An algebra problem by Alessandro Fenu

Using AM-GM independently on the terms $a, 1$ then $b, 1$ and then $c, 1$ we obtain

$\begin{cases} \frac{a+1}{2}\geq \sqrt{a}\Rightarrow (a+1)\geq 2\cdot \sqrt{a} \\ \frac{b+1}{2}\geq \sqrt{b}\Rightarrow (b+1)\geq 2\cdot \sqrt{b} \\ \frac{c+1}{2}\geq \sqrt{c}\Rightarrow (c+1)\geq 2\cdot \sqrt{c}. \end{cases}$

Multiplying all of these together we get :

$(a+1)(b+1)(c+1)\geq 8\cdot \sqrt{a\cdot b \cdot c}$ ,

or, namely, using the fact that $(a+1)(b+1)(c+1)=8 \Rightarrow abc\leq 1$