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A bijection with close adjacent values

Source: Serbia TST 2024, P4

May 18, 2024
algebra

Problem Statement

Let f:N→Nf: \mathbb{N} \rightarrow \mathbb{N} be a bijection and let kk be a positive integer such that ∣f(x+1)−f(x)∣≤k|f(x+1)-f(x)| \leq k for all positive integers xx. Show that there exists an integer dd, such that f(x)=x+df(x)=x+d for infinitely many positive integers xx.
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