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For each P inside the triangle ABC, let A(P), B(P), and

In riangleABC with ngleC = 60o, prove that c

The opposite sides of the reentrant hexagon AFBDCE in-

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

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

Prove that for every positive integer n coprime to 10 there

A fair coin is tossed repeatedly until there is a run of an odd

Let ABC and DEF be acute-angled triangles. Write d = EF,

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

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

Triangle ABC is given for which BC = AC + 1

Given that n elements a1, a2, . . . , an are organized into n pairs

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

Numbers un,k (1 \leqk \leqn) are defined as follows:

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

Find all plane triangles whose sides have integer length and

Prove that the perpendiculars drawn from the midpoints of the

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

Call a four-digit number (xyzt)B in the number system with

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

Let x = p, y = q, z = r, w = s be the unique solution of the

For any positive integer n consider all representations n =

Let M, N, P be the midpoints of the sides BC, CA, AB of a

In a group of interpreters each one speaks one or several foreign

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

Given a positive integer n, find the greatest integer p with the

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

Let O be an interior point of a tetrahedron A1A2A3A4. Let

Prove that there exist infinitely many natural numbers a

Let {un} be the Fibonacci sequence, i.e., u0 = 0, u1 = 1,

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

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

Assuming that the roots of x3+px2+qx+r = 0 are all real and

Let n > 1 be a natural number, a \geq1 a real number, and

Six points P1, . . . , P6 are given in 3-dimensional space such that

A set G with elements u, v, w, . . . is a group if the following

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

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

A polynomial P(x) has degree at most 2k, where k = 0, 1,

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

Find a function f(x) defined for all real values of x such that

Let n be a positive integer. Prove that the number of ways

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

Let n = 2k −1, where k \geq6 is an integer. Let T be the set

A set M is formed of

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

Let x1, x2, x3, x4, x5, x6 be given integers, not divisible by 7.

Find the positions of three points A, B, C on the boundary of

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

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

Suppose medians ma and mb of a triangle are orthogonal.

A circle k = (S, r) is given and a hexagon AA′BB′CC′ inscribed

Prove that 1

Let arphi(n, m), m ̸= 1, be the number of positive integers less

Simplify

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

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

Suppose that positive real numbers x1, x2, x3 satisfy

Prove that for any triangle with sides a, b, c and area P the

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

(a) What is the maximal number of acute angles in a convex

Find the maximum value of

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

Prove that if P(x) = (x−a)kQ(x), where k is a positive integer,

Suppose that points X, Y, Z are located on sides BC, CA,

Find all real solutions of the system of equations

Let A1, A2, . . . , An+1 be positive integers such that (Ai, An+1)

Find the conditions on the positive real number a such that

Find the integer solutions of the equation

Let A and B be points on the circle \gamma. A point C, different

Prove that for every triangle the following inequality holds:

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

For each pair of positive integers k and n, let Sk(n) be the

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

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

Let S be the point of intersection of the two lines l1 : 7x−5y +

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

Let A, B, C be points on the sides B1C1, C1A1, A1B1 of a

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

Let a1, a2, a3, b1, b2, b3 be positive real numbers. Prove that

We are given n mass points of equal mass in space. We define

The points S(i, j) with integer Cartesian coordinates 0 < i \leqn,

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

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

Given the expression

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

Poldavia is a strange kingdom. Its currency unit is the bourbaki

Let S be a subset of the real numbers with the following

Consider the square ABCD in which a segment is drawn

Let a and b be two positive real numbers. If x is a real solution

Let Sn = {1, 2, . . ., n} and fn : Sn oSn be defined inductively

Let a and b be two natural numbers that have an equal number

It is proposed to partition the set of positive integers into two

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

Suppose that 1985 points are given inside a unit cube. Show

Through a point O on the diagonal BD of a parallelogram

A system of n numbers x1, x2, . . . , xn is given such that

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

To each pair (x, y) of distinct elements of a finite set X a number

Prove that