1. Given $0<x, y<1$. Find the maximum value of
$$
\frac{x y(1-x-y)}{(x+y)(1-x)(1-y)}
$$

(Liu Shixiong)
1. When $x=y=\frac{1}{3}$, the value of the algebraic expression is $\frac{1}{8}$.

Below is the proof: For any $0 < x, y, z < 1$ such that $x + y + z = 1$. Then
$$
\begin{aligned}
& \frac{x y(1-x-y)}{(x+y)(1-x)(1-y)} \\
= & \frac{x y z}{(x+y)(y+z)(z+x)} \\
\leqslant & \frac{x y z}{2 \sqrt{x y} \cdot 2 \sqrt{y z} \cdot 2 \sqrt{z x}}=\frac{1}{8} .
\end{aligned}
$$

In conclusion, the maximum value sought is $\frac{1}{8}$.