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

Solve the equation |x2 −1| + |x2 −4| = mx as a function of the

Let X and Y be two sets of points in the plane and M be a set

A turtle runs away from an UFO with a speed of 0.2 m/s. The

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

Denote by xn(p) the multiplicity of the prime p in the canonical

(a) Solve the equation

The intersection of a plane with a regular tetrahedron with

Let P be the set of rectangular parallelepipeds that have at

We are given a circle K with center S and radius 1 and a square

Given an equilateral triangle ABC of side a in a plane, let

The equation

Find all positive integers x such that the product of all digits

Let ABC, AA1A2, BB1B2, CC1C2 be four equilateral triangles

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

Let ABC be an equilateral triangle and \Gamma the semicircle

For which arrangements of two infinite circular cylinders does

The set {1, 2, . . ., 49} is divided into three subsets. Prove that

Seventeen cities are served by four airlines. It is noted that

Let E be a finite set of points in space such that E is not

On a chessboard (8 imes 8 squares with sides of length 1) two

Let ABCD be a quadrilateral inscribed in a circle with diam-

In the Martian language every finite sequence of letters of

Given a triangle ABC, let R be the radius of its circumcir-

Prove the inequality

Triangle ABC is given for which BC = AC + 1

If x, y, z are real numbers satisfying the relations x+y+z = 1

In a multiple choice test there were 4 questions and 3 possible

In a plane, a circle with center O and radius R and two points

Let p be a prime number greater than 5. Let V be the collection

Show that the set S of natural numbers n for which 3/n

A regular 14-gon with side length a is inscribed in a circle of

Prove that the number 191976 + 761976:

A wheel consists of a fixed circular disk and a mobile circular

A fox stands in the center of the field which has the form of an

Three mutually nonparallel lines li (i = 1, 2, 3) are given

Let AB and CD be two perpendicular chords of a circle with

Consider all the sums of the form

If A1A2 . . . An is a regular n-gon (n \geq3), how many different

Let f(x) = a sin2 x+b sin x+c, where a, b, and c are real num-

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

Determine all continuous functions f such that

Consider the equation x4 + ax3 + bx2 + ax + 1 = 0 with real

Let a, b, c be natural numbers such that a+b+c = 2pq(p30+q30),

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

Solve in the set of real numbers the equation 3x3 −[x] = 3,

Solve the equation

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

Are there integers m and n such that

Let K be a convex set in the xy-plane, symmetric with respect

Let a1, . . . , an be distinct positive integers that do not contain

Let ABCD be a cyclic quadrilateral. Show that the centroids of

Find a natural number n such that for all prime numbers p, n

Given the equation

Let n be a positive integer. Show that (

Consider the set Q2 of points in R2, both of whose coordinates

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

The function f(n) is defined on the nonnegative integers n by:

The colonizers of a spherical planet have decided to build N

The circle k and its diameter AB are given. Find the locus of

A finite number of parallel segments in the plane are given with

We consider the infinite chessboard covering the whole plane.

Does there exist an integer z that can be written in two different

Prove that if for a polynomial P(x, y) we have

Let E be a finite set, PE the family of its subsets, and f a

In space, n points (n \geq3) are given. Every pair of points

Let a1, a2, . . . , an be positive real numbers. Prove the inequality

Let f1(x) = x3 +a1x2 +b1x+c1 = 0 be an equation with three

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

Let Li, i = 1, 2, 3, be line segments on the sides of an equilateral

Evaluate

Let Ax, By be two noncoplanar rays with AB as a common per-

Simplify

We define a binary operation ⋆in the plane as follows: Given

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

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

Let P be a prime number and n a natural number. Prove that

Find the total number of different integers that the function

Points D and E are chosen on the sides AB and AC of the

Prove that in a Euclidean plane there are infinitely many

Let G be the centroid of the triangle OAB.

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

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

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

(a) Prove that for a, b, c, d \inR, m \in[1, +\infty) with am + b =

Let S \subset[0, 1] be a set of 5 points with {0, 1} \subsetS. The graph

A two-person game is played with nine boxes arranged in a

An (n2 +n+1) imes(n2 +n+1) matrix of zeros and ones is given.

If A, B, C, and D are four distinct points in space, prove that

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

The faces of a convex polyhedron are six squares and eight

Let A be a set of positive integers such that for any two elements

Prove that the equation 4x+6x = 9x has no rational solutions.

Show that tan 7◦30′ =

Determine an equation of third degree with integral coefficients

Three disks of diameter d are touching a sphere at their centers.

Is it possible to partition 3-dimensional Euclidean space into

A triangle ABC is given. Each side of ABC is divided into equal

A pack of 2n cards contains n different pairs of cards. Each

How many real solutions are there to the equation x =