Question 15
Let $\overline{AB}$ be a chord of a circle $\omega,$ and let $P$ be a point on the chord $\overline{AB}.$ Circle $\omega_1$ passes through $A$ and $P$ and is internally tangent to $\omega.$ Circle $\omega_2$ passes through $B$ and $P$ and is internally tangent to $\omega.$ Circles $\omega_1$ and $\omega_2$ intersect at points $P$ and $Q.$ Line $PQ$ intersects $\omega$ at $X$ and $Y.$ Assume that $AP = 5,$ $PB = 3,$ $XY = 11,$ and $PQ^2 = \frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$