BG-MO 1995 Problem 5

Let $A = \{1,2,...,m + n\}$ , where $m,n$ are positive integers, and let the function f : $A \to A$ be defined by: $f(m) = 1$ , $f(m+n) = m+1$ and $f(i) = i+1$ for all the other $i$ . (a) Prove that if $m$ and $n$ are odd, then there exists a function $g : A \to A$ such that $g(g(a)) = f(a)$ for all $a \in A$ . (b) Prove that if $m$ is even, then there is a function $g : A\to A$ such that $g(g(a))=f(a)$ for all $a \in A$ is and only if $n = m$ .