USA TSTST #0k5e — USA TSTST · USAMO Guide - Olympiad Mathematics Learning Platform

USA TSTST #0k5e

USA TSTSTHard

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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

Let $ABC$ be an acute triangle with incenter $I$ , circumcenter $O$ , and circumcircle $\Gamma$ . Let $M$ be the midpoint of $\overline{AB}$ . Ray $AI$ meets $\overline{BC}$ at $D$ . Denote by $\omega$ and $\gamma$ the circumcircles of $\triangle BIC$ and $\triangle BAD$ , respectively. Line $MO$ meets $\omega$ at $X$ and $Y$ , while line $CO$ meets $\omega$ at $C$ and $Q$ . Assume that $Q$ lies inside $\triangle ABC$ and $\angle AQM = \angle ACB$ . Consider the tangents to $\omega$ at $X$ and $Y$ and the tangents to $\gamma$ at $A$ and $D$ . Given that $\angle BAC \neq 60^\circ$ , prove that these four lines are concurrent on $\Gamma$ .