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2025 SSMO Speed Round Problems/Problem 9

Revision as of 11:10, 9 September 2025 by Pinkpig (talk | contribs) (Created page with "==Problem== Let <math>ABC</math> be a triangle. The point <math>P</math> lies on side <math>BC,</math> the point <math>Q</math> lies on side <math>AB,</math> and the point <m...")
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Problem

Let $ABC$ be a triangle. The point $P$ lies on side $BC,$ the point $Q$ lies on side $AB,$ and the point $R$ lies on side $AC$ such that $PQ = BQ,$ $CR = PR,$ and $\angle APB < 90^\circ$. Let $H$ be the foot of the altitude from $A$ to $BC$. Given that $BP = 3,$ $CP = 5,$ and $[AQPR] = \tfrac{3}{7} \cdot [ABC],$ the value of $BH \cdot CH$ can be expressed in the form $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution

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