HMMT Spring 2021 #0kgv

HMMT Spring 2021Hard

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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.

Triangle trigonometryTrigonometryAngle chasingDistance chasing

Problem

Problem:

Acute triangle $ABC$ has circumcircle $\Gamma$ . Let $M$ be the midpoint of $BC$ . Points $P$ and $Q$ lie on $\Gamma$ so that $\angle APM = 90^{\circ}$ and $Q \neq A$ lies on line $AM$ . Segments $PQ$ and $BC$ intersect at $S$ . Suppose that $BS = 1$ , $CS = 3$ , $PQ = 8 \sqrt{\frac{7}{37}}$ , and the radius of $\Gamma$ is $r$ . If the sum of all possible values of $r^2$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$ , compute $100a + b$ .

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