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

Find the average of the quantity

Let c be the inscribed circle of the triangle ABC, d a line tan-

Let I, H, O be the incenter, centroid, and circumcenter of the

The diagonals of a convex 18-gon are colored in 5 different

Let us consider a variable polygon with 2n sides (n \inN) in a

Let M be the circumcenter of a triangle ABC. The line through

The sequence (an) of real numbers is defined as follows:

Let n be an integer that is not divisible by any square greater

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

Determine all pairs of positive integers (x, y) satisfying the

Find necessary and sufficient conditions on given positive num-

Let P be the set of rectangular parallelepipeds that have at

Find the least natural number k such that for any n \in[0, 1]

Let P, Q, R be polynomials with real coefficients, satisfying

Let f be a function defined on the set of pairs of nonzero

Let l denote the length of the smallest diagonal of all rectangles

Connecting the vertices of a regular n-gon we obtain a closed

Let a, b, and c denote the three sides of a billiard table in the

Let n and k be natural numbers and a1, a2, . . . , an positive real

(a) Let ABC be a triangle with AB = 12 and AC = 16. Suppose M is the

Show that tan 7◦30′ =

Let b1, b2, . . . , b1989 be positive real numbers such that the

Which of the numbers 1, 2, . . ., 1983 has the largest number of

Prove that if the equation x4 + ax3 + bx + c = 0 has all its

Given any integer m > 1 prove that there exist infinitely

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

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

Given a regular convex 2m-sided polygon P, show that there is

For a \geq0, b \geq0, c \geq0, d \geq0, prove the inequality

In 3-dimensional space a point O is given and a finite set A

For each nonnegative integer n, Fn(x) is a polynomial in x of

Find all real solutions of the system of equations

Let there be given an acute angle ngleAOB = 3lpha, where OA =

In an urn there are n balls numbered 1, 2, . . . , n. They are

A sequence a1, a2, a3, . . . is defined recursively by a1 = 1 and

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

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

Determine all continuous functions f such that

In how many ways can 1, 2, . . . , 2n be arranged in a 2 imes n

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

Find the set of all a \inR for which there is no infinite sequence

Let M be an interior point of tetrahedron V ABC. Denote

Given a positive integer k, find the least integer nk for which

A square 2n imes 2n grid is given. Let us consider all possible

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

In the plane a circle C of unit radius is given. For any line l

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

The numbers 1, 2, 3, . . ., 64 are placed on a chessboard, one

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

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

Decompose into real factors the expression 1 −sin5 x−cos5 x.

Two straight lines perpendicular to each other meet each side

A be an infinite set of positive integers such that every n \inA is

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

If a1, a2, . . . , an are real constants, and if

Let ABCD be a convex quadrilateral whose diagonals AC and

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

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

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

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

Prove that the number 191976 + 761976:

Let A be an n imesn matrix whose elements are nonnegative real

Consider 37 distinct points in space, all with integer coordi-

Simplify

Four swallows are catching a fly. At first, the swallows are

For given positive integers r, v, n let S(r, v, n) denote the num-

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

If n is even, prove that

Let A, B, C be three points with integer coordinates in the

Let l be tangent to the circle k at B. Let A be a point on k

Let a and b be arbitrary integers. Prove that if k is an integer

If

Find the natural number n with the following properties:

It is given that x = −2272, y = 103 +102c+10b+a, and z = 1

The area of a triangle is S and the sum of the lengths of its

Two cyclists leave simultaneously a point P in a circular run-

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

Consider a sequence of numbers (a1, a2, . . . , a2n). Define the

Find the last eight digits of the binary development of 271986.

Given the side a and the corresponding altitude ha of a triangle

Given two distinct numbers b1 and b2, their product can be

Find all integer solutions of the equation

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

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

All the irreducible positive rational numbers such that the prod-

Let G be the centroid of the triangle OAB.

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

Given n points A1, A2, . . . , An, no three collinear, show that

Let n be a positive integer. Show that (

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

Evaluate sec′′ \pi

In a plane we are given two distinct points A, B and two lines

In the set Sn = {1, 2, . . ., n} a new multiplication a∗b is defined

Given real numbers xi (i = 1, 2, . . . , 4x + 2) such that

Given two points A, B outside of a given plane P, find the

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

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

The sphere inscribed in tetrahedron ABCD touches the sides

Let ABC be an equilateral triangle. Let D, E, F, M, N, and