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Given a segment AB of the length 1, define the set M of points

Let ABCD be a quadrilateral. Let A′BCD′ be the reflection

Let ABC be an equilateral triangle with side length equal to a

Let n and k be positive integers such that 1 \leqn \leqN + 1,

There are n points on a flat piece of paper, any two of them

(a) Find the number of ways 500 can be represented as a sum of

Let f(x) = xm + a1xm−1 + \cdot \cdot \cdot + am−1x + am and g(x) =

Let d \geq1 be an integer that is not the square of an integer.

Construct a triangle ABC given its side a = BC, its circum-

A desert expedition camps at the border of the desert, and

Into every lateral face of a quadrangular pyramid a circle is

Let f : [0, 1] o[0, 1] satisfy f(0) = 0, f(1) = 1 and

Let A, B, C be angles of a triangle. Prove that

Let a, b, c be positive real numbers and p, q, r complex numbers.

Each of the numbers x1, x2, . . . , xn equals 1 or −1 and

Prove that the system of equations

Let k be a positive integer and Mk the set of all the integers

Find the largest integer not exceeding $1992

The integers 1, 2, . . ., n2 are placed on the fields of an n imes n

On the circle with center O and radius 1 the point A0 is

Given n real numbers 0 < t1 \leqt2 \leq\cdot \cdot \cdot \leqtn < 1, prove that

Let M be a set, and A, B, C given subsets of M. Find a

Let

In the system of base n2 + 1 find a number N with n different

Let \alpha + \beta + \gamma = \pi. Prove that

A sequence of real numbers x0, x1, x2, . . . is defined as follows:

Is it possible to put 100 (or 200) points on a wooden cube such

Solve the system

For each nonzero complex number z, let arg z be the unique

Given m+n numbers ai (i = 1, 2, . . . , m), bj (j = 1, 2, . . ., n),

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

Prove that there is a positive integer n such that the decimal

Let [x] denote the greatest integer less than or equal to x. Let lpha

Let a \inR and let z1, z2, . . . , zn be complex numbers of mod-

For any positive integer n we denote by F(n) the number of

What is the greatest number of balls of radius 1/2 that can be

Prove that if a person a has infinitely many descendants (chil-

Suppose that positive real numbers x1, x2, x3 satisfy

Let E = {1, 2, . . ., 16} and let M be the collection of all

The sequences a0, a1, . . . and b0, b1, . . . are defined by the equal-

Prove that there exist distinct natural numbers m1, m2, . . . ,

A “number triangle” (tnk) (0 \leqk \leqn) is defined by tn,0 =

The decimal number 13101 is given. It is instead written as a

Let P1(x), P2(x), . . . , Pn(x) be polynomials with real coefficients.

By h(n), where n is an integer greater than 1, let us denote the

Let P be a set of n points and S a set of l segments. It is

In a chess tournament there are n \geq5 players, and they have

By \omega(n), where n is an integer greater than 1, let us denote

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

Find all possible finite sequences {n0, n1, n2, . . . , nk} of integers

Solve the equation

Let 2n + 3 points be given in the plane in such a way that

The base of an inclined prism is a triangle ABC. The per-

Prove the inequality

Consider the sequence (cn):

Let g(k) be the number of partitions of a k-element set M, i.e.,

The triangles A0B0C0 and A′B′C′ have all their angles

Let mj > 0 for j = 1, 2, . . ., n and a1 \leq\cdot \cdot \cdot \leqan < b1 \leq\cdot \cdot \cdot \leq

Let (an)n\geq0 and (bn)n\geq0 be two sequences of natural numbers.

Let ABCD be a regular tetrahedron and Z an isometry map-

We are given a circle K and a point P lying on a line g. Construct

The sides a, b, c of a triangle ABC form an arithmetic progression;

Prove that for arbitrary positive numbers the following in-

A set of n standard dice are shaken and randomly placed in a

Find the greatest number c such that for all natural numbers

A plane rectangular grid is given and a “rational point” is

Let Z be a set of points in the plane. Suppose that there exists

In the triangle ABC, let B1 be on AC, E on AB, G on BC,

Let ABC be a triangle, O its circumcenter, S its centroid, and

The points A1, A2, . . . , A1983 are set on the circumference of a

Prove that there exist infinitely many natural numbers a

The positive integer n has the property that in any set of n

Establish the maximum and minimum values that the sum

Prove that the equation in x

Prove the following statement: If a polynomial f(x) with

Let p be a prime number and a1, a2, . . . , a(p+1)/2 different nat-

T is a given triangle with vertices P1, P2, P3. Consider an arbi-

For any angle lpha with 0 < lpha < 180◦, we call a closed convex

We are given n (n \geq5) circles in a plane. Suppose that every

If n is even, prove that

Let f : R oR be of the form f(x) = x + psilon sin x, where

Superchess is played on on a 12 imes 12 board, and it uses su-

Consider the set E consisting of pairs of integers (a, b), with a \geq

Let S be a set of positive integers n1, n2, . . . , n6 and let n(f)

Prove that the sequence 5, 12, 19, 26, 33, . . . contains no term

In riangleABC with ngleC = 60o, prove that c

On the sides AB and AC of triangle ABC two points K and

One Martian, one Venusian, and one Human reside on Pluton.

Let A1, A2, . . . , A29 be 29 different sequences of positive integers.

Prove that there exists a natural number k0 such that for

Let Ak (1 \leqk \leqh) be n-element sets such that each two

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

Let S be an infinite set of integers containing zero and such

An infinite increasing sequence of positive integers nj (j =

In connection with a convex pentagon ABCDE we consider

(a) Consider a circle K with diameter AB, a circle L tangent to AB and

Let (an)n\inN be the sequence of integers defined recursively by

The set X has 1983 members. There exists a family of subsets

Let g(n) be defined as follows:

Let ABC be a triangle with inradius r and circumradius R.