2022 SSMO Speed Round Problems/Problem 9

Problem

Consider a triangle $ABC$ such that $AB=13$ , $BC=14$ , $CA=15$ and a square $WXYZ$ such that $Y$ and $Z$ lie on $\overleftrightarrow{BC}$ , $W$ lies on $\overleftrightarrow{AB}$ , and $X$ lies on $\overleftrightarrow{CA}$ . Suppose further that $W$ , $X$ , $Y$ , and $Z$ are distinct from $A$ , $B$ , and $C$ . Let $O$ be the center of $WXYZ$ . If $AO$ intersects $BC$ at $P$ , then the sum of all values of $\frac{BP}{CP}$ can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .

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2022 SSMO Speed Round Problems/Problem 9

Problem

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