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f(f(f(...f(x)))...)=g(x)+a where 1-1 f appears 2009 times

Source: Mathcenter Contest / Oly - Thai Forum 2010 R1 p6 https://artofproblemsolving.com/community/c3196914_mathcenter_contest

November 15, 2022
algebrafunctional equation

Problem Statement

Find all a∈Na\in\mathbb{N} such that exists a bijective function g:N→Ng :\mathbb{N} \to \mathbb{N} and a function f:N→Nf:\mathbb{N}\to\mathbb{N}, such that for all x∈Nx\in\mathbb{N}, f(f(f(...f(x)))...)=g(x)+af(f(f(...f(x)))...)=g(x)+a where ff appears 20092009 times.
(tatari/nightmare)
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