MathDB

Irritated function

Source: Switzerland Final Round 2021 P6

February 24, 2021

functionalgebranice problem

Problem Statement

Let

\mathbb{N}

be the set of positive integers. Let

f: \mathbb{N} \rightarrow \mathbb{N}

be a function such that for every positive integer

n \in \mathbb{N}

f(n) -n<2021 \text{and} f^{f(n)}(n) =n Prove that

f(n)=n

for infinitely many

n \in \mathbb{N}