MathNet 2011 Question 51

Positive integers $a_1,a_2,\ldots$ satisfy $(m+n)\mid(a_m+a_n)$ for all positive integers $m,n$, and $a_n\le n^{100}$ for every positive integer $n$. Prove there is a polynomial $f$ with $f(n)=a_n$ for all positive integers $n$.