## Task A-2.1.

How many integers $n$ are there for which the inequality $x^{2}+n x+100>0$ holds for all real numbers $x$?
## Solution.

The discriminant of the quadratic function $f(x)=x^{2}+n x+100$ is $D=n^{2}-400$.

Since the leading coefficient is positive, for the desired inequality to hold, it is necessary that $D<0$.

Thus, the inequality $n^{2}<400$ must hold, i.e., $|n|<20$.

Finally, the integers $-19,-18, \ldots,-1,0,1,2, \ldots, 19$ satisfy the condition of the problem, which totals 39.

Note: For the last 1 point in the scoring scheme, it is not necessary to list the numbers $n$.