1. (12th "Hope Cup" Invitational Training Question) In the quadratic trinomial $a x^{2}+b x+c$, $a>100$, how many integer values of $x$ at most can make the absolute value of the quadratic trinomial not exceed 50?
1. Let $f(x)=a x^{2}+b x+c, a>100$, then there are at most two different integers $x$ such that $|f(x)| \leqslant 50$ holds, for example, $f(x)=101 x^{2}-52$. When $x= \pm 1$, $|f(x)|=49 \leqslant 50$.

In fact, if not, then there exist at least 3 integers $x_{1}100$ contradiction, hence there are at most two different integers $x$ such that $|f(x)| \leqslant 50$ holds.