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IMO Shortlist 2012, Algebra 6

Source: IMO Shortlist 2012, Algebra 6

July 29, 2013
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Problem Statement

Let f:N→Nf: \mathbb{N} \rightarrow \mathbb{N} be a function, and let fmf^m be ff applied mm times. Suppose that for every n∈Nn \in \mathbb{N} there exists a k∈Nk \in \mathbb{N} such that f2k(n)=n+kf^{2k}(n)=n+k, and let knk_n be the smallest such kk. Prove that the sequence k1,k2,…k_1,k_2,\ldots is unbounded.
Proposed by Palmer Mebane, United States
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