CMO 2014 Problem 3

Prove that: there exists only one function $f:\mathbb{N^*}\to\mathbb{N^*}$ satisfying: i) $f(1)=f(2)=1$ ; ii) $f(n)=f(f(n-1))+f(n-f(n-1))$ for $n\ge 3$ . For each integer $m\ge 2$ , find the value of $f(2^m)$ .