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Let {an}\infty

Find the maximum value that the quantity 2m + 7n can have

Let S be a circle, and lpha = {A1, . . . , An} a family of open arcs

Let there be 3399 numbers arbitrarily chosen among the first

You are given an algebraic system admitting addition and

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

Let Pn = (19 + 92)(192 + 922) \cdot \cdot \cdot (19n + 92n) for each positive

Let us consider a variable polygon with 2n sides (n \inN) in a

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

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

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

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

Determine all functions f : R oR satisfying the following two

Prove that if the equation x4 + ax3 + bx + c = 0 has all its

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

Let a and b be arbitrary integers. Prove that if k is an integer

Prove that the product of two sides of a triangle is always

The points A, B, C are in this order on line D, and AB = 4BC.

If a1, a2, . . . , an denote the lengths of the sides of an arbitrary

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

A fixed point A inside a circle is given. Consider all chords

In an urn there are one ball marked 1, two balls marked 2, and

A rhombus with its incircle is given. At each vertex of the

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

In a school six different courses are taught: mathematics,

In making Euclidean constructions in geometry it is permit-

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

Three faces of a tetrahedron are right triangles, while the fourth

Let p, q, and r be the angles of a triangle, and let a = sin 2p,

Let n be a natural number. Solve in integers the equation

In a plane, three pairwise intersecting circles C1, C2, C3 with

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

A regular tetrahedron A1B1C1D1 is inscribed in a regular

A be an infinite set of positive integers such that every n \inA is

In an urn there are n balls numbered 1, 2, . . . , n. They are

Let a, b, c denote the lengths of the sides BC, CA, AB, respec-

Given a triangle ABC and a plane \pi having no common points

Find the number of permutations a1, . . . , an of the set

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

Four swallows are catching a fly. At first, the swallows are

Prove that the center of the sphere circumscribed around a

Prove that if a diagonal is drawn in a quadrilateral inscribed

Determine the sixth number after the decimal point in the

In a Cartesian coordinate system, the circle C1 has center

Construct a scalene triangle such that

From each of the vertices of a regular n-gon a car starts to

Denote by an the greatest number that is not divisible by 3

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

A directed graph (any two distinct vertices joined by at most

Let n be an integer greater than 1. In the Cartesian coordinate

If 0 \leqa \leqb \leqc \leqd, prove that

In the plane 4000 points are given such that each line passes

Find the least natural number k such that for any n \in[0, 1]

Consider the square ABCD in which a segment is drawn

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

Let A and B be fixed distinct points on the X axis, none of

A figure of area 1 is cut out from a sheet of paper and divided

Find all pairs of integers (p, q) for which all roots of the trino-

If the inradius of a triangle is half of its circumradius, prove

Find the radius of the circle circumscribed about the isosceles

Prove the following statement: If r1 and r2 are real numbers

Prove that for all x1, x2, . . . , xn \inR the following inequality

A cylindrical container has height 6 cm and radius 4 cm. It

The distance between the centers of the circles k1 and k2 with

We are given n points in space. Some pairs of these points

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

Let ABC be an isosceles triangle, AB = AC, ngleA = 20◦. Let

Does there exist an infinite number of sets C consisting of 1983

Let p(x, y) and q(x, y) be polynomials in two variables such

Let there be given an acute angle ngleAOB = 3lpha, where OA =

If a, b, c, d are integers such that ad is odd and bc is even, prove

Diagonals of a convex quadrilateral ABCD intersect at a

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

Let n + 1 (n \geq1) positive integers be given such that for each

Let m and n denote integers greater than 1, and let u(n) be

If a, b, c, d are real numbers such that a2 + b2 + c2 + d2 \leq1,

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

Let P be a convex 1986-gon in the plane. Let A, D be interior

Which natural numbers can be expressed as the difference of

Let F be the family of all k-element subsets of the set

A function f has the following property: If k > 1, j > 1,

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

Construct a triangle ABC given the side AB and the distance

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

All edges and all diagonals of regular hexagon A1A2A3A4A5A6

The diagonals of a convex 18-gon are colored in 5 different

Prove the existence of a unique sequence {un} (n = 0, 1, 2 . . .)

The solid S is defined as the intersection of the six spheres with

Determine the least possible value of the natural number n

A balance has a left pan, a right pan, and a pointer that moves

Let xn = 22n + 1 and let m be the least common multiple of

Let A and B be positions of two ships M and N, respectively,

On the three distinct lines a, b, and c three points A, B, and

A square ABCD is given. A line passing through A intersects

Find, with proof, the smallest real number C with the following

Let AB be a segment of unit length and let C, D be variable

Consider two quadrilaterals ABCD and A′B′C′D′ in an affine

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

Show that the equation

Let k be one of the integers 2, 3, 4 and let n = 2k −1. Prove