<s>[INST] Maximum of given expression? Suppose $a,b,c>0$ and further that $a^{2}  +  b^{2} + c^{2}=2abc + 1 $.
The problem is to find $\max  \big(a-2bc\big)  \big(b-2ca\big)  \big(c-2ab\big) $.
Give me some help. I've tried $X=a-2bc$, $Y=b-2ca$, $Z=c-2ab$ 
which yields $X^2 + Y^2 + Z^2 = 1-2XYZ$, but $\frac12$ is not the maximum because 
$XYZ=0$.
Can someone give me an elegant solution? 
 [/INST] let $$x=a-2bc,y=b-2ca,z=c-2ab$$
since
$$a^2+b^2+c^2=2abc+1\Longrightarrow x^2+y^2+z^2+2xyz=1$$
so we only find $xyz$ maximu,since
$$1=x^2+y^2+z^2+2xyz\ge 3(x^2y^2z^2)^{\frac{1}{3}}+2xyz\Longrightarrow xyz\le\dfrac{1}{8}$$
so
$$(a-2bc)(b-2ac)(c-2ab)\le\dfrac{1}{8}$$
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