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Sequence of Points on a Circle

Source: 2012 USAJMO Day 2 #4

April 25, 2012
floor functioninductionanalytic geometryirrational numberAMCUSAJMO

Problem Statement

Let α\alpha be an irrational number with 0<α<10<\alpha < 1, and draw a circle in the plane whose circumference has length 11. Given any integer n≥3n\ge 3, define a sequence of points P1,P2,…,PnP_1, P_2, \ldots , P_n as follows. First select any point P1P_1 on the circle, and for 2≤k≤n2\le k\le n define PkP_k as the point on the circle for which the length of arc Pk−1PkP_{k-1}P_k is α\alpha, when travelling counterclockwise around the circle from Pk−1P_{k-1} to PkP_k. Suppose that PaP_a and PbP_b are the nearest adjacent points on either side of PnP_n. Prove that a+b≤na+b\le n.
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