HMMT February

Since $a-b \mid P(a)-P(b)$ for any integer polynomial $P$ and integers $a$ and $b$, we require that $10 \mid P(0)-P(-10)$ and $9 \mid P(0)-P(9)$. So, we are looking for an integer $a$ near 0 for which $$a \equiv 5 \bmod 10, a \equiv 2 \bmod 9$$ The smallest such positive integer is 65, and the smallest such negative integer is -25. This is achievable, for example, if $P(x)=2 x^{2}+3 x-25$, so our answer is 25.