Problem

Source: Iranian TST 2019, second exam day 2, problem 5

Tags: combinatorial geometry, combinatorics, perimeter



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