Harvard-MIT November Tournament #0j32

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.

Problem

Problem:

$ABC$ is a right triangle with $\angle A = 30^{\circ}$ and circumcircle $O$ . Circles $\omega_1$ , $\omega_2$ , and $\omega_3$ lie outside $ABC$ and are tangent to $O$ at $T_1$ , $T_2$ , and $T_3$ respectively and to $AB$ , $BC$ , and $CA$ at $S_1$ , $S_2$ , and $S_3$ , respectively. Lines $T_1 S_1$ , $T_2 S_2$ , and $T_3 S_3$ intersect $O$ again at $A'$ , $B'$ , and $C'$ , respectively. What is the ratio of the area of $A'B'C'$ to the area of $ABC$ ?