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 $\frac{1}{\sin \alpha}+ \frac{1}{\sin \beta} \ge \frac{8}{ 3+2 \ cos\gamma}$ .

A polygon is called good if it satisfies the following conditions: (i) All its angles are in $(0,\pi)$ or in $(\pi ,2\pi)$, (ii) It is not self-intersecing, (iii) For any three sides, two are parallel and equal. Find all $n$ for which there exists a good $n$-gon.

Find the greatest $n$ for which there exist $n$ lines in space, passing through a single point, such that any two of them form the same angle.