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Source: 2023 AIME II / 12

February 16, 2023
AMCAIMEAIME II

Problem Statement

In △ABC\triangle ABC with side lengths AB=13AB=13, BC=14BC=14, and CA=15CA=15, let MM be the midpoint of BC‾\overline{BC}. Let PP be the point on the circumcircle of △ABC\triangle ABC such that MM is on AP‾\overline{AP}. There exists a unique point QQ on segment AM‾\overline{AM} such that ∠PBQ=∠PCQ\angle PBQ = \angle PCQ. Then AQAQ can be written as mn\frac{m}{\sqrt{n}}, where mm and nn are relatively prime positive integers. Find m+nm+n.
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