Harvard-MIT Mathematics Tournament #0ift

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.

Problem

Problem:

Triangle $ABC$ has incircle $\omega$ which touches $AB$ at $C_1$ , $BC$ at $A_1$ , and $CA$ at $B_1$ . Let $A_2$ be the reflection of $A_1$ over the midpoint of $BC$ , and define $B_2$ and $C_2$ similarly. Let $A_3$ be the intersection of $AA_2$ with $\omega$ that is closer to $A$ , and define $B_3$ and $C_3$ similarly. If $AB=9$ , $BC=10$ , and $CA=13$ , find $\left[A_3 B_3 C_3\right] / [ABC]$ . (Here $[XYZ]$ denotes the area of triangle $XYZ$ .)