<s>[INST] Minimum value of: $x^7(yz-1)+y^7(zx-1)+z^7(xy-1)$ 
$x$, $y$ and $z$ are positive reals such that $x+y+z=xyz$. Find the minimum value of:
  $$x^7(yz-1)+y^7(zx-1)+z^7(xy-1)$$

I put it in the form $x^6y +x^6z+y^6x+y^6z+z^6x +z^6y$. I tried AM-GM but it's not helping.
 [/INST] $$x^6y +x^6z+y^6x+y^6z+z^6x +z^6y \ge 6 (xyz)^{7/3}$$
Also $xyz = x+y+z \ge 3 \sqrt[3]{xyz} \implies xyz \ge 3\sqrt3$.
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