Question 11
Find all functions $f:\mathbb N_{\geq1}\to\mathbb N_{\geq0}$ satisfying both conditions:
1. $f(xy)=f(x)+f(y)$ for all integers $x,y\geq1$;
2. there are infinitely many integers $n\geq1$ such that $f(k)=f(n-k)$ for every integer $k$ with $1\leq k\leq n-1$.
Here $\mathbb N_{\geq0}$ and $\mathbb N_{\geq1}$ denote the nonnegative and positive integers, respectively.
1. $f(xy)=f(x)+f(y)$ for all integers $x,y\geq1$;
2. there are infinitely many integers $n\geq1$ such that $f(k)=f(n-k)$ for every integer $k$ with $1\leq k\leq n-1$.
Here $\mathbb N_{\geq0}$ and $\mathbb N_{\geq1}$ denote the nonnegative and positive integers, respectively.