2014 AIME II Problems/Problem 14

14. In $△ABC, AB=10, ∠A=30∘$ (Error compiling LaTeX. Unknown error_msg), and $∠C=45∘$ (Error compiling LaTeX. Unknown error_msg). Let H, D, and M be points on the line BC such that AH⊥BC, ∠BAD=∠CAD, and BM=CM. Point N is the midpoint of the segment HM, and point P is on ray AD such that PN⊥BC. Then , where m and n are relatively prime positive integers. Find m+n.