BALKAN 2000

Let $ABC$ be an acute-angled triangle and $D$ the midpoint of $BC$ . Let $E$ be a point on segment $AD$ and $M$ its projection on $BC$ . If $N$ and $P$ are the projections of $M$ on $AB$ and $AC$ then the interior angule bisectors of $\angle NMP$ and $\angle NEP$ are parallel.

How many $1 \times 10\sqrt 2$ rectangles can be cut from a $50\times 90$ rectangle using cuts parallel to its edges?

Show that for any $n$ we can find a set $X$ of $n$ distinct integers greater than 1, such that the average of the elements of any subset of $X$ is a square, cube or higher power.