2020 IMO Shortlist G8

Let $\Gamma$ and $I$ be the circumcircle and the incenter of an acute-angled triangle $A B C$. Two circles $\omega_{B}$ and $\omega_{C}$ passing through $B$ and $C$, respectively, are tangent at $I$. Let $\omega_{B}$ meet the shorter arc $A B$ of $\Gamma$ and segment $A B$ again at $P$ and $M$, respectively. Similarly, let $\omega_{C}$ meet the shorter arc $A C$ of $\Gamma$ and segment $A C$ again at $Q$ and $N$, respectively. The rays $P M$ and $Q N$ meet at $X$, and the tangents to $\omega_{B}$ and $\omega_{C}$ at $B$ and $C$, respectively, meet at $Y$. Prove that the points $A, X$, and $Y$ are collinear. (Netherlands)

令$\Gamma$ 和$I$ 为锐角三角形$A B C$ 的外接圆和内心。分别经过$B$和$C$的两个圆$\omega_{B}$和$\omega_{C}$在$I$处相切。让$\omega_{B}$与$\Gamma$的较短弧$A B$相交，并分别在$P$和$M$处再次分段$A B$。类似地，让$\omega_{C}$与$\Gamma$的较短弧$A C$相交，并分别在$Q$和$N$处再次分段$A C$。射线$P M$ 和$Q N$ 相交于$X$，$\omega_{B}$ 和$\omega_{C}$ 的切线分别在$B$ 和$C$ 相交于$Y$。证明点 $A、X$ 和 $Y$ 共线。 (荷兰)