ELMO SL 2012 N2

For positive rational $x$ , if $x$ is written in the form $p/q$ with $p, q$ positive relatively prime integers, define $f(x)=p+q$ . For example, $f(1)=2$ .

a) Prove that if $f(x)=f(mx/n)$ for rational $x$ and positive integers $m, n$ , then $f(x)$ divides $|m-n|$ .

b) Let $n$ be a positive integer. If all $x$ which satisfy $f(x)=f(2^nx)$ also satisfy $f(x)=2^n-1$ , find all possible values of $n$ .

Anderson Wang.