Question 118
Let $\mathcal{P}$ be a simple polygon (non-self-intersecting) with perimeter $36$ that lies in a circle with radius $1$ and does not pass through its center. Prove that there is either a radius of this circle that intersects $\mathcal{P}$ at least $6$ times, or there is a second circle, concentric with this circle, that has at least $6$ common points with $\mathcal{P}$.