USAMO PROBELMS 2

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.