FKMO 1998

Let $D$ , $E$ , $F$ be points on the sides $BC$ , $CA$ , $AB$ respectively of a triangle $ABC$ . Lines $AD$ , $BE$ , $CF$ intersect the circumcircle of $ABC$ again at $P$ , $Q$ , $R$ , respectively.Show that: $$\frac{AD}{PD}+\frac{BE}{QE}+\frac{CF}{RF}\geq 9$$ and find the cases of equality.

Let $I$ be the incenter of triangle $ABC$ , $O_1$ a circle through $B$ tangent to $CI$ , and $O_2$ a circle through $C$ tangent to $BI$ . Prove that $O_1$ , $O_2$ and the circumcircle of $ABC$ have a common point.