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International Mathematics Competition
for University Students
2016

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IMC2016: Day 2, Problem 8

8. Let n be a positive integer, and denote by Zn the ring of integers modulo n. Suppose that there exists a function f:ZnZn 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 xZn.

Prove that n\equiv 2 \pmod4.

Proposed by Ander Lamaison Vidarte, Berlin Mathematical School, Germany

        

IMC
2016

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