Question 14
The incircle $\omega$ of triangle $ABC$ is tangent to $\overline{BC}$ at $X.$ Let $Y \neq X$ be the other intersection of $\overline{AX}$ with $\omega.$ Points $P$ and $Q$ lie on $\overline{AB}$ and $\overline{AC},$ respectively, so that $\overline{PQ}$ is tangent to $\omega$ at $Y.$ Assume that $AP = 3,$ $PB = 4,$ $AC = 8,$ and $AQ = \frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$