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Harvard-MIT November Tournament #0iwh

Harvard-MIT November TournamentHard

From module Power of a Point

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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 theorem

Problem

Problem:

A circle ω1\omega_{1} of radius 1515 intersects a circle ω2\omega_{2} of radius 1313 at points PP and QQ. Point AA is on line PQPQ such that PP is between AA and QQ. RR and SS are the points of tangency from AA to ω1\omega_{1} and ω2\omega_{2}, respectively, such that the line ASAS does not intersect ω1\omega_{1} and the line ARAR does not intersect ω2\omega_{2}. If PQ=24PQ = 24 and ∠RAS\angle RAS has a measure of 90∘90^{\circ}, compute the length of ARAR.

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