brain

Let ABCD be a rhombus with angle ngleA = 60◦. Let E be a

Let x1, x2, . . . , xn (n \geq1) be real numbers such that 0 \leqxj \leq\pi,

Father has left to his children several identical gold coins.

Given a polynomial

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

On the sides AB and AC of triangle ABC two points K and

Suppose that the sides AB and DC of a convex quadrilateral

Let A be a set of polynomials with real coefficients and let

Determine all functions f : R oR satisfying the following two

Determine all continuous functions f such that

Prove that there exist 78 lines in the plane such that they have

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

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

Decide whether it is possible to color the 1984 natural numbers

Let g(x) be a fixed polynomial and define f(x) by f(x) =

In a triangle ABC, the incircle touches the sides BC, CA, AB

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

Prove that the two last digits of 999 and 9999

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

Given a regular convex 2m-sided polygon P, show that there is

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

Let I and J be the centers of the incircle and the excircle in

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

Let P be the set of rectangular parallelepipeds that have at

We are given twelve coins, one of which is a fake with a different

If in a convex quadrilateral ABCD, E and F are the midpoints

Prove that

Let p be a prime odd number. Is it possible to find p−1 natural

The satellites A and B circle the Earth in the equatorial plane

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

The circles c1 and c2 are tangent at the point A. A straight

For each \lambda (0 < \lambda < 1 and \lambda ̸= 1/n for all n = 1, 2, 3, . . .)

Let F be the correspondence associating with every point P =

Let (an)\infty

Prove that for any natural number n, the number

Show that if 994 integers are chosen from 1, 2, . . . , 1992 and

Determine all positive roots of the equation xx = 1/

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

Let c, s be real functions defined on R\{0} that are nonconstant

Let four points Ai (i = 1, 2, 3, 4) in the plane determine four

Given the side a and the corresponding altitude ha of a triangle

If A1A2 . . . An is a regular n-gon (n \geq3), how many different

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

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

Let a, b, and c denote the three sides of a billiard table in the

To every natural number k, k \geq2, there corresponds a sequence

If n is a natural number, prove that

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

Let [x] denote the greatest integer less than or equal to x. Let lpha

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

Does there exist a number lpha (0 < lpha < 1) such that there is an

A town has a road network that consists entirely of one-way

Given a triangle ABC, let R be the radius of its circumcir-

Let ABC be an equilateral triangle and \Gamma the semicircle

Let ABCDS be a pyramid with four faces and with ABCD

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

Let n and z be integers greater than 1 and (n, z) = 1. Prove:

A sequence (an)N

A two-person game is played with nine boxes arranged in a

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

In the system of base n2 + 1 find a number N with n different

Construct the circle that is tangent to three given circles.

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

(a) Solve the equation

Consider the set Q2 of points in R2, both of whose coordinates

Let ABC be a triangle with interior angle bisectors AA1,

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

Given the equation

Find a necessary and sufficient condition on the natural num-

Four faces of tetrahedron ABCD are congruent triangles whose

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

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

For two given triangles A1A2A3 and B1B2B3 with areas ∆A

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

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

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

We wish to construct a matrix with 19 rows and 86 columns,

Given an isosceles triangle ABC with a right angle at C,

Let M be the set of the lengths of an octahedron whose sides

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

The altitude from a vertex of a given tetrahedron intersects

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

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

Thirty-four countries participated in a jury session of the IMO,

Find all integer solutions of the equation

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

An infinite set of rectangles in the Cartesian coordinate

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

In a plane a finite number of equal circles are given. These circles

Let M be a set of points in a plane with at least two elements.

Two families of parallel lines are given in the plane, consisting

A collection of 2n letters contains 2 each of n different letters.

Let Ak (1 \leqk \leqh) be n-element sets such that each two

Assume that two parallelograms P, P ′ of equal areas have sides

Through the circumcenter O of an arbitrary acute-angled trian-

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

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

Show that there do not exist more than 27 half-lines (or rays)

Let f(x) = ax2 + bx + c and g(x) = cx2 + bx + a. If |f(0)| \leq1,

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