52. Let positive real numbers $x, y, z$ satisfy $x+y+z=x y z$, find the minimum value of $x^{7}(y z-1)+y^{7}(z x-1)+$ $z^{7}(x y-1)$. (2003 China National Training Team Problem)
52. Since $x, y, z > 0$, and $x + y + z = xyz$, we have $z(xy - 1) = x + y$, similarly $y(zx - 1) = z + x$, and $x(yz - 1) = y + z$. By the AM-GM inequality, we have $xyz = x + y + z \geqslant 3 \cdot \sqrt[3]{xyz} \Rightarrow xyz \geqslant 3 \sqrt{3}$, with equality holding if and only if $x = y = z = \sqrt{3}$. Therefore,
$$\begin{array}{l}
x^{7}(yz - 1) + y^{2}(zx - 1) + z^{7}(xy - 1) = \\
x^{6}(y + z) + y^{6}(z + x) + z^{6}(x + y) \geqslant \\
6 \sqrt[6]{x^{6} y \cdot x^{6} z \cdot y^{6} z \cdot y^{6} x \cdot z^{6} x \cdot z^{6} y} \geqslant 6 \sqrt[6]{x^{14} \cdot y^{14} \cdot z^{14}} \geqslant 162 \sqrt{3}
\end{array}$$