<s>[INST] Prove that $(a^2 + 1)(b^2 + 1)(c^2 + 1) \ge (a + b)(b + c)(c + a)$ for $a, b, c \in \mathbb{R}$ How to prove that $(a^2 + 1)(b^2 + 1)(c^2 + 1) \ge (a + b)(b + c)(c + a)$ for $a, b,  c \in \mathbb{R}$ ? I have tried AM-GM but with no effect.
 [/INST] \begin{eqnarray*}
(abc-1)^2 + \frac{1}{2}\sum_{perms} a^2(b-1)^2 \geq 0.
\end{eqnarray*}
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