2013 USAMO Problems/Problem 2
For a positive integer plot
equally spaced points around a circle. Label one of them
, and place a marker at
. One may move the marker forward in a clockwise direction to either the next point or the point after that. Hence there are a total of
distinct moves available; two from each point. Let
count the number of ways to advance around the circle exactly twice, beginning and ending at
, without repeating a move. Prove that
for all
.