BALKAN-SL 2009 A6

We denote the set of nonzero integers and the set of non-negative integers with $\mathbb Z^*$ and $\mathbb N_0$ , respectively. Find all functions $f:\mathbb Z^* \to \mathbb N_0$ such that: $a)$ $f(a+b)\geq min(f(a), f(b))$ for all $a,b$ in $\mathbb Z^*$ for which $a+b$ is in $\mathbb Z^*$ . $b)$ $f(ab)=f(a)+f(b)$ for all $a,b$ in $\mathbb Z^*$ .