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Find the relations among the angles of the triangle ABC whose

Given an equilateral triangle ABC, let M be an arbitrary point

Let 0 \leqx1 \leqx2 \leq\cdot \cdot \cdot \leqxn \leq1. Prove that for all A \geq1

A set M is formed of

Let a and b be integers. Prove that 2a2−1

For which natural numbers n do there exist n natural numbers

Let N be a point inside the triangle ABC. Through the mid-

Prove that 2147 −1 is divisible by 343.

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

Find all integer solutions of the equation

An infinite set of rectangles in the Cartesian coordinate

Evaluate (cos(\pi/4) + i sin(\pi/4))10 in two different ways and

Three concentric circles with common center O are cut by a

The four circumcircles of the four faces of a tetrahedron have

In a triangle ABC, ngleBAC = 100◦, AB = AC. A point

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

Consider h + 1 chessboards. Number the squares of each board

Given n points in the plane such that no three of them

Let O be a point on the oriented Euclidean plane and (i, j)

Let f(x) = (x −a1)(x −a2) \cdot \cdot \cdot (x −an) −2, where n \geq3

For given positive integers r, v, n let S(r, v, n) denote the num-

For a positive integer n, let 6(n) be the natural number whose

Let n be an integer > 1. In a circular arrangement of n lamps

Prove that the polynomial x4 + \lambdax3 + µx2 + \nux + 1 has no

Let r1, . . . , rn be the radii of n spheres. Call S1, S2, . . . , Sn the

Let C be the curve determined by the equation y = x3 in the

A regular octagon P is given whose incircle k has diameter 1.

Prove that for an arbitrary pair of vectors f and g in the

Let a and b be two nonnegative integers. Denote by H(a, b)

Find all numbers lpha for which the equation

Let (an)\infty

An urn contains balls of k different colors; there are ni balls

Find the last two digits of a sum of eighth powers of 100

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

In a plane three points P, Q, R, not on a line, are given. Let

Given a triangle ABC and external points X, Y , and Z such

Suppose that the sides AB and DC of a convex quadrilateral

A finite set of points P in the plane has the following prop-

ABCD is a parallelogram; AB = a, AD = 1, lpha is the size

Two equilateral triangles are inscribed in a circle with radius

Prove that

Suppose that n numbers x1, x2, . . . , xn are chosen randomly

Prove the following assertion: If c1, c2, . . . , cn (n \geq2) are real

Two perpendicular chords are drawn through a given interior

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

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

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

Let K1, . . . , Kn be nonnegative integers. Prove that

The expressions a + b + c, ab + ac + bc, and abc are called the

Find for every value of n a set of numbers p for which the fol-

The bisectors of the exterior angles of a pentagon B1B2B3B4B5

For every sequence (x1, x2, . . . , xn) of the numbers {1, 2, . . ., n}

In riangleABC, the inscribed circle is tangent to side BC at X.

(a) Let ABC be a triangle with AB = 12 and AC = 16. Suppose M is the

Let C be the circumcircle of the square with vertices (0, 0),

Is it possible to choose a set of 100 (or 200) points on the

A sequence (an)\infty

Prove that 1

Prove that for every triangle the following inequality holds:

Which of the numbers 1, 2, . . ., 1983 has the largest number of

Consider a cube C and two planes \sigma, \tau, which divide Euclidean

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

Prove that for all X > 1 there exists a triangle whose sides

M = (ai,j), i, j = 1, 2, 3, 4, is a square matrix of order four.

In a plane we are given two distinct points A, B and two lines

We are given n > 3 points in the plane, no three of which lie on

Given the vertex A and the centroid M of a triangle ABC,

A family of sets A1, A2, . . . , An has the following properties:

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

Let a, b, c be integers. Prove that there are integers p1, q1, r1,

Let ABCD and A′B′C′D′ be two squares in the same plane and

In a school, n children numbered 1 to n are initially arranged in

There are 1979 equilateral triangles: T1, T2, . . . , T1979. A side of

Determine whether there exist 100 distinct lines in the plane

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

Let n > 1 and xi \inR for i = 1, . . . , n. Set Sk = xk

Given a polynomial

Find the natural number n with the following properties:

Let C1, C2 be circles of radius 1/2 tangent to each other and

A circle \gamma is drawn and let AB be a diameter. The point C

The Fibonacci sequence is defined by

Let a real number \lambda > 1 be given and a sequence (nk) of positive

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

A line parallel to the side BC of a triangle ABC meets AB

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

Prove the inequality

Let A and B be two finite disjoint sets of points in the plane

Determine whether there exist 1976 nonsimilar triangles with

An Egyptian number is a positive integer that can be expressed

Let ABCD be a convex quadrilateral. DA and CB meet at

A “large” circular disk is attached to a vertical wall. It rotates

Let ABC be a nonequilateral triangle. Prove that there exist

The sphere inscribed in tetrahedron ABCD touches the sides

The segment AB perpendicularly bisects CD at X. Show that,

Let n numbers x1, x2, . . . , xn be chosen in such a way that

Let ABCDS be a pyramid with four faces and with ABCD

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

(a) Prove that if 0 \leqa0 \leqa1 \leqa2, then

Prove the inequalities

For n \inN, let f(n) be the number of positive integers k \leqn