一、(Full marks 20 points) Find all positive real numbers $a$ such that the equation $x^{2}-a x+4 a=0$ has only integer roots.

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Translation:

I. (Full marks 20 points) Find all positive real numbers $a$ such that the equation $x^{2} - a x + 4 a = 0$ has only integer roots.
Let the two integer roots be $x, y(x \leqslant y)$,
then $\left\{\begin{array}{l}x+y=a>0, \\ x y=4 a>0 .\end{array}\right.$
$$
\therefore \frac{a}{2} \leqslant y \leqslant a, 4 \leqslant x \leqslant 8 \text {. }
$$

Also, it can be deduced that $x \neq 4$,
$$
\therefore a=\frac{x^{2}}{x-4} \text {. }
$$

Since $x$ is an integer,
$$
\begin{aligned}
\therefore x & =5 \text { when, } a=25, y=20 ; \\
x & =6 \text { when, } a=18, y=12 ; \\
x & =7 \text { when, } a \text { is not an integer; } \\
x & =8 \text { when, } a=16, y=8 .
\end{aligned}
$$

Thus, $a=25$ or 18 or 16 are the solutions.