US-TST 2008 Problem 7

Let $ABC$ be a triangle with $G$ as its centroid. Let $P$ be a variable point on segment $BC$ . Points $Q$ and $R$ lie on sides $AC$ and $AB$ respectively, such that $PQ \parallel AB$ and $PR \parallel AC$ . Prove that, as $P$ varies along segment $BC$ , the circumcircle of triangle $AQR$ passes through a fixed point $X$ such that $\angle BAG = \angle CAX$ .