FKMO 2009 Problem 2

$ABC$ is an obtuse triangle. (angle $B$ is obtuse) Its circumcircle is $O$ . A tangent line for $O$ passing $C$ meets with $AB$ at $B_1$ . Let $O_1$ be a circumcenter of triangle $AB_1C$ . $B_2$ is a point on the segment $BB_1$ . Let $C_1$ be a contact point of the tangent line for $O$ passing $B_2$ , which is more closer to $C$ . Let $O_2$ be a circumcenter of triangle $AB_2C_1$ . Prove that if $OO_2$ and $AO_1$ is perpendicular, then five points $O,O_2,O_1,C_1,C$ are concyclic.