2014 AIME II Problems/Problem 14

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14. In $△ABC, AB=10, ∠A=30∘$ (Error compiling LaTeX. ! Package inputenc Error: Unicode char \u8:△ not set up for use with LaTeX.), and $∠C=45∘$ (Error compiling LaTeX. ! Package inputenc Error: Unicode char \u8:∠ not set up for use with LaTeX.). 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 $AP^2=m/n$, where m and n are relatively prime positive integers. Find m+n.

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