15th BALKAN MATHEMATICAL OLYMPIAD
NICOSIA, CYPRUS, 1998

Problem 1

Find the number of different terms of the finite sequence [k2/1998], where k = 1, 2,... , 1997 and [x] denotes the integer part of x.

Problem 2

If n>= 2 is an integer and 0 < a1 < a2 ... < a2n+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 y2 = x5 - 4 has no integer solutions.