USAJMO 2012 Problem 4

Let $\alpha$ be an irrational number with $0<\alpha < 1$ , and draw a circle in the plane whose circumference has length $1$ . Given any integer $n\ge 3$ , define a sequence of points $P_1, P_2, \ldots , P_n$ as follows. First select any point $P_1$ on the circle, and for $2\le k\le n$ define $P_k$ as the point on the circle for which the length of arc $P_{k-1}P_k$ is $\alpha$ , when travelling counterclockwise around the circle from $P_{k-1}$ to $P_k$ . Suppose that $P_a$ and $P_b$ are the nearest adjacent points on either side of $P_n$ . Prove that $a+b\le n$ .