IMO 1974 LL POL25

Let f : R oR be of the form f(x) = x + psilon sin x, where

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IMO 1974 LL POL25

Origin: POL

Problem

Let f : R \toR be of the form f(x) = x + \epsilon sin x, where 0 < |\epsilon| \leq1. Define for any x \inR, xn = f ◦\cdot \cdot \cdot ◦f    n times (x). Show that for every x \inR there exists an integer k such that limn\to\inftyxn = k\pi.