<s>[INST] Maximize $(a-1)(b-1)(c-1)$ knowing that : $a+b+c=abc$. If : $a,b,c>0$, and : $a+b+c=abc$, then find the maximum of $(a-1)(b-1)(c-1)$.

I noted that : $a+b+c\geq 3\sqrt{3}$, I believe that the maximum is at : $a=b=c=\sqrt{3}$. (Can you give hints).
 [/INST] Let $a+b+c=3u$, $ab+ac+bc=3v^2$ and $abc=w^3$.
Hence, the condition does not depend on $v^2$ and we need to find a maximum of
$$abc-ab-ac-bc+a+b+c-1$$ or
$$6u-3v^2-1,$$
which is linear function of $v^2$, 
which says that it's enough to find a maximum for an extremal value of $v^2$,
which happens for equality case of two variables.
Let $b=a$.
Hence, $c=\frac{2a}{a^2-1}$, where $a>1$ and we need to find $\max\limits_{a>1}f$, where
$$f(a)=(a-1)^2\left(\frac{2a}{a^2-1}-1\right)^2,$$ 
which gives that 
$$\max\limits_{a>1}f=-10+6\sqrt3.$$
Done!
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