<s>[INST] Find Max value of this expression: $P=(x-2yz)(y-2zx)(z-2xy)$ Let $x$, $y$ and $z$ be positives and satisfying $x^2+y^2+z^2=2xyz+1$ . 
Find a maximum of this expression:
$$P=(x-2yz)(y-2zx)(z-2xy).$$
 [/INST] Set $y=x$ and $z=1$. Then $x^2+y^2+z^2=2x^2+1=2xyz+1$ and
$$(x-2yz)(y-2zx)(z-2xy)=(-x)(-x)(1-2x^2)=x^2(1-2x^2)$$
which clearly has no minimum...
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