Question 61
Let $p$ be a circle with center $K$ passing through $M$, $q$ a semicircle with diameter $KM$ and $L$ a point inside the segment $KM$. A line through $L$ perpendicular to $KM$ intersects $q$ at point $Q$ and $p$ at points $P_1, P_2$ such that $P_1Q \gt P_2Q$. Line $MQ$ intersects $p$ for the second time at $R \ne M$. Prove that areas $S_1, S_2$ of triangles $MP_1Q, P_2RQ$ satisfy
$$
1 \lt \frac{S_1}{S_2} \lt 3 + \sqrt{8}.
$$
$$
1 \lt \frac{S_1}{S_2} \lt 3 + \sqrt{8}.
$$