Question 173
There are 2022 equally spaced points on a circular track $\gamma$ of circumference 2022. The points are labeled $A_1, A_2, \dots, A_{2022}$ in some order, each label used once. Initially, Bunbun the Bunny begins at $A_1$. She hops along $\gamma$ from $A_1$ to $A_2$, then from $A_2$ to $A_3$, until she reaches $A_{2022}$, after which she hops back to $A_1$. When hopping from $P$ to $Q$, she always hops along the shorter of the two arcs $\overarc{PQ}$ of $\gamma$; if $\overarc{PQ}$ is a diameter of $\gamma$, she moves along either semicircle.
Determine the maximal possible sum of the lengths of the 2022 arcs which Bunbun traveled, over all possible labellings of the 2022 points.
Determine the maximal possible sum of the lengths of the 2022 arcs which Bunbun traveled, over all possible labellings of the 2022 points.