Question 29
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
$$
\min \left\{ \frac{PA_1}{AA_1}, \frac{QB_1}{BB_1}, \frac{RC_1}{CC_1} \right\} \le 1.
$$
$$
\min \left\{ \frac{PA_1}{AA_1}, \frac{QB_1}{BB_1}, \frac{RC_1}{CC_1} \right\} \le 1.
$$