Question 97
Let $\Gamma$ and $I$ be the circumcircle and incenter of an acute triangle $ABC$. Two circles $\omega_B$ and $\omega_C$, passing through $B$ and $C$ respectively, are tangent at $I$. Let $\omega_B$ meet the shorter arc $AB$ of $\Gamma$ and the segment $AB$ again at $P$ and $M$, respectively. Similarly, let $\omega_C$ meet the shorter arc $AC$ of $\Gamma$ and the segment $AC$ again at $Q$ and $N$, respectively. The rays $PM$ and $QN$ meet at $X$, and the tangents to $\omega_B$ and $\omega_C$ at $B$ and $C$, respectively, meet at $Y$. Prove that $A,X,Y$ are collinear.