IR-TST 2019 Problem 11

Let $P$ be a simple polygon completely in $C$ , a circle with radius $1$ , such that $P$ does not pass through the center of $C$ . The perimeter of $P$ is $36$ . Prove that there is a radius of $C$ that intersects $P$ at least $6$ times, or there is a circle which is concentric with $C$ and have at least $6$ common points with $P$ .

Proposed by Seyed Reza Hosseini