CMO 2012

In the triangle $ABC$ , $\angle A$ is biggest. On the circumcircle of $\triangle ABC$ , let $D$ be the midpoint of $\widehat{ABC}$ and $E$ be the midpoint of $\widehat{ACB}$ . The circle $c_1$ passes through $A,B$ and is tangent to $AC$ at $A$ , the circle $c_2$ passes through $A,E$ and is tangent $AD$ at $A$ . $c_1$ and $c_2$ intersect at $A$ and $P$ . Prove that $AP$ bisects $\angle BAC$ .

$A$ $c_1$ $c_2$ $P$ $B$ $E$ $C$ $D$

Let $f(x)=(x + a)(x + b)$ where $a,b>0$ . For any reals $x_1,x_2,\ldots ,x_n\geqslant 0$ satisfying $x_1+x_2+\ldots +x_n =1$ , find the maximum of $F=\sum\limits_{1 \leqslant i < j \leqslant n} {\min \left\{ {f({x_i}),f({x_j})} \right\}}$ .