Harvard-MIT Mathematics Tournament #0in4

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:

Let $\omega$ denote the incircle of triangle $ABC$ . The segments $BC$ , $CA$ , and $AB$ are tangent to $\omega$ at $D$ , $E$ , and $F$ , respectively. Point $P$ lies on $EF$ such that segment $PD$ is perpendicular to $BC$ . The line $AP$ intersects $BC$ at $Q$ . The circles $\omega_1$ and $\omega_2$ pass through $B$ and $C$ , respectively, and are tangent to $AQ$ at $Q$ ; the former meets $AB$ again at $X$ , and the latter meets $AC$ again at $Y$ . The line $XY$ intersects $BC$ at $Z$ . Given that $AB=15$ , $BC=14$ , and $CA=13$ , find $\lfloor XZ \cdot YZ\rfloor$ .