### 15^{th} BALKAN MATHEMATICAL OLYMPIAD
NICOSIA, CYPRUS, 1998

### Problem 1

Find the number of different terms of the finite sequence
[*k*^{2}/1998], where *k* = 1, 2,... , 1997 and
[*x*] denotes the integer part of *x*.
### Problem 2

If *n*>= 2 is an integer and 0 < *a*_{1} <
*a*_{2} ... < *a*_{2n+1}
are real numbers, prove the inequality:

### Problem 3

Let *S* be the set of all points inside and on the border of the
triangle *ABC* without one inside point *T*. Prove that *S*
can be represented as a union of closed segments no two of which have a
point in common. (A closed segment contains both of its ends.)
### Problem 4

Prove that the equation *y*^{2} = *x*^{5} - 4
has no integer solutions.