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Prove the following assertion: The four altitudes of a tetrahe-

The diagonals of a quadrilateral ABCD are perpendicular:

B1 (CAN 2)

For any positive integer x0, three sequences {xn}, {yn}, and

Let n > 6 and a1 < a2 < \cdot \cdot \cdot < ak be all natural numbers

Find all functions f from the reals to the reals such that

What is the smallest positive integer t such that there exist

Let a be a rational number with 0 < a < 1 and suppose that

An international society has its members in 6 different

In the triangle ABC, with ∡A = 60◦, a parallel IF to AC

Let F(n) be the set of polynomials P(x) = a0+a1x+\cdot \cdot \cdot+anxn,

Find all functions f : R oR satisfying the equation

S4 (NZL) Suppose that x1, x2, x3, . . . are positive real numbers for which

Let n be a positive integer. How many integer solutions

Every point with integer coordinates in the plane is the

Let … be a point inside … such that

A nonempty set A of real numbers is called a B3-set if the

Let … be the real polynomial function

Let n, k be positive integers such that n is not divisible by

The quadrilateral ABCD has the following properties:

Three persons A, B, C, are playing the following game: A k-

Let A, B, C, and D be distinct points on a line, in that

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

Does there exist a function s: Q … such that if x

Given a nonequilateral triangle ABC, the vertices listed coun-

We call a positive integer alternate if its decimal digits

Let f(x) be a monic polynomial of degree 1991 with integer

An infinite square grid is colored in the chessboard pattern.

Let N = {1, 2, . . ., n}, n \geq2. A collection F = {A1, . . . , At}

The following operation is allowed on a finite graph: Choose

Let ABC be a triangle for which there exists an interior

For a triangle ABC, let k be its circumcircle with radius r. The

Find all positive integer solutions x, y, z of the equation 3x +

Prove:

Define a k-clique to be a set of k people such that every pair

For a positive integer n define a sequence of zeros and ones

A pentagonal prism A1A2 . . . A5B1B2 . . . B5 is given. The

Let c be a positive integer. The sequence {fn} is defined as

Let a0 = 1994 and an+1 =

Let n be a positive integer. A sequence of n positive integers

Let S be a convex quadrilateral ABCD and O a point inside

The circle S has center O, and BC is a diameter of S.

Let f(x) = xn where n is a fixed positive integer and x =

Find all functions f : R oR such that

For points A1, . . . , A5 on the sphere of radius 1, what is the

How many words with n digits can be formed from the alphabet

Let f and g be two integer-valued functions defined on the set

Let Z denote the set of all integers. Prove that for any integers

II 4 (FIN 3)IMO2 Let riangleABC be a triangle. Prove that there exists a

Let ABC be a triangle, Ωits incircle and Ωa, Ωb, Ωc three

Find all the functions f : R oR that satisfy

Let {ak}\infty

Find all natural numbers n for which 28 + 211 + 2n is a perfect

The point M inside the convex quadrilateral ABCD is such

Let p be an odd prime and n a positive integer. In the

Prove that there exists a unique triangle whose side

II 6 (USS 1) In a certain language words are formed using an alphabet

Find all positive integers x and y such that x+y2+z3 = xyz,

Denote by S the set of all primes p such that the decimal

Let ABCDEF be a convex hexagon with AB = BC =

Let m and n be positive integers. The set A = {a1, a2, . . . ,

Let O be the circumcenter of an acute-angled triangle ABC

Prove that on the coordinate plane it is impossible to draw a

Consider on the first quadrant of the trigonometric circle the

Prove that there exists a four-coloring of the set M =

Let k be a positive integer. Prove that there are infinitely

(IND 3′)IMO1 Given a circle with two chords AB, CD that meet at E, let

Let … have orthocenter …, and let … be a point on its circumcircle, distinct from …, …, …. Let … be the foot of the…

There are 2n words of length n over the alphabet {0, 1}. Prove

Let O be an interior point of acute triangle ABC. Let A1

Let … be given, and define recursively

A circle O with center O on base BC of an isosceles triangle

Prove that every positive rational number can be repre-

Let A = (a1, a2, . . . , a2001) be a sequence of positive integers.

Let ABC be a triangle with semiperimeter s and inradius

A biologist watches a chameleon. The chameleon catches

A wobbly number is a positive integer whose digits in base 10

Let R+ be the set of all positive real numbers. Find all

Given … (…) points in space such that every three of them form a triangle with one angle greater than or equal to …,…

Let ABCD be a convex quadrilateral and O the intersection of

Let g : C \toC, w \inC, a \inC, w3 = 1 (w ̸= 1). Show that

Prove that for all positive real numbers a, b, c,

Let P be a cubic polynomial given by P(x) = ax3+bx2+cx+

Find a set A of positive integers such that for any infinite

Prove that the set {1, 2, . . ., 1989} can be expressed as the

Prove that N = 5125−1

The bisectors of angles A, B, C of a triangle ABC meet its cir-

1a.(CZS 3) The positive integers x1, . . . , xn, n \geq3, satisfy x1 < x2 <

Let x1, x2, . . . , xn be real numbers satisfying the conditions

For a given triangle ABC, let X be a variable point on

Let … be a real number and … a real function defined on all of …, satisfying for all …,

Let ABC be an acute-angled triangle with AB ̸= AC.

Let n be an integer, n \geq3. Let x1, x2, . . . , xn be real numbers

For what values of n does there exist an n imes n array of entries

Let m positive integers a1, . . . , am be given. Prove that there

Let lpha(n) be the number of digits equal to one in the binary

Two identically oriented equilateral triangles, ABC with center

Let m and n be nonnegative integers. Prove that m!n!(m+

A4 (BUL 2) Determine all real values of the parameter a for which the

(FIN 2‘) Let E be a finite set of points such that E is not contained in