USAMO 2013 Problem 1

In triangle $ABC$ , points $P$ , $Q$ , $R$ lie on sides $BC$ , $CA$ , $AB$ respectively. Let $\omega_A$ , $\omega_B$ , $\omega_C$ denote the circumcircles of triangles $AQR$ , $BRP$ , $CPQ$ , respectively. Given the fact that segment $AP$ intersects $\omega_A$ , $\omega_B$ , $\omega_C$ again at $X$ , $Y$ , $Z$ , respectively, prove that $YX/XZ=BP/PC$ .