BALKAN-SL 2009 A2

Let $ABCD$ be a square and points $M$ $\in$ $BC$ , $N \in CD$ , $P$ $\in$ $DA$ , such that $\angle BAM$ $=$ $x$ , $\angle CMN$ $=$ $2x$ , $\angle DNP$ $=$ $3x$ [list=i] [] Show that, for any $x \in (0, \tfrac{\pi}{8} )$ , such a configuration exists [] Determine the number of angles $x \in ( 0, \tfrac{\pi}{8} )$ for which $\angle APB =4x$