HMMT November 2022 #0kqa

HMMT November 2022Hard

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Understand the scale

Scale 0.5: Easiest math competition problems, often solvable even for early middle school students without experience. 1: Problems strictly for beginners, often at a moderate middle school level. 1.5: Problems for stronger beginner students, on the level of the middling problems in most middle school contests. 2: For motivated beginners, harder questions from the previous categories. 2.5: More advanced beginner problems, hardest questions from previous categories. 3: Early intermediate problems that require more creative thinking. 3.5: Tougher early intermediate problems that consistently stretch into higher-level creative thinking or conceptual knowledge. 4: Intermediate-level problems. 4.5: Upper intermediate problems approaching the upper bound for non-invitational sprint math competitions. 5: More difficult AIME problems or simple proof-based Olympiad-style problems. 6: High-level AIME-styled questions or introductory Olympiad-level questions. 7: Tougher Olympiad-level questions, may require more technical knowledge. 8: High-level Olympiad-level questions. 9: Expert Olympiad-level questions. 9.5: The hardest problems appearing on Olympiads which the strongest students could reasonably solve. 10: Historically notorious problems, generally unsuitable for even very hard competitions because they are exceedingly tedious, long, and difficult.

Distance chasingAngle chasingHomothety

Problem

Problem: Let $ABC$ be a triangle with $AB=13$ , $BC=14$ , and $CA=15$ . Pick points $Q$ and $R$ on $AC$ and $AB$ such that $\angle CBQ=\angle BCR=90^{\circ}$ . There exist two points $P_{1} \neq P_{2}$ in the plane of $ABC$ such that $\triangle P_{1}QR$ , $\triangle P_{2}QR$ , and $\triangle ABC$ are similar (with vertices in order). Compute the sum of the distances from $P_{1}$ to $BC$ and $P_{2}$ to $BC$ .

Proposed by: Ankit Bisain Answer: 48

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