2007 USAMO Problems/Problem 6

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.

Solution

2007 USAMO (Problems • Resources)
Preceded by Problem 5	Followed by Last Question
1 • 2 • 3 • 4 • 5 • 6
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2007 USAMO Problems/Problem 6

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Solution