RU-SPB 2009 Problem 94

Points $A_1$ and $C_1$ are on $BC$ and $AB$ of acute-angled triangle $ABC$ . $AA_1$ and $CC_1$ intersect in $K$ . Circumcircles of $AA_1B,CC_1B$ intersect in $P$ - incenter of $AKC$ . Prove, that $P$ - orthocenter of $ABC$