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2 Functions

Source: Junior Olympiad of Malaysia Shortlist 2015 C3

July 17, 2015
functioncombinatorics

Problem Statement

Let n≥2 n\ge 2 be a positive integer and S={1,2,⋯ ,n} S= \{1,2,\cdots ,n\} . Let two functions f:S→{1,−1} f:S \rightarrow \{1,-1\} and g:S→S g:S \rightarrow S satisfy:
i) f(x)f(y)=f(x+y),∀x,y∈S f(x)f(y)=f(x+y) , \forall x,y \in S \\ ii) f(g(x))=f(x),∀x∈S f(g(x))=f(x) , \forall x \in S \\ iii) f(x+n)=f(x),∀x∈Sf(x+n)=f(x) ,\forall x \in S\\ iv) g g is bijective.\\
Find the number of pair of such functions (f,g) (f,g) for every nn.
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