**Problem 1. (BUL 1)**

An integer sequence is defined by

Prove that 2

**Problem 2. (BUL 3)**

Let n be a positive integer. Find the number of odd
coefficients of the polynomial

The triangle ABC is inscribed in a circle. The interior bisectors of the angles A, B and C meet the circle again at A', B' and C', respectively. Prove that the area of triangle A'B'C' is greater than or equal to the area of triangle ABC.

**Problem 4. (CZE 1)**

An n x n chessboard (n ³ 2) is
numbered by the numbers 1, 2, ..., n^{2} (and
every number occurs). Prove that there exist two neighbouring
(with common edge) squares such that their numbers differ by
at least n.

**Problem 5. (CZE 2)**

Let n be an even positive integer. Let A_{1},
A_{2}, ..., A_{n+1} be sets having n elements
each such that any two of them have exactly one element in
common while every element of their union belongs to at least
two of the given sets. For which n can one assign to every
element of the union one of the numbers 0 and 1 in such a
manner that each of the sets has exactly n/2 zeros?.

**Problem 6. (CZE 3)**

In a given tetrahedron ABCD let K and L be the centres of
edges AB and CD, respectively. Prove that every plane that
contains the line KL divides the tetrahedron into two parts
of equal volume.

**Problem 7. (FRA 2)**

Let a the greatest positive root of the equation x^{3}
- 3x^{2} + 1 = 0. Show that [a^{1788}] and
[a^{1988}] are both divisible by 17. ( [x] denotes the
integer part of x).

**Problem 8. (FRA 3)**

Let u_{1}, u_{2}, ..., u_{m} be m
vectors in the plane, each of length £
1, with zero sum. Show that one can rearrange
u_{1}, u_{2}, ..., u_{m} as a sequence
v_{1}, v_{2}, ..., v_{m} such that each
partial sum v_{1}, v_{1} + v_{2},
v_{1} + v_{2} + v_{3}, ...,
v_{1} + v_{2} + ... + v_{m} has length
less than or equal to Ö5.

**Problem 9. (FRG 1)**

Let *a* and *b* be two positive integers such that ab + 1 divides
a^{2} + b^{2}. Show that (a^{2} +
b^{2})/(ab + 1) is a perfect square.

**Problem 10. (GDR 1)**

Let *N* = {1, 2, ..., n}, n ³ 2.
A collection *F* = {A_{1}, ..., A_{t}}
of subsets A_{i} Í *N*,
i = 1, ..., t, is said to be separating, if for every pair {x,y}
Í *N*, there is a set A_{i}
Î *F* so that A_{i}
Ç {x,y} contains just one element.
*F* is said to be covering, if every element of *N*
is contained in at least one set A_{i}
Î *F*. What is the smallest
value f(n) of t, so that there is a set
*F* = {A_{1}, ..., A_{t}} which is
simultaneously separating and covering.

**Problem 11. (GDR 3)**

The lock on a safe consist of 3 wheels, each of which may be set
in 8 different positions. Due to a defect in the safe mechanism
the door will open if any two of the three wheels are in the
correct position. What is the smallest number of combinations
which must be tried if one is to guarantee being able to open
the safe (assumming the "right combination" is not known)?.

**Problem 12. (GRE 2)**

In a triangle ABC, choose any points K Î
BC, L Î AC, M
Î AB, N Î
LM, R Î MK and F
Î KL. If E_{1}, E_{2},
E_{3}, E_{4}, E_{5}, E_{6} and E
denotes the areas of the triangles AMR, CKR, BKF, ALF, BNM, CLN
and ABC, respectively. Show that

In a right-angled triangle ABC let AD be the altitude drawn to the hypotenuse and let the straight line joining the incentres of the triangles ABD, ACD intersect the sides AB, AC at the points K, L respectively. If E and E

**Problem 14. (HUN 1)**

For what values of n does there exist an n x n array of entries
-1, 0 or 1 such that the 2n sums obtaines by summing the elements
of the rows and the columns are all different?.

