2023 WSMO Speed Round Problems/Problem 10

Problem

Consider acute triangle $ABC$ , $H$ is the orthocenter. Extend $AH$ to meet $BC$ at $D$ . The angle bisector of $\angle{ABH}$ meets the midpoint of $AD$ , $M$ . If $AB=10, BH=4$ , then the area of $ABC$ is $\frac{a\sqrt{b}}{c},$ for squarefree $b$ and relatively prime positive integers $a$ and $c.$ Find $a+b+c.$

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2023 WSMO Speed Round Problems/Problem 10

Problem

Solution