9 Given $x \geqslant 0, y \geqslant 0, z \geqslant 0, x+y+z=1$, find the maximum value of $x^{2} y^{2}+y^{2} z^{2}+z^{2} x^{2}+x^{2} y^{2} z^{2}$.
9. Suppose $x \geqslant y \geqslant z$, then
$$\begin{aligned}
& x^{2} y^{2}+y^{2} z^{2}+z^{2} x^{2}+x^{2} y^{2} z^{2} \\
= & x^{2} \cdot\left[(1-x)^{2}-2 y z\right]+y^{2} z^{2}+x^{2} y^{2} z^{2} \\
= & x^{2}(1-x)^{2}-2 x^{2} y z+y^{2} z^{2}+x^{2} y^{2} z^{2} \\
\leqslant & \frac{1}{16}-\left(x^{2} y z-y^{2} z^{2}\right)-\left(x^{2} y z-x^{2} y^{2} z^{2}\right) \\
\leqslant & \frac{1}{16} .
\end{aligned}$$

When $x=y=\frac{1}{2}, z=0$, the equality holds.