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Iterating function and fixed point

Source: Japan Mathematical Olympiad Finals 2018 Q3

February 13, 2018
functionalgebra

Problem Statement

Let S={1,2,…,999}S=\{1,2,\ldots ,999\}. Consider a function f:S→Sf: S\to S, such that for any n∈Sn\in S, fn+f(n)+1(n)=fnf(n)(n)=n.f^{n+f(n)+1}(n)=f^{nf(n)}(n)=n. Prove that there exists a∈Sa\in S, such that f(a)=af(a)=a. Here fk(n)=f(f(…f⏟k(n)…))f^k(n) = \underbrace{f(f(\ldots f}_{k}(n)\ldots)).
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