ILL 1989 Problem 14

For a triangle $ABC,$ let $k$ be its circumcircle with radius $r.$ The bisectors of the inner angles $A, B,$ and $C$ of the triangle intersect respectively the circle $k$ again at points $A', B',$ and $C'.$ Prove the inequality

$$16Q^3 \geq 27 r^4 P,$$

where $Q$ and $P$ are the areas of the triangles $A'B'C'$ and $ABC$ respectively.