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Let P(x) be a polynomial with integer coefficients such that

Find all plane triangles whose sides have integer length and

Let a real number \lambda > 1 be given and a sequence (nk) of positive

Show that the sequence {an}n\geq1 defined by an = [n

Suppose that positive real numbers x1, x2, x3 satisfy

Let four points Ai (i = 1, 2, 3, 4) in the plane determine four

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

Let a1, a2, a3, b1, b2, b3, c1, c2, c3 be nine strictly positive real

Determine all positive roots of the equation xx = 1/

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

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

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

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

Prove that the system of equations

The four circumcircles of the four faces of a tetrahedron have

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

One hundred red points and one hundred blue points are

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

A convex quadrilateral ABCD with sides AB = a, BC = b,

The segment AB perpendicularly bisects CD at X. Show that,

Let a, b be coprime integers. Show that the equation ax2 +

Given a polynomial

Show that if n runs through all positive integers, f(n) =

Let S be the unit circle with center O and let P1, P2, . . . , Pn

Given positive integers k, m, n with km \leqn and nonnegative

Without using any tables, find the exact value of the product

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

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

One hundred convex polygons are placed on a square with edge

In a room there are nine men. Among every three of them there

The Fibonacci sequence is defined by

Prove that for all X > 1 there exists a triangle whose sides

For a point O inside a triangle ABC, denote by A1, B1, C1

Numbers d(n, m), with m, n integers, 0 \leqm \leqn, ae defined

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

We wish to construct a matrix with 19 rows and 86 columns,

Determine whether there exist 100 distinct lines in the plane

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

Solve the set of simultaneous equations

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

Let ABCD be a tetrahedron and O its incenter, and let the

Find values of the parameter u for which the expression

Let ABCD be a regular tetrahedron. To an arbitrary point

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

Which natural numbers can be expressed as the difference of

Let {An | n = 1, 2, . . .} be a set of points in the plane such

Prove the inequalities

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

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

Several segments, which we shall call white, are given, and

Find the number of five-digit numbers with the following

An alphabet consists of n letters. What is the maximal length

In the plane a point O and a sequence of points P1, P2, P3, . . .

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

A square hole of depth h whose base is of length a is given.

Let f1 = (a1, a2, . . . , an), n > 2, be a sequence of integers.

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

The function F is a one-to-one transformation of the plane into

In the Martian language every finite sequence of letters of

Consider a cube C and two planes \sigma, \tau, which divide Euclidean

Let E be the set of all bijective mappings from R to R satisfying

The equation

A circle K with radius r, a point D on K, and a convex

Construct a triangle given the three exradii.

Prove that in any parallelepiped the sum of the lengths of the

The ternary expansion x = 0.10101010 . . . is given. Give the

Let squares be constructed on the sides BC, CA, AB of a trian-

Let m be a positive integer and define f(m) to be the number

Find a necessary and sufficient condition on the natural num-

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

Let O be the center of a circle. Let OU, OV be perpendicular

Find, with proof, all solutions of the equation 1

If p is a prime number greater than 2 and a, b, c integers not

The expressions a + b + c, ab + ac + bc, and abc are called the

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

Consider a polynomial P(x) = ax2 + bx + c with a > 0 that

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

Let a, 0 < a < 1, be a real number and f a continuous function

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

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

If n is a natural number, prove that

Integers a1, a2, . . . , an satisfy |ak| = 1 and

The real numbers lpha1, lpha2, lpha3, . . . , lphan are positive. Let us denote

Prove the inequality

Given a circle, construct a chord that is trisected by two given

There are n \geq2 people in a room. Prove that there exist two

Let Q be a square with side length 6. Find the smallest integer

Prove that 2147 −1 is divisible by 343.

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

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

Let there be given a circle with center S and radius 1 in the plane,

We match sets M of points in the coordinate plane to sets M∗

Given a set of 1988 points in the plane, no three points of the

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

Given any triangle ABC and any positive integer n, we say

You are given an algebraic system admitting addition and

Let ABCD be a convex quadrilateral whose diagonals intersect

Let g(x) be a fixed polynomial and define f(x) by f(x) =

The square ABCD is to be decomposed into n triangles

Let ABC be a triangle with angles \alpha, \beta, \gamma commensurable with