<s>[INST] problem on solving equations with three variables Find all positive real numbers $x,y,z$ which satisfy the following equations simultaneously.
    $x^3+y^3+z^3=x+y+z$
    $x^2+y^2+z^2=xyz$
 [/INST] The equations do not have positive solutions. It follows directly from AM-GM inequalities.
Recall that $x,y,z> 0$ then
$x+y+z=x^3+y^3+z^3\geq3xyz=3(x^2+y^2+z^2)$
$3(x^2+y^2+z^2)-(x+y+z)^2=(x-y)^2+(y-z)^2+(z-x)^2\geq 0$
Hence $x+y+z\geq (x+y+z)^2$
then $x+y+z\leq 1$.
But $x,y,z>0$, so $x,y,z$ are strictly smaller than 1.
So $x^3<x$, $y^3<y$, $z^3<z$, which means $x^3+y^3+z^3<x+y+z$, a contradiction!
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