IR-MO 2002 Problem 9

Let $M$ and $N$ be points on the side $BC$ of triangle $ABC$ , with the point $M$ lying on the segment $BN$ , such that $BM = CN$ . Let $P$ and $Q$ be points on the segments $AN$ and $AM$ , respectively, such that $\measuredangle PMC =\measuredangle MAB$ and $\measuredangle QNB =\measuredangle NAC$ . Prove that $\measuredangle QBC =\measuredangle PCB$ .