BALKAN 1998

Consider the finite sequence $\left\lfloor \frac{k^2}{1998} \right\rfloor$ , for $k=1,2,\ldots, 1997$ . How many distinct terms are there in this sequence?

Greece

Let $n\geq 2$ be an integer, and let $0 < a_1 < a_2 < \cdots < a_{2n+1}$ be real numbers. Prove the inequality $$\sqrt[n]{a_1} - \sqrt[n]{a_2} + \sqrt[n]{a_3} - \cdots + \sqrt[n]{a_{2n+1}} < \sqrt[n]{a_1 - a_2 + a_3 - \cdots + a_{2n+1}}.$$ Bogdan Enescu, Romania

Let $\mathcal S$ denote the set of points inside or on the border of a triangle $ABC$ , without a fixed point $T$ inside the triangle. Show that $\mathcal S$ can be partitioned into disjoint closed segemnts.

Yugoslavia

Prove that the following equation has no solution in integer numbers: $$x^2 + 4 = y^5.$$ Bulgaria