2013 USAMO Problems/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 $YX/XZ=BP/PC$