BALKAN 1997

Suppose that $O$ is a point inside a convex quadrilateral $ABCD$ such that $$OA^2 + OB^2 + OC^2 + OD^2 = 2\mathcal A[ABCD] ,$$ where by $\mathcal A[ABCD]$ we have denoted the area of $ABCD$ . Prove that $ABCD$ is a square and $O$ is its center.

Yugoslavia

Let $S = \{A_1,A_2,\ldots ,A_k\}$ be a collection of subsets of an $n$ -element set $A$ . If for any two elements $x, y \in A$ there is a subset $A_i \in S$ containing exactly one of the two elements $x$ , $y$ , prove that $2^k\geq n$ .

Yugoslavia

The circles $\mathcal C_1$ and $\mathcal C_2$ touch each other externally at $D$ , and touch a circle $\omega$ internally at $B$ and $C$ , respectively. Let $A$ be an intersection point of $\omega$ and the common tangent to $\mathcal C_1$ and $\mathcal C_2$ at $D$ . Lines $AB$ and $AC$ meet $\mathcal C_1$ and $\mathcal C_2$ again at $K$ and $L$ , respectively, and the line $BC$ meets $\mathcal C_1$ again at $M$ and $\mathcal C_2$ again at $N$ . Prove that the lines $AD$ , $KM$ , $LN$ are concurrent.

Greece