Challenging Geometric Inequality: Prove $\leq R^{3}$

[ishfaqhaque]

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
$8P_{A}Q_{A}\cdot P_{B}Q_{B}\cdot P_{C}Q_{C}\leq R^{3}\;$

[Moon]

whoa...man...this is not a very good problem to post here. (However, it can be a open challenge for Zubaer, Dhononjoy, Urmi, and some other guys...)

BTW you have defied a rule of the forum. You're not using a good topic.
Read: viewtopic.php?p=13&f=9#p13

[ishfaqhaque]

where are DJ, Urmi, ZU, Pranon, promi and others? all of you should be able to solve this

[Moon]

Yup...this is ridiculous. In our time, we did not get problems from the seniors (even after begging). Now people don't even try these problems...huh...

Oh...Tanvir vi, and Nayel vi used to give us a few problems.