = a1a2 . . . an for which 9 imes a1a2 . . . an =

Alice has two urns. Each urn contains four balls and on each

Father has left to his children several identical gold coins.

A is a 2m-digit positive integer each of whose digits is 1. B is

The integers 1 through 1000 are located on the circumference

Let x1, x2, x3, x4, and x5 be positive integers satisfying

Let n be a natural number, n \geq2, and let \varphi be Euler’s function;

Find values of the parameter u for which the expression

Find values of n \inN for which the fraction 3n−2

Solve the set of simultaneous equations

Prove that the equation

Two moving bodies M1, M2 are displaced uniformly on two

For a real number x, let [x] denote the greatest integer not

Let c, s be real functions defined on R\{0} that are nonconstant

(a) Prove that (a1 +a2 +\cdot \cdot \cdot+ak)2 \leqk(a2

Let x1, x2, . . . , xn (n \geq1) be real numbers such that 0 \leqxj \leq\pi,

Let arphi(n, m), m ̸= 1, be the number of positive integers less

Given that x1+x2+x3 = y1+y2+y3 = x1y1+x2y2+x3y3 = 0,

Prove the trigonometric inequality cos x < 1 −x2

Is it possible to cover a rectangle of dimensions m imes n with

In a plane a finite number of equal circles are given. These circles

Let two glasses, numbered 1 and 2, contain an equal quantity

Which regular polygons can be obtained (and how) by cutting

Every x, 0 \leqx \leq1, admits a unique representation x =

The equation x3 + ax2 + bx + c = 0 has three (not necessarily

Let a and b be integers and n a positive integer. Prove that

Find the largest positive real number p (if it exists) such that

Prove that a triangle with angles \alpha, \beta, \gamma, circumradius R, and

Let k be a positive integer. Define u0 = 0, u1 = 1, and

Let P(x) be a polynomial such that the following inequalities

Let l, l′ be two lines in 3-space and let A, B, C be three points

For 
hi : N oZ let us define M
hi = {f : N oZ; f(x) >

We consider a game on an infinite chessboard similar to that of

A subset S of the set of integers 0, . . . , 99 is said to have

A collection of 2n letters contains 2 each of n different letters.

Let n be a positive integer. Prove that the number of ways

Suppose \alphai > 0, \betai > 0 for 1 \leqi \leqn (n > 1) and that

Let n points be given arbitrarily in the plane, no three of

A parabola P1 with equation x2 −2py = 0 and parabola P2

Let Ka, Kb, Kc with centers Oa, Ob, Oc be the excircles of a

Let p \geq2 be a natural number. Prove that there exists an

Find all plane triangles whose sides have integer length and

Let Sn = {1, 2, . . ., n} and fn : Sn oSn be defined inductively

Does there exist a 2n-digit number a2na2n−1 . . . a1 (for an

Simplify

In a triangle ABC for which 6(a + b + c)r2 = abc, we consider

Let p be a prime. For which k can the set {1, 2, . . ., k} be

Given two numbers x0 and x1, let lpha and eta be coefficients

Through the circumcenter O of an arbitrary acute-angled trian-

If x is a positive rational number, show that x can be uniquely

Determine the sum of all positive integers whose digits (in base

Let {un} be the sequence defined by its first two terms u0, u1

The opposite sides of the reentrant hexagon AFBDCE in-

We are given an n imes n board, where n is an odd number. In

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

A sequence (un) of integers is defined for n \geq0 by u0 = 0,

The sequence an,k, k = 1, 2, 3, . . ., 2n, n = 0, 1, 2, . . ., is defined

Consider the expansion

From a square board 11 squares long and 11 squares wide, the

Let A, B, and C denote the angles of a triangle. If sin2 A +

Let a1, a2, . . . , an be positive numbers, mg = (a1a2 \cdot \cdot \cdot an)1/n