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Arithmetic progression

Source: RMO 2004, 10th grade, problem 1

March 6, 2005
functionarithmetic sequencealgebra unsolvedalgebra

Problem Statement

Let f:R→Rf : \mathbb{R} \to \mathbb{R} be a function such that ∣f(x)−f(y)∣≤∣x−y∣|f(x)-f(y)| \leq |x-y|, for all x,y∈Rx,y \in \mathbb{R}. Prove that if for any real xx, the sequence x,f(x),f(f(x)),…x,f(x),f(f(x)),\ldots is an arithmetic progression, then there is a∈Ra \in \mathbb{R} such that f(x)=x+af(x)=x+a, for all x∈Rx \in \mathbb R.
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