$8 \cdot 90$ Let the sequence of non-zero real numbers $x_{1}, x_{2}, x_{3}, \cdots$ satisfy
$$
x_{n}=\frac{x_{n-2} x_{n-1}}{2 x_{n-2}-x_{n-1}}, n=3,4,5, \cdots
$$

Find the necessary and sufficient condition that $x_{1}$ and $x_{2}$ must satisfy so that the sequence has infinitely many integer terms.
［Solution］Since
$$
\begin{aligned}
x_{n} & =\frac{x_{n-2} x_{n-1}}{2 x_{n-2}-x_{n-1}}=\frac{x_{n-2}}{2 x_{n-2}-\frac{x_{n-3} x_{n-2}}{2 x_{n-3}-x_{n-2}}} \cdot \frac{x_{n-3} x_{n-2}}{2 x_{n-3}-x_{n-2}} \\
& =\frac{x_{n-3} x_{n-2}}{3 x_{n-3}-2 x_{n-2}},
\end{aligned}
$$

it is easy to prove by induction that for $n \geqslant 3$ we have
$$
x_{n}=\frac{x_{1} x_{2}}{(n-1) x_{1}-(n-2) x_{2}}=\frac{x_{1} x_{2}}{n\left(x_{1}-x_{2}\right)+2 x_{2}-x_{1}} .
$$

If $x_{1}=x_{2}$ is a non-zero integer, it is clear from (1) that
$$
x_{n}=x_{1}, n=1,2, \cdots
$$

i.e., every term of $\left\{x_{n}\right\}$ is a non-zero integer. Conversely, if $\left\{x_{n}\right\}$ has infinitely many integer terms, it is clear from (1) that $x_{1}=x_{2}$, and further, $x_{1}=x_{2}$ must be a non-zero integer.

In summary, the necessary and sufficient condition for the non-zero sequence $\left\{x_{n}\right\}$ to have infinitely many integer terms is that $x_{1}=$ $x_{2}$ is a non-zero integer.