Team Selection Test for IMO 2011

Let $O$ be the center of $\Gamma$. Since $ABC$ is an acute triangle $O$ lies inside $ABC$. Assume that $K$ lies on the same side of the lines $AO$ and $BO$ as $C$, and on the same side of the bisector of the line segment $AB$ as $B$. Then $KA\ge OA$.

Let $\omega$ be the circle passing through $K$ and tangent to $\Gamma$ at $A$. Then $\Gamma$ and $\omega$ are homothetic with center $A$ and ratio $PA/A_{1}A$. Since $KA\ge OA$, $O$ lies inside $\omega$ and the homothety ratio is at most $2$. Hence $P{A}_1/A{A}_1\le 1$.