Harvard-MIT Mathematics Tournament #0izb

Harvard-MIT Mathematics TournamentHard

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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.

TangentsRadical axis theoremSpiral similarityAngle chasingTriangle trigonometry

Problem

Problem:

Circles $\omega_{1}$ and $\omega_{2}$ intersect at points $A$ and $B$ . Segment $PQ$ is tangent to $\omega_{1}$ at $P$ and to $\omega_{2}$ at $Q$ , and $A$ is closer to $PQ$ than $B$ . Point $X$ is on $\omega_{1}$ such that $PX \parallel QB$ , and point $Y$ is on $\omega_{2}$ such that $QY \parallel PB$ . Given that $\angle APQ = 30^{\circ}$ and $\angle PQA = 15^{\circ}$ , find the ratio $AX / AY$ .

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