2012 USAJMO Problems/Problem 4

Problem

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 $P_1$ , $P_2$ , $\dots$ , $P_n$ 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 $P_k$ . Supose that $P_a$ and $P_b$ are the nearest adjacent points on either side of $P_n$ . Prove that $a + b \le n$ .

2012 USAJMO (Problems • Resources)
Preceded by Problem 2	Followed by Problem 4
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2012 USAJMO Problems/Problem 4

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