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Let k, m, and n be positive integers such that m+k + 1 is

Find the least integer n with the following property: For any

Given a finite sequence of complex numbers c1, c2, . . . , cn, show

Let n be a positive integer, n \geq2, and consider the polynomial

Find the triples of positive integers x, y, z satisfying

In the triangle ABC, let D, E, and F be the midpoints of the

Let ABCD be a rhombus with angle ngleA = 60◦. Let E be a

Let Sn = {1, . . . , n} and let f be a function that maps every

Which fraction p/q, where p, q are positive integers less than

On a one-way street, an unending sequence of cars of width a,

Find the number of positive integers n satisfying arphi(n) | n such

Let O be a point on a nondegenerate conic. A right angle with

Find digits x, y, z such that the equality

Let r > 1 be a real number, and let n be the largest integer

Let ABCD be a regular tetrahedron. To an arbitrary point

Congruent rectangles with sides m (cm) and n (cm) are

Prove that for every positive integer n coprime to 10 there

The circles (R, r) and (P, ho), where r > ho, touch externally

Solve the system of equations

Prove that

Prove that in every convex hexagon of area S one can draw

There are six circles inside a fixed circle, each tangent to

Let (u1, . . . , un) be an ordered ntuple. For each k, 1 \leqk \leqn,

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

The quadrilateral A1A2A3A4 is cyclic and its sides are a1 =

Let d and p be two real numbers. Find the first term of an arith-

Fibonacci numbers are defined as follows: F1 = F2 = 1, Fn+2 =

Find all integer solutions of the equation

We are given a bag of sugar, a two-pan balance, and a weight of

Find the greatest integer A for which in any permutation of

Let E1, E2, and E3 be three mutually intersecting ellipses, all

Let T be the set of all lattice points (i.e., all points with

Let A be a set of positive integers such that no positive integer

Let ABC and A′B′C′ be any two coplanar triangles. Let L be

A regular pentagon A1A2A3A4A5 with side length s is given.

A game consists in pushing a flat stone along a sequence of

Find eight positive integers n1, n2, . . . , n8 with the follow-

We are given a finite collection of segments in the plane, of

For a positive real number p, find all real solutions to the equation

The square ABCD is to be decomposed into n triangles

Let PQ be a line segment of constant length \lambda taken on the

Let M be a finite set and P = {M1, M2, . . . , Mk} a partition

Let v1, v2, . . . , v1989 be a set of coplanar vectors with |vr| \leq1

For positive numbers a, b, c define A = (a + b + c)/3, G =

Solve the system of simultaneous equations

A variable tetrahedron ABCD has the following properties:

Let I and J be the centers of the incircle and the excircle in

Find whether among all quadrilaterals whose interiors lie inside

Let S be a unit circle and K a subset of S consisting of several

How many tangents to the curve y = x3 −3x (y = x3 + px)

An accurate 12-hour analog clock has an hour hand, a minute

How many permutations a1, a2, . . . , an of {1, 2, . . ., n} are

Prove that if x, y, z > 1 and 1

Given natural numbers k and n, k \leqn, n \geq3, find the set

Prove that a regular polygon with an odd number of edges

Let N = B1 \cup\cdot \cdot \cdot\cupBq be a partition of the set N of all positive

Find all x for which for all n,

Given a finite number of angular regions A1, . . . , Ak in a plane,

Given a quadrangle of sides a, b, c, d and area S, show that S \leq

Let ABC be an arbitrary scalene triangle. Define \Sigma to be the

Let Q+ denote the set of nonnegative rational numbers. Show

A regular n-gonal truncated pyramid is circumscribed around

Let a, b, x, y be positive integers such that a and b have no

Let M be the set of all functions f with the following proper-

(a) Given a tetrahedron ABCD and its four altitudes (i.e.,

Let f and g be functions from the set A to the same set A.

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

A ball K of radius r is touched from the outside by mutually

Let f(x) be a periodic function of period T > 0 defined over R.

In the set of 20 elements {1, 2, 3, 4, 5, 6, 7, 8, 9, 0, A, B, C,

Let a1, a2, . . . , an be n real numbers such that 0 < a \leqak \leqb

In a company of n persons, each person has no more than d

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

A circle K centered at (0, 0) is given. Prove that for every vector

If a, b, c are side lengths of a triangle, prove that

Let ABC be a triangle. Denote by a, b, and c the lengths of

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

Define the functions f, F : N oN, by

Let a0, a1, a2 be determined with a0 = 0, an+1 = 2an + 2n.

Let a cube of side 1 be given. Prove that there exists a point

Find all cubic polynomials x3 + ax2 + bx + c admitting the

Determine the smallest positive integer m such that 529n +m\cdot

Let a, b, c be positive real numbers, 0 < a \leqb \leqc. Prove that

One country has n cities and every two of them are linked by a

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

Let (an), n = 0, 1, . . ., be a sequence of real numbers such that

We are given a triangle ABC and three rectangles R1, R2, R3

In a triangle, a symmedian is a line through a vertex that is

Let log2

Through a point O on the diagonal BD of a parallelogram

Find all real solutions of the system of equations

For every integer r > 1 find the smallest integer h(r) > 1

The triangle ABC has a right angle at C. The point P is

Outside an arbitrary triangle ABC, triangles ADB and BCE

Find the maximal number of regions into which a sphere can

Let there be given three circles K1, K2, K3 with centers

Four faces of tetrahedron ABCD are congruent triangles whose

Assume that two parallelograms P, P ′ of equal areas have sides

A sequence of real numbers x1, x2, . . . , xn is given such that

Find all solutions (x, y) \inZ2 of the equation