USA IMO TST #0kft — USA IMO TST · USAMO Guide - Olympiad Mathematics Learning Platform

USA IMO TST #0kft

USA IMO TSTHard

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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

Two circles $\Gamma_1$ and $\Gamma_2$ have common external tangents $l_1$ and $l_2$ meeting at $T$ . Suppose $l_1$ touches $\Gamma_1$ at $A$ and $l_2$ touches $\Gamma_2$ at $B$ . A circle $\Omega$ through $A$ and $B$ intersects $\Gamma_1$ again at $C$ and $\Gamma_2$ again at $D$ , such that quadrilateral $ABCD$ is convex. Suppose lines $AC$ and $BD$ meet at point $X$ , while lines $AD$ and $BC$ meet at point $Y$ . Show that $T$ , $X$ , $Y$ are collinear.