KMO 2016 Problem 3

Acute triangle $\triangle ABC$ has area $S$ and perimeter $L$ . A point $P$ inside $\triangle ABC$ has $dist(P,BC)=1, dist(P,CA)=1.5, dist(P,AB)=2$ . Let $BC \cap AP = D$ , $CA \cap BP = E$ , $AB \cap CP= F$ . Let $T$ be the area of $\triangle DEF$ . Prove the following inequality. $$\left( \frac{AD \cdot BE \cdot CF}{T} \right)^2 > 4L^2 + \left( \frac{AB \cdot BC \cdot CA}{24S} \right)^2$$