Question 30
Determine all functions $f$ defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions:
(i) $f(n) \neq 0$ for at least one $n$;
(ii) $f(x y)=f(x)+f(y)$ for every positive integers $x$ and $y$;
(iii) there are infinitely many positive integers $n$ such that $f(k)=f(n-k)$ for all $k\lt n$.
(i) $f(n) \neq 0$ for at least one $n$;
(ii) $f(x y)=f(x)+f(y)$ for every positive integers $x$ and $y$;
(iii) there are infinitely many positive integers $n$ such that $f(k)=f(n-k)$ for all $k\lt n$.