USAMO 2010 Problem 2

There are $n$ students standing in a circle, one behind the other. The students have heights $h_1 < h_2 < \dots < h_n$. If a student with height $h_k$ is standing directly behind a student with height $h_{k-2}$ or less, the two students are permitted to switch places. Prove that it is not possible to make more than $\binom{n}{3}$ such switches before reaching a position in which no further switches are possible.

(This problem was suggested by Kiran Kedlaya, Travis Schedler, and David Speyer.)