IMC 2016 Problem 8

Let $n$ be a positive integer, and denote by $\mathbb{Z}_n$ the ring of integers modulo $n$ . Suppose that there exists a function $f:\mathbb{Z}_n\to\mathbb{Z}_n$ satisfying the following three properties:

(i) $f(x)\neq x$ ,

(ii) $f(f(x))=x$ ,

(iii) $f(f(f(x+1)+1)+1)=x$ for all $x\in\mathbb{Z}_n$ .

Prove that $n\equiv 2 \pmod4$ .

(Proposed by Ander Lamaison Vidarte, Berlin Mathematical School, Germany)