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IMC2016: Day 2, Problem 88. Let n be a positive integer, and denote by Zn the ring of integers modulo n. Suppose that there exists a function f:Zn→Zn satisfying the following three properties: (i) f(x)≠x, (ii) f(f(x))=x, (iii) f(f(f(x+1)+1)+1)=x for all x∈Zn. Prove that n\equiv 2 \pmod4. Proposed by Ander Lamaison Vidarte, Berlin Mathematical School, Germany | |||||||||
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