18.2. (Jury, Sweden, 79). Find the maximum value of the product $x^{2} y^{2} z^{2} u$ given that $x, y, z, u \geqslant 0$ and

$$
2 x+x y+z+y z u=1
$$
18.2. Using the theorem of means, we have

$$
\sqrt[4]{2 x^{2} y^{2} z^{2} u}=\sqrt[4]{2 x \cdot x y \cdot z \cdot y z u} \leqslant \frac{2 x+x y+z+y z u}{4}=\frac{1}{4}
$$

i.e., $x^{2} y^{2} z^{2} u \leqslant 1 / 512$. Equality is achieved if $2 x=x y=z=y z u=1 / 4$, i.e., when $x=1 / 8, y=2, z=1 / 4, u=1 / 2$. Thus, the maximum value is $1 / 512$.