MK-MO 1996

Let $ABCD$ be a parallelogram which is not a rectangle and $E$ be the point in its plane such that $AE \perp AB$ and $CE \perp CB$ . Prove that $\angle DEA = \angle CEB$ .

Let $P$ be the set of all polygons in the plane and let $M : P \to R$ be a mapping that satisfies: (i) $M(P) \ge 0$ for each polygon $P$ , (ii) $M(P) = x^2$ if $P$ is an equilateral triangle of side $x$ , (iii) If a polygon $P$ is partitioned into polygons $S$ and $T$ , then $M(P) = M(S)+ M(T)$ , (iv) If polygons $P$ and $T$ are congruent, then $M(P) = M(T )$ . Determine $M(P)$ if $P$ is a rectangle with edges $x$ and $y$ .

Prove that if $\alpha, \beta, \gamma$ are angles of a triangle, then $\frac{1}{\sin \alpha}+ \frac{1}{\sin \beta} \ge \frac{8}{ 3+2 \ cos\gamma}$ .