N1. Let $\mathbb{Z}_{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ such that
$$
m^{2}+f(n) \mid m f(m)+n
$$
for all positive integers $m$ and $n$.
Answer. $f(n)=n$.
Solution 1. Setting $m=n=2$ tells us that $4+f(2) \mid 2 f(2)+2$. Since $2 f(2)+22 m^{2}$ be a prime number. Note that $p>m f(m)$ also. Plugging in $n=p-m f(m)$ we learn that $m^{2}+f(n)$ divides $p$. Since $m^{2}+f(n)$ cannot equal 1 , it must equal $p$. Therefore $p-m^{2}=f(n) \mid n=p-m f(m)$. But $p-m f(m)|f(n)-n|$. It follows that $f$ is the identity function.