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IMC 2016, Problem 8

Source: IMC 2016

July 28, 2016
IMCIMC 2016college contestsabstract algebrafunction

Problem Statement

Let nn be a positive integer, and denote by Zn\mathbb{Z}_n the ring of integers modulo nn. Suppose that there exists a function f:Zn→Znf:\mathbb{Z}_n\to\mathbb{Z}_n satisfying the following three properties:
(i) f(x)≠xf(x)\neq x,
(ii) f(f(x))=xf(f(x))=x,
(iii) f(f(f(x+1)+1)+1)=xf(f(f(x+1)+1)+1)=x for all x∈Znx\in\mathbb{Z}_n.
Prove that n≡2(mod4)n\equiv 2 \pmod4.
(Proposed by Ander Lamaison Vidarte, Berlin Mathematical School, Germany)
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