ZA-SAMO 2007 Problem 5

Let $Z$ and $R$ denote the sets of integers and real numbers, respectively. Let $f: Z \rightarrow R$ be a function satisfying: (i) $f(n) \ge 0$ for all $n \in Z$ (ii) $f(mn)=f(m)f(n)$ for all $m,n \in Z$ (iii) $f(m+n) \le max(f(m),f(n))$ for all $m,n \in Z$

(a) Prove that $f(n) \le 1$ for all $n \in Z$ (b) Find a function $f: Z \rightarrow R$ satisfying (i), (ii),(iii) and $0<f(2)<1$ and $f(2007) = 1$