###
16^{th} BALKAN MATHEMATICAL OLYMPIAD
OHRID, MACEDONIA, 1999

### Problem 1

Given an acute triangle *ABC*, let *D* be the midpoint
of the arc *BC* of the circumcircle around the triangle ABC,
not containing the point *A*. The points which are symmetric
to *D* with respect to the line *BC* and the circumcentre *O*
are denoted by *E* and *F*, respectively. Finally, let *K*
be the midpoint of the segment [*EA*]. Prove that:
*a)* The circle passing through the midpoints of the sides of
the triangle *ABC*, also passes through *K*;
*b)* the line passing through *K* and the midpoint of the
segment [*BC*] is perpendicular to the line *AF*.
### Problem 2

Let *p* > 2 be a prime number, such that 3 divides *p* - 2.
Let:

*S* = { *y*^{2} - *x*^{3} - 1 |
*x* and *y* are integers, 0 =< *x*,
*y* =< *p* - 1}.
Prove that at most *p*-1 elements of the set *S* are divisible
by *p*.

### Problem 3

Let *ABC* be an acute triangle, and let *M*,
*N* and *P* be the feet of the perpendiculars drawn
from the centriod *G* of the triangle *ABC*
towards its sides *AB*, *BC* and *CA*,
respectively. Prove that:

### Problem 4

Let 0=< *x*_{1}=< *x*_{2}
=< *x*_{3}=< ...
=< *x*_{n}=< ...
be a non-decreasing sequence of non-negative integers, such that for every
*k*, *k* >= 0, the number of terms of the sequence which are
less than or equal to *k* is finite, and let that number be denoted
by *y*_{k}. Prove that for all positive integers *m* and
*n* it holds: