1. Find all integer solutions to the equation
$$
\frac{x y z}{w}+\frac{y z w}{x}+\frac{z w x}{y}+\frac{w x y}{z}=4
$$

(Supplied by Wang Fumin)
1. Notice that, $\frac{x y z}{w} 、 、 \frac{y z w}{x} 、 、 \frac{z w x}{y} 、 、 \frac{w x y}{z}$ have the same sign.
Thus, they are all positive.
Hence, by the AM-GM inequality, we have
$$
\begin{array}{l}
4=\frac{x y z}{w}+\frac{y z w}{x}+\frac{z w x}{y}+\frac{w x y}{z} \\
\geqslant 4 \sqrt[4]{(x y z w)^{2}} \geqslant 4 .
\end{array}
$$

Equality holds if and only if
$$
x y z w=1,|x|=|y|=|z|=|w|=1
$$

Thus, all solutions to the equation are
$$
\begin{array}{l}
(x, y, z, w) \\
=(1,1,1,1),(-1,-1,-1,-1), \\
\quad(-1,-1,1,1),(-1,1,-1,1), \\
\quad(-1,1,1,-1),(1,-1,-1,1), \\
\\
(1,-1,1,-1),(1,1,-1,-1) .
\end{array}
$$