Question 177
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.
$$
8P_AQ_A \cdot P_BQ_B \cdot P_CQ_C \le R^3,
$$
with equality if and only if triangle $ABC$ is equilateral.