**Problem 15. (ICE 1)**

Let ABC be an acute-angled triangle. Three lines L_{A}
, L_{B} and L_{C} are constructed through the
vertices A, B and C, respectively according to the following
description: Let H be the foot of the altitude drawn from the
vertex A to the side BC; let S_{A} be the circle with
diameter AH; let S_{A} meet the sides AB and AC at M
and N respectively, where M and N are distinct from A; then
L_{A} is the line through A perpendicular to MN. The
lines L_{B} and L_{C} are constructed similarly.
Prove that L_{A}
, L_{B} and L_{C} are concurrent.

**Problem 16. (IRE 1)**

Show that the solution set of the inequality

is a union of disjoint intervals, the sum of whose lengths is 1988.

**Problem 17. (ISA 2)**

In the convex pentagon ABCDE, the sides BC, CD, DE are equal.
Moreover each diagonal is parallel to a side (AC is parallel
to DE, BD is parallel to AE, etc.). Prove that ABCDE is a
regular pentagon.

**Problem 18. (LUX 1)**

Consider 2 concentric circles of radii R and r (R > r) with
centre O. Fix P on the small circle and consider the variable
chord PA of the small circle. Points B and C lie on the large
circle; B, P, C are collinear and BC is perpendicular to AP.

- i) For which value(s) of ÐOPA
is the sum BC
^{2}+ CA^{2}+ AB^{2}extremal? - ii) What are the possible positions of the midpoints U of BA and V of AC as ÐOPA varies?

Let f(n) be a function defined on the set of all positive integers and having its values in the same set. Suppose that f ( f(n) + f(m) ) = m + n for all positive integers n,m. Find all possible values for f (1988).

**Problem 20. (MON 4)**

Find the least natural number n such that, if the set
{1, 2, ..., n} is arbitrarily divided into two non-intersecting
subsets, then one of the subsets contains 3 distinct numbers
such that the product of two of them equals the third.

**Problem 21. (POL 4)**

Forty-nine students solve a set of 3 problems. The score for
each problem is a whole number of points from 0 to 7. Prove
that there exist two students A and B such that, for each
problem, A will score at least as many points as B.

**Problem 22. (ROK 2)**

Let p be the product of two consecutive integers greater
than 2. Show that there no integers x_{1},
x_{2}, ..., x_{p} satisfying the equation

or show there are only two values of p for which there are
integers x_{1}, x_{2}, ..., x_{p}
satisfying

where all the sums are taken from i = 1 to i = p.

**Problem 23. (SIN 2)**

Let Q be the centre of the inscribed circle of a
triangle ABC. Prove that for any point P,

**Problem 24. (SWE 2)**

Let
{a_{k}}_{1}^{¥
} be a sequence of non-negative real numbers such that

A positive integer is called a

**Problem 26. (UNK 2)**

A function *f* defined on the positive integers (and
taking positive integer values) is given by:

f (2n) = f (n),

f (4n+1) = 2 f(2n+1) - f(n),

f (4n+3) = 3 f(2n+1) - 2f(n),

**Problem 27. (UNK 4)**

The triangle ABC is acute-angled. L is any line in the plane
of the triangle and u, v, w are the lengths of the perpendiculars
from A, B, C respectively to L. Prove that

where D is the area of the triangle, and determine the lines L for which equality holds.

**Problem 28. (UNK 5)**

The sequence {a_{n}} of integers is defined by

Prove that a_{n} is odd for all n > 1.

**Problem 29. (USA 3)**

A number of signal lights are equally spaced along one-way
railroad track, labeled in order 1, 2, ..., N (N
³ 2). As a safety rule,
a train is not allowed to pass a signal if any
other train is in motion on the length of track between it and the
following signal. However, thers is no limit to the number of
trains that can be parked motionless at a signal, one behind
the other. (Assume the trains have zero length.)

A series of K freight trains must be driven from Signal 1 to
Signal N. Each train travels at a distinct but constant speed at
all times when it is not blocked by the safety rule.
Show that, regardless of the order
in which the trains are arranged, the same time will elapse
between the first train's departure from Signal 1 and the last
train's arrival at Signal N.

**Problem 30. (USS 1)**

A point M is chosen on the side AC of the triangle ABC in such a
way that the radii of the circles inscribed in the triangles
ABM and BMC are equal. Prove that

**Problem 31. (USS 2)**

Around a circular table an even number of persons have a
discussion. After break they sit again around the circular
table in a different order. Prove that there are at least two
people such that the number of participants sitting between them
before and after the break is the same.