Question 83
Let $n \ge 3$ be a positive integer. A positive integer $m \ge n+1$ is called $n$-colorable if and only if each vertex of a regular $m$-gon can be colored with one of $n$ colors so that any $n+1$ consecutive vertices of the regular $m$-gon contain all $n$ colors.
Prove that for every $n \ge 3$ there are only finitely many positive integers that are not $n$-colorable, and find the largest one.
Prove that for every $n \ge 3$ there are only finitely many positive integers that are not $n$-colorable, and find the largest one.