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exactly one of the equalities f(g(x)) = x and g(f(x)) = x holds

Source: Dutch IMO TST1 2019 p2

January 10, 2020
functionalgebra

Problem Statement

Write SnS_n for the set {1,2,...,n}\{1, 2,..., n\}. Determine all positive integers nn for which there exist functions f:Sn→Snf : S_n \to S_n and g:Sn→Sng : S_n \to S_n such that for every xx exactly one of the equalities f(g(x))=xf(g(x)) = x and g(f(x))=xg(f(x)) = x holds.
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