NZ-NZMO 2019 Problem 6

Let $V$ be the set of vertices of a regular 21-gon. Given a non-empty subset $U$ of $V$ , let $m(U)$ be the number of distinct lengths that occur between two distinct vertices in $U$ . What is the maximum value of $\frac{m(U)}{|U|}$ as $U$ varies over all non-empty subsets of $V$ ?