Let $K$ be a point in the interior of an acute triangle $ABC$ and $ARBPCQ$ be a convex hexagon whose vertices lie on the circumcircle $\Gamma$ of the triangle $ABC.$ Let $A_1$ be the second point where the circle passing through $K$ and tangent to $\Gamma$ at $A$ intersects the line $AP.$ The points $B_1$ and $C_1$ are defined similarly. Prove that
\[ \min\left\{\frac{PA_1}{AA_1}, \: \frac{QB_1}{BB_1}, \: \frac{RC_1}{CC_1}\right\} \leq 1.\]
Official solution:
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 $KA \geq OA.$ 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 $PA/A_1A.$ Since $KA \geq OA, \: O$ lies inside $\omega$ and the homothety ratio is at most $2.$ Hence $PA_1/AA_1 \leq 1.$