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nice problem for Iran

Source: Iranian Third Round 2020 Algebra exam Problem3

November 20, 2020
functional equationSubsetsalgebra

Problem Statement

find all kk distinct integers a1,a2,...,aka_1,a_2,...,a_k such that there exists an injective function ff from reals to themselves such that for each positive integer nn we have {fn(x)−x∣x∈R}={a1+n,a2+n,...,ak+n}\{f^n(x)-x| x \in \mathbb{R} \}=\{a_1+n,a_2+n,...,a_k+n\}.
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