2014 AIME II Problems/Problem 14

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14. In △ABC, AB=10, ∠A=30∘, and ∠C=45∘. Let H, D, and M be points on the line $<math>\overline{BC}$</math> such that $<math>\overline{AH}⊥$ (Error compiling LaTeX. ! Package inputenc Error: Unicode char \u8:⊥ not set up for use with LaTeX.)\overline{BC}$$ (Error compiling LaTeX. ! Missing $ inserted.), $$ (Error compiling LaTeX. ! Missing $ inserted.)∠BAD=∠CAD$$ (Error compiling LaTeX. ! Missing $ inserted.), and $$ (Error compiling LaTeX. ! Missing $ inserted.)BM=CM$$ (Error compiling LaTeX. ! Missing $ inserted.). Point $$ (Error compiling LaTeX. ! Missing $ inserted.)N$$ (Error compiling LaTeX. ! Missing $ inserted.) is the midpoint of the segment $$ (Error compiling LaTeX. ! Missing $ inserted.)\overline{HM}$$ (Error compiling LaTeX. ! Missing $ inserted.), and point $$ (Error compiling LaTeX. ! Missing $ inserted.)P$$ (Error compiling LaTeX. ! Missing $ inserted.) is on ray AD such that $$ (Error compiling LaTeX. ! Missing $ inserted.)\overline{PN}⊥\overline{BC}$$ (Error compiling LaTeX. ! Missing $ inserted.). Then $$ (Error compiling LaTeX. ! Missing $ inserted.)AP^2=\dfrac{m}{n}$</math>, where m and n are relatively prime positive integers. Find m+n.

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