Harvard-MIT November Tournament #0j2o

Harvard-MIT November 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.

TangentsHomothetyDistance chasingAngle chasing

Problem

Problem:

$AB$ is a diameter of circle $O$ . $X$ is a point on $AB$ such that $AX = 3 BX$ . Distinct circles $\omega_1$ and $\omega_2$ are tangent to $O$ at $T_1$ and $T_2$ and to $AB$ at $X$ . The lines $T_1 X$ and $T_2 X$ intersect $O$ again at $S_1$ and $S_2$ . What is the ratio $\frac{T_1 T_2}{S_1 S_2}$ ?

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