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Consider the set A = {0, 1, 2, . . ., 9} and let (B1, B2, . . . , Bk)

Prove that a triangle with angles \alpha, \beta, \gamma, circumradius R, and

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

(a) Prove that (a1 +a2 +\cdot \cdot \cdot+ak)2 \leqk(a2

One Martian, one Venusian, and one Human reside on Pluton.

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

Let a, b, c be positive real numbers, 0 < a \leqb \leqc. Prove that

Two nonzero integers x, y (not necessarily positive) are such

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

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

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

Let f : [0, 1] o[0, 1] satisfy f(0) = 0, f(1) = 1 and

A set of n standard dice are shaken and randomly placed in a

Prove that

Let n points be given arbitrarily in the plane, no three of

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

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

Which regular polygons can be obtained (and how) by cutting

Every x, 0 \leqx \leq1, admits a unique representation x =

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

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

Let the polynomials

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

Let O be a point on a nondegenerate conic. A right angle with

Evaluate sec′′ \pi

Let Ka, Kb, Kc with centers Oa, Ob, Oc be the excircles of a

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

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

Prove that the equation

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

Find the largest positive real number p (if it exists) such that

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

Given a finite number of angular regions A1, . . . , Ak in a plane,

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

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

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

Let A1A2A3A4 be a quadrilateral inscribed in a circle C. Show

The integers 1 through 1000 are located on the circumference

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

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

A sequence (un) of integers is defined for n \geq0 by u0 = 0,

Solve the system of simultaneous equations

Find, with argument, the integer solutions of the equation

Given a segment AB of the length 1, define the set M of points

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

The ternary expansion x = 0.10101010 . . . is given. Give the

A is a 2m-digit positive integer each of whose digits is 1. B is

An accurate 12-hour analog clock has an hour hand, a minute

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

Let n and p be two integers such that 2p \leqn. Prove the

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

We are given a finite collection of segments in the plane, of

A polygon (not necessarily convex) with vertices in the lattice

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

Show that for every natural number n, n

Let X be an arbitrary nonempty set contained in the plane and

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

Find all real numbers \lambda such that the equation

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

Prove that in a Euclidean plane there are infinitely many

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

Let ABC be an equilateral triangle and \Gamma the semicircle

A square hole of depth h whose base is of length a is given.

Let n be an integer that is not divisible by any square greater

Let C1 and C2 be circles in the same plane, P1 and P2 arbitrary

If 0 < k \leq1 and ai are positive real numbers, i = 1, 2, . . . , n,

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

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

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

Let E be the set of all triangles whose only points with integer

For any positive integer n we denote by F(n) the number of

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

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

By \omega(n), where n is an integer greater than 1, let us denote

Solve the equation

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

Given n real numbers 0 < t1 \leqt2 \leq\cdot \cdot \cdot \leqtn < 1, prove that

Let ABC be a triangle with interior angle bisectors AA1,

Show that for nonnegative real numbers a, b and integers n \geq2,

Prove the trigonometric inequality cos x < 1 −x2

Let p \geq2 be a natural number. Prove that there exists an

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

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

An infinite set of rectangles in the Cartesian coordinate

Consider 37 distinct points in space, all with integer coordi-

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

Two persons, X and Y , play with a die. X wins a game if the

A variable tetrahedron ABCD has the following properties:

Prove that for every positive integer n coprime to 10 there

We are given n (n \geq5) circles in a plane. Suppose that every

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

Let Zm,n be the set of all ordered pairs (i, j) with i \in

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

The sequence (an) of real numbers is defined as follows:

Let m be a positive integer and define f(m) to be the number

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

Several segments, which we shall call white, are given, and

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

Prove that if a diagonal is drawn in a quadrilateral inscribed

In a Cartesian coordinate system, the circle C1 has center