Question 6
Let $ABC$ be an acute triangle with $\omega$, $S$, and $R$ being its incircle, circumcircle, and circumradius, respectively. Circle $\omega_{A}$ is tangent internally to $S$ at $A$ and tangent externally to $\omega$. Circle $S_{A}$ is tangent internally to $S$ at $A$ and tangent internally to $\omega$. Let $P_A$ and $Q_A$ denote the centers of $\omega_A$ and $S_A$, respectively. Define points $P_B$, $Q_B$, $P_C$, $Q_C$ analogously. Prove that
$$ 8 P_A Q_A \cdot P_B Q_B \cdot P_C Q_C \le R^3 $$
with equality if and only if triangle $ABC$ is equilateral.
Proposed by Sung-Yoon Kim; source discussion: https://aops.com/community/p825515
$$ 8 P_A Q_A \cdot P_B Q_B \cdot P_C Q_C \le R^3 $$
with equality if and only if triangle $ABC$ is equilateral.
Proposed by Sung-Yoon Kim; source discussion: https://aops.com/community/p825515