Let $\mathbb{N}$ be the set of all positive integers. The function $f: \mathbb{N} \to \mathbb{N}$ satisfies $f(1)=3, f(mn)=f(m)f(n)-f(m+n)+2$ for all $m,n \in \mathbb{N}$. Prove that $f$ does not exist. Comment: The original problem asked for the value of $f(2006)$, which obviously does not exist when $f$ does not. This was probably a mistake by the Olympiad committee. Hence the modified problem.