VMO 2010 Problem 3

In plane,let a circle $(O)$ and two fixed points $B,C$ lies in $(O)$ such that $BC$ not is the diameter.Consider a point $A$ varies in $(O)$ such that $A\neq B,C$ and $AB\neq AC$ .Call $D$ and $E$ respective is intersect of $BC$ and internal and external bisector of $\widehat{BAC}$ , $I$ is midpoint of $DE$ .The line that pass through orthocenter of $\triangle ABC$ and perpendicular with $AI$ intersects $AD,AE$ respective at $M,N$ .

1/Prove that $MN$ pass through a fixed point

2/Determint the place of $A$ such that $S_{AMN}$ has maxium value