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Let ABC be an isosceles triangle with right angle at point A.

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

Given a line p and a triangle rianglein the plane, construct an

Let ABCD be any quadrilateral. A square is constructed on

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

Let AA′, BB′, CC′ be the bisectors of the angles of a triangle

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

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

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

Let p, q and r be three lines in space such that there is no plane

The runs of a decimal number are its increasing or decreasing

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

(a) Show that the set N of all natural numbers can be parti-

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

If n is a natural number, prove that

Prove the following statement: There does not exist a pyramid

A boy at point A wants to get water at a circular lake and

Let A, B, C, D be four arbitrary distinct points in space.

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

Given a graph with n vertices and a positive integer m that is

Let N = {1, 2, . . ., n}, n \geq3. To each pair i, j of elements of N,

Let a regular (2n + 1)-gon be inscribed in a circle of radius r.

A real-valued function f on Q satisfies the following conditions

Given a unit cube, find the locus of the centroids of all tetra-

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

Given a square ABCD, let P, Q, R, and S be four variable

Prove that the perpendiculars drawn from the midpoints of the

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

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

Let ak be positive numbers such that a1 \geq1 and ak+1 −ak \geq1

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

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

Let (an)n\inN be the sequence of integers defined recursively by

Find all (real) solutions of the system

Construct a triangle given the three exradii.

Prove that for a > b2,

Let a, b, c, d be a permutation of the numbers 1, 9, 8, 4 and let

Let n be a natural number not greater than 44. Prove that for

For a finite set E of cardinality n \geq3, let f(n) denote the

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

Consider the binomial coefficients

The polynomial P(x) = a0xk + a1xk−1 + \cdot \cdot \cdot + ak, where

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

Notice that in the fraction 16

Prove that there are infinitely many positive integers that

Decompose the number 51985−1 into a product of three integers,

Find all integers x, y, z that satisfy

Find the conditions on the positive real number a such that

(Alternative to GBR 2) Prove that there exists, for n \geq4, a

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

Find the number of lines dividing a given triangle into two parts

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

Let S be the set of all sequences {ai | 1 \leqi \leq7, ai = 0 or 1}.

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

Let S be a set of n2 + 1 closed intervals (n a positive integer).

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

Let F be the correspondence associating with every point P =

Determine positive integers p, q, and r such that the diagonal

Prove the inequality

A right-angled triangle OAB has its right angle at the point B.

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

The incenter of a triangle is the midpoint of the line seg-

Suppose we have a pack of 2n cards, in the order 1, 2, . . . , 2n. A

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

Let \theta1, \theta2, . . . , \thetan be real numbers such that sin \theta1 + \cdot \cdot \cdot +

Let a, b, c be nonnegative integers such that a \leqb \leqc, 2b ̸=

Given n real numbers a1 \leqa2 \leq\cdot \cdot \cdot \leqan, define

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

Let ABCD be a convex quadrilateral whose diagonals AC and

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

Let A, B, C, D be points in space. If for every point M on the

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

Prove that a convex polyhedron all of whose faces are equilat-

An arithmetic function is a real-valued function whose do-

Consider a finite set of vectors in space {a1, a2, . . . , an} and

For any positive integer n consider all representations n =

Let N = {1, 2, 3, . . .}. For real x, y, set S(x, y) = {s | s =

Let two circles A and B with unequal radii r and R, respec-

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

Evaluate sec′′ \pi

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

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

It is well known that the binomial coefficients

Let P, Q, R be the polynomials with real or complex coefficients

Given that

A hexahedron ABCDE is made of two regular congruent tetra-

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

Find, with proof, all solutions of the equation 1

Let ABCD be a rectangle, AB = a, BC = b. Consider the

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

Let lpha(n) be the number of pairs (x, y) of integers such that

Show that there exists a convex polyhedron with all its vertices

A tetrahedron is inscribed in a sphere of radius 1 such that the

If p is a prime greater than 3, show that at least one of the

Let ABC be an arbitrary triangle and let S1, S2, . . . , S7 be

We consider n real variables xi (1 \leqi \leqn), where n is an

Let b \geq2 be a positive integer.

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

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

Consider the integer d = ab−1