Question 95
Determine all functions $f$ defined on the positive integers and taking nonnegative integer values, satisfying the following three conditions.
(1) $f(n)\ne0$ for at least one positive integer $n$.
(2) $f(xy)=f(x)+f(y)$ for all positive integers $x,y$.
(3) There are infinitely many positive integers $n$ such that $f(k)=f(n-k)$ for every positive integer $k\lt n$.
(1) $f(n)\ne0$ for at least one positive integer $n$.
(2) $f(xy)=f(x)+f(y)$ for all positive integers $x,y$.
(3) There are infinitely many positive integers $n$ such that $f(k)=f(n-k)$ for every positive integer $k\lt n$.