BALKAN 1991

Let $ABC$ be an acute triangle inscribed in a circle centered at $O$ . Let $M$ be a point on the small arc $AB$ of the triangle's circumcircle. The perpendicular dropped from $M$ on the ray $OA$ intersects the sides $AB$ and $AC$ at the points $K$ and $L$ , respectively. Similarly, the perpendicular dropped from $M$ on the ray $OB$ intersects the sides $AB$ and $BC$ at $N$ and $P$ , respectively. Assume that $KL=MN$ . Find the size of the angle $\angle{MLP}$ in terms of the angles of the triangle $ABC$ .

Show that there are infinitely many noncongruent triangles which satisfy the following conditions: i) the side lengths are relatively prime integers; ii)the area is an integer number; iii)the altitudes' lengths are not integer numbers.

A regular hexagon of area $H$ is inscribed in a convex polygon of area $P$ . Show that $P \leq \frac{3}{2}H$ . When does the equality occur?

Prove that there is no bijective function $f : \left\{1,2,3,\ldots \right\}\rightarrow \left\{0,1,2,3,\ldots \right\}$ such that $f(mn)=f(m)+f(n)+3f(m)f(n)$ .