9. (16 points) Given the function
$$
f(x)=10 x^{2}+m x+n(m, n \in \mathbf{Z})
$$

has two distinct real roots in the interval $(1,3)$. Find the maximum possible value of $f(1) f(3)$.
$$
\begin{array}{l}
\text { II. 9. Let } f(x)=10\left(x-x_{1}\right)\left(x-x_{2}\right) \text {. Then } \\
f(1) f(3) \\
=100\left(1-x_{1}\right)\left(1-x_{2}\right)\left(3-x_{1}\right)\left(3-x_{2}\right) \\
\leqslant 100\left(\frac{x_{1}-1+3-x_{1}}{2}\right)^{2}\left(\frac{x_{2}-1+3-x_{2}}{2}\right)^{2} \\
=100,
\end{array}
$$

When $x_{1}=x_{2}=2$, the equality holds in the above expression.
Thus, $f(1) f(3)<100$.
Since $m, n \in \mathbf{Z}$, it follows that $f(1) f(3) \in \mathbf{Z}$.
Therefore, $f(1) f(3) \leqslant 99$.
Take $f(x)=10 x^{2}-39 x+38$, whose roots are $2$ and $\frac{19}{10}$, satisfying the conditions. In this case,
$$
f(1)=9, f(3)=11, f(1) f(3)=99 \text {. }
$$