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Cycle conditions in group of permutations determine f(2021)

Source: 2021 Macedonian Team Selection Test P4

May 30, 2021

combinatoricsalgebra

Problem Statement

Let

S=\{1, 2, 3, \dots 2021\}

and

f:S \to S

be a function such that

f^{(n)}(n)=n

for each

n \in S

. Find all possible values for

f(2021)

. (Here,

f^{(n)}(n) = \underbrace{f(f(f\dots f(}_{n \text{ times} }n)))\dots))

.)
Proposed by Viktor Simjanoski