Question 14
In $\triangle ABC,$ $AB = 10,$ $\angle A = 30^\circ,$ and $\angle C = 45^\circ.$ Let $H,$ $D,$ and $M$ be points on line $\overline{BC}$ such that $\overline{AH} \perp \overline{BC},$ $\angle BAD = \angle CAD,$ and $BM = CM.$ Point $N$ is the midpoint of segment $\overline{HM},$ and point $P$ is on ray $AD$ such that $\overline{PN} \perp \overline{BC}.$ Then $AP^2 = \frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$