HMMT Spring 2021 Guts Round #0khr

HMMT Spring 2021 Guts RoundHard

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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.

Problem

Problem:

Two circles $\Gamma_{1}$ and $\Gamma_{2}$ of radius $1$ and $2$ , respectively, are centered at the origin. A particle is placed at $(2,0)$ and is shot towards $\Gamma_{1}$ . When it reaches $\Gamma_{1}$ , it bounces off the circumference and heads back towards $\Gamma_{2}$ . The particle continues bouncing off the two circles in this fashion.

If the particle is shot at an acute angle $\theta$ above the $x$ -axis, it will bounce $11$ times before returning to $(2,0)$ for the first time. If $\cot \theta = a - \sqrt{b}$ for positive integers $a$ and $b$ , compute $100a + b$ .