2020 IMO Shortlist N5

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 $k<n$. (Croatia)

确定在所有正整数集合上定义的所有函数 $f$ 并取非负整数值，满足三个条件： (i) $f(n) \neq 0$ 对于至少一个 $n$； (ii) 对于每个正整数 $x$ 和 $y$，$f(x y)=f(x)+f(y)$； (iii) 存在无限多个正整数 $n$，对于所有 $k<n$，$f(k)=f(n-k)$。 (克罗地亚)