USAMO #0inr — USAMO · USAMO Guide - Olympiad Mathematics Learning Platform

USAMO #0inr

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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 $\omega, \Omega$ , and $R$ being its incircle, circumcircle, and circumradius, respectively. Circle $\omega_A$ is tangent internally to $\Omega$ at $A$ and tangent externally to $\omega$ . Circle $\Omega_A$ is tangent internally to $\Omega$ at $A$ and tangent internally to $\omega$ . Let $P_A$ and $Q_A$ denote the centers of $\omega_A$ and $\Omega_A$ , respectively. Define points $P_B, Q_B, P_C, Q_C$ analogously. Prove that

8P_AQ_A \cdot P_BQ_B \cdot P_CQ_C \le R^3,

with equality if and only if triangle $ABC$ is equilateral.