BALKAN 1996

Let $O$ be the circumcenter and $G$ be the centroid of a triangle $ABC$ . If $R$ and $r$ are the circumcenter and incenter of the triangle, respectively, prove that $$OG \leq \sqrt{ R ( R - 2r ) } .$$ Greece

Let $p$ be a prime number with $p>5$ . Consider the set $X = \left\{p - n^2 \mid n\in \mathbb{N} ,\ n^2 < p\right\}$ . Prove that the set $X$ has two distinct elements $x$ and $y$ such that $x\neq 1$ and $x\mid y$ .

Albania

In a convex pentagon $ABCDE$ , the points $M$ , $N$ , $P$ , $Q$ , $R$ are the midpoints of the sides $AB$ , $BC$ , $CD$ , $DE$ , $EA$ , respectively. If the segments $AP$ , $BQ$ , $CR$ and $DM$ pass through a single point, prove that $EN$ contains that point as well.

Yugoslavia

Suppse that $X=\{1,2, \ldots, 2^{1996}-1\}$ , prove that there exist a subset $A$ that satisfies these conditions:

a) $1\in A$ and $2^{1996}-1\in A$ ;

b) Every element of $A$ except $1$ is equal to the sum of two (possibly equal) elements from $A$ ;

c) The maximum number of elements of $A$ is $2012$ .

Romania