Question 64
Two circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. Line $l$ is tangent to $\omega_1$ at $P$ and to $\omega_2$ at $Q$ so that $A$ is closer to $l$ than $B$. Let $X$ and $Y$ be points on major arcs $\widehat{PA}$ (on $\omega_1$) and $\widehat{AQ}$ (on $\omega_2$), respectively, such that $AX/PX = AY/QY = c$. Extend segments $PA$ and $QA$ through $A$ to $R$ and $S$, respectively, such that $AR = AS = c \cdot PQ$. Given that the circumcenter of triangle $ARS$ lies on line $XY$, prove that $\angle XPA = \angle AQY$.
(This problem was suggested by Delong Meng.)
(This problem was suggested by Delong Meng.)