Problem

Source: Balkan MO ShortList 2009 A2

Tags:



Let $ABCD$ be a square and points $M$ $\in$ $BC$, $N \in CD$, $P$ $\in$ $DA$, such that $\angle BAM$ $=$ $x$, $\angle CMN$ $=$ $2x$, $\angle DNP$ $=$ $3x$ 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 $\angle APB =4x$