KZ-MO 2003 Problem 4

Let the inscribed circle $\omega$ of triangle $ABC$ touch the side $BC$ at the point $A '$ . Let $AA '$ intersect $\omega$ at $P \neq A$ . Let $CP$ and $BP$ intersect $\omega$ , respectively, at points $N$ and $M$ other than $P$ . Prove that $AA ', BN$ and $CM$ intersect at one point.