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

The circle inscribed in the triangle A1A2A3 is tangent to

A, B, C, D, E are points on a circle O with radius equal to r.

With the medians of an acute-angled triangle another triangle is

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

Given f(x) \leqx for all real x and

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

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

Let there be given three circles K1, K2, K3 with centers

Given integers a1, . . . , a10, prove that there exists a nonzero

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

Let (an)\infty

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

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

Given 5 points in the plane, no three of which are collinear, prove

Five points lie on the surface of a ball of unit radius. Find the

Let a, b, x, y be positive integers such that a and b have no

Let f be a function from the real numbers to the real numbers

Two concentric circles have radii R and r respectively. Determine

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

A line l is drawn through the intersection point H of the

Suppose tan lpha = p/q, where p and q are integers and q ̸= 0.

Let there be given two sequences of integers fi(1), fi(2), . . .

A sequence of numbers an, n = 1, 2, . . ., is defined as follows:

For each point X of a given polytope, denote by f(X) the sum

Prove the inequality

Prove that the perpendiculars drawn from the midpoints of the

The function ϕ(x, y, z), defined for all triples (x, y, z) of real

Show that the triangle whose angles satisfy the equality

Determine the sixth number after the decimal point in the

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

Find all real numbers \lambda such that the equation

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

Let a0, a1, a2 be determined with a0 = 0, an+1 = 2an + 2n.

Given a ring G in the plane bounded by two concentric circles

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

Prove that the numbers A, B, and C are equal, where we

Let O be the midpoint of the axis of a right circular cylinder.

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

In a triangle ABC, let H be its orthocenter, O its circumcenter,

Two perpendicular chords are drawn through a given interior

Five distinct numbers are drawn successively and at random

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

Find all functions f defined for all x that satisfy the condition

Prove that for any triangle ABC there exists a point P in the

Consider the square ABCD in which a segment is drawn

(a) Given a tetrahedron ABCD and its four altitudes (i.e.,

Find all real values of the parameter a for which the system of

(Variant of GDR 4) Given a nonconstant function f : R+ oR

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

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

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

Let a, b, c, d be positive integers such that ab = cd and a + b =

Let a be a real number such that 0 < a < 1, and let n be a

Given a sphere K, determine the set of all points A that are

Let ABC be an equilateral triangle and \Gamma the semicircle

Let ABC be a triangle. Denote by a, b, and c the lengths of

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

From a square board 11 squares long and 11 squares wide, the

Prove that there are infinitely many positive integers n for

Let p be a prime. For which k can the set {1, 2, . . ., k} be

A rectangle ABCD is given whose sides have lengths 3 and

Suppose that f is a real function defined for 0 \leqx \leq1 having

A pentagon ABCDE inscribed in a circle for which BC < CD

We are given a triangle ABC and three rectangles R1, R2, R3

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

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

Let f : N oN be such that

Let a and b be integers and n a positive integer. Prove that

The perpendicular line issued from the center of the circum-

Given the equation

Let us define u0 = 0, u1 = 1 and for n \geq0, un+2 = aun+1+bun,

Triangle ABC is given for which BC = AC + 1

Let m and n be natural numbers with m > n. Prove that

A hexahedron ABCDE is made of two regular congruent tetra-

A ball K of radius r is touched from the outside by mutually

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

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

Prove that a pyramid A1A2 . . . A2k+1S with equal lateral edges

Let (ai)i\inN be a strictly increasing sequence of positive real

Let Fn be the nth Fibonacci number, defined by F1 = F2 = 1

In the set of 20 elements {1, 2, 3, 4, 5, 6, 7, 8, 9, 0, A, B, C,

Note that 83 −73 = 169 = 132 and 13 = 22 + 32. Prove that

A convex planar polygon M with perimeter l and area S is given.

Consider the expansion

The bisectors of the angles B, C of a triangle ABC intersect

If p and q are distinct prime numbers, then there are integers

Let P1(x, y) and P2(x, y) be two relatively prime polynomials

In a school six different courses are taught: mathematics,

Prove that the volume V and the lateral area S of a right circular

The equation x3 + ax2 + bx + c = 0 has three (not necessarily

Three of the roots of the equation x4 −px3 + qx2 −rx + s = 0

Let S be a k-element set.

Prove that if x, y, z are real numbers such that x2+y2+z2 = 2,

Evaluate

Let z be an integer > 1 and let M be the set of all numbers

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

In making Euclidean constructions in geometry it is permit-

Let a1, a2, . . . , an (n \geq2) be a sequence of integers. Show that

Find the sum of the fiftieth powers of all sides and diagonals of