**Problem 1. (AUS 3)**

The integer 9 can be written as a sum of two consecutive integers:
9 = 4 + 5; moreover it can be written as a sum of (more than one)
consecutive positive integers in exactly two ways, namely 9 =
4 + 5 = 2 + 3 + 4. Is there an integer which can be written as
a sum of 1990 consecutive positive integers and which can be
written as a sum of (more than one) consecutive positive integers
in exactly 1990 ways?.

**Problem 2. (CAN 1)**

Given n countries with 3 representatives each, m committees A(1),
A(2), ..., A(m) are called a cycle if:

- each committee has n members, one from each country,
- no two committees have the same membership,
- for i = 1, 2, 3, ..., m, committee A(i) and committee A(i+1) have no member in common where A(m+1) denotes A(1),
- If 1 < |i-j| < m-1, then committees A(i) and A(j) have at least one member in common.

On a circle 2n - 1 (n ³ 3) different points are given. Find the minimum of natural number N with the property that whenever N of the given points are coloured black, there exist two black points such that the interior of one of the corresponding arcs contains exactly n of the given 2n - 1 points.

On a circle 2n - 1 (n ³ 3) different points are given, from which n are coloured black. Prove that one may find two black points such that one of the corresponding arcs contains exactly n of the given 2n - 1 points.

Assume that the set of all positive integers is descomposed into r (disjoint) subsets A

Given the triangle ABC with no side equal to another side, let G, K and H be its centroid, incenter and orthocenter, respectively. Prove that ÐGKH > 90°.

Two players A and B play a game in which they choose numbers alternately according to the following rule.

At the begining, an initial natural number n_{0}
> 0 is given. Having known n_{2k}, the player
A may choose any n_{2k+1} Î
**N**, such that

It is stipulated that, player A wins the game if he (she) succeeds in choose the number 1990, and player B wins if he (she) succeeds in choosing 1.

For which initial n_{0} the player A can manage to win
the game, for which n_{0} player B can manage to win,
and for which n_{0} players A and B would go to a tie?.

**Problem 7. (HEL 2)**

Let f(0) = f(1) = 0 and f(n+2) = 4^{n+2}.f(n+1) -
16^{n+1}.f(n) + n.2^{n2}, n = 0, 1, 2, 3, ... .
Show that the numbers f(1989), f(1990), f(1991) are
divisible by 13.

**Problem 8. (HUN 1)**

For a given positive integer k denote the square of the sum
of its digits by f_{1}(k) and let

The incenter of the triangle ABC is K. The midpoint of AB is C

A plane cuts a right circular cone in two parts. The plane is tangent to the circumference of the base of the cone and passes through the midpoint of the altitude. Find the ratio of the volume of the smaller part to the volume of the whole cone.

Given a circle with two chords AB, CD which meet at E. Let M be a pint of chord AB other than E. Draw the circle through D, E and M. The tangent line to the circle DEM at E meets the lines BC, AC at F, G, respectively. Given (AM/AB) = l, find (GE/EF).

Triangle ABC is acute-angled with circumcircle G and orthocenter H; also AB ¹AC. Let AH meet BC and G at D and E, respectively. The tangent to circle (DEF) at D meets the lines AB, AC at M, L, respectively. Prove that MD = DL.

Let ABC be a triangle and L the line through C parallel to the side AB. Let be the internal bisector of the angle at A meet the side BC at D and the line L at E and let the internal bisector of the angle at B meet the side AC at F and the line L at G. If GF = DE, prove that AC = BC.

An eccentric mathematician has a ladder with n rungs which he always ascends and descends in the following way: when he ascends each step he takes covers b rungs of the ladder, where a and b are fixed positive integers. By a sequence of ascending and descending steps he can climb from ground level to the top rung of the ladder and come back down to ground level again. Find, with proof, the minimum value of n, expressed in terms of a and b.

On the coordinate plane a rectangle with vertices (0,0), (m,0), (0,n), (m,n) is given where both m and n are odd integers.

The rectangle is partitioned into triangles in such a way that:

- each triangle in the partition has at least one side (to be called "good" side) which lies on a line of the form x = j or y = k where j and k are integers and the altitude on this side has length 1,
- each "bad" side (i.e. a side of any triangle in the partition which is not a "good" one) is a common side of two triangles in the partition.

Determine for which positive integers k, the set

Is there a 1990-gon with the following properties (1) and (2)?.

- All angles are equal;
- The lengths of the 1990 sides are a permutation of the
numers 1
^{2}, 2^{2}, ..., 1989^{2}, 1990^{2}.

Unit cubes are made into beads by drilling a hole through them along a diagonal. The beads are put on a string in such a way that they can move freely in space under the restriction that the vertices of two neighbooring cubes are touching. Let A be the begin-vertex and B be the end-vertex. Let there be p x q x r cubes on the string (p, q, r ³ 1).

- (a) Determine for which values of p, q and r it is possible to build a block with dimensions p, q and r. Give reasons for your answer.
- (b) The same question as (a) with the extra condition that A = B.

Let a, b be natural numbers with 1 £ a £ b, and M = (a + b)/2. Define the function f:

f(n) = n - b, if n ³ M

Let P be a point inside a regular tetrahedron T of unit volume. The four planes passing through P and parallel to the faces of T partition T into 14 pieces. Let f (P) be the joint volume of those pieces which are neither a tetrahedron nor a parallelepiped (i.e., pieces adjacent to an edge but not to a vertex). Find the exact bounds for f (P) as P varies over T.

Prove that every integer k (>1) has a positive multiple which is less than k

Let n be a composite natural number and p be a proper divisor of n. Find the binary representation of the smallest natural number N such that ((1 + 2

Ten localities are served by two international airlines such that there exist a direct service (without stops) between any two of these localities and all airline schedules are both ways. Prove that at least one of the airlines can offer two disjoint round trips containing each an odd number of landings.

FInd all positive integers n having the property that (2

Let a,b,c,d be nonnegative real numbers such that ab + bc + cd + da = 1. Show that

Let Q

Let P be a cubic polynomial with rational coefficients, and let q

Find all natural numbers n for which every natural number whose decimal representation has n-1 digits 1 and one digit 7 is prime.

Prove that on a coordinate plane it is impossible to draw a closed broken line such that

- coordinates of each vertex are rational,
- the length of its every edge is equal to 1,
- the line has an odd number of vertices.