Question 23
Let $n$ be a positive integer, $n \ge 3$. In a regular $n$-gon, one draws a maximal set of diagonals, no two of which intersect in the interior of the $n$-gon. Every diagonal is labelled with the number of sides of the $n$-gon between the endpoints of the diagonal along the shortest path. Find the maximum value of the sum of the labels.
*Answer:* $\frac{n^2-9}{4}$ for odd $n$, $\frac{n^2-8}{4}$ for even $n$.
*Answer:* $\frac{n^2-9}{4}$ for odd $n$, $\frac{n^2-8}{4}$ for even $n$.