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Ouroboros functions

Source: India TST 2023 Day 1 P2

July 9, 2023
algebrafunctional equationfunction

Problem Statement

Let g:N→Ng:\mathbb{N}\to \mathbb{N} be a bijective function and suppose that f:N→Nf:\mathbb{N}\to \mathbb{N} is a function such that:
[*] For all naturals xx, f(⋯(f⏟x2023  f’s(x)))=x.\underbrace{f(\cdots (f}_{x^{2023}\;f\text{'s}}(x)))=x. [*] For all naturals x,yx,y such that x∣yx|y, we have f(x)∣g(y)f(x)|g(y).
Prove that f(x)=xf(x)=x.
Proposed by Pulkit Sinha
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