<s>[INST] Inequality with constraint I've been trying to prove the following inequality without success.
 For $a,b,c \in \mathbb{R}$ such that $abc=1$, prove that:
 $$\frac{1}{a^2+a+1}+\frac{1}{b^2+b+1} + \frac{1}{c^2+c+1} \geq 1$$
 [/INST] Multiply both sides of the inequality by $(a^2 + a + 1)(b^2 + b + 1)(c^2 + c + 1)$ to get
$$
\begin{align*}
&(b^2 + b + 1)(c^2 + c + 1) + (a^2 + a + 1)(c^2 + c + 1) + (a^2 + a + 1)(b^2 + b + 1)\\ 
\geq &(a^2 + a + 1)(b^2 + b + 1)(c^2 + c + 1).
\end{align*}
$$
Suffer through the algebra (or enlist Maple's help) to simplify this to
$$
a^2 + b^2 + c^2 - ab - ac - bc \geq 0.
$$
Note that during the simplification, we use the constraint $abc = 1$ several times.
Now use F'OlaYinka's observation: The inequality above is true if and only if
$$
2(a^2 + b^2 + c^2 - ab - ac - bc) \geq 0.
$$
The lefthand side of this expression can be factored as
$$
(a-b)^2 + (a-c)^2 + (b-c)^2.
$$
Since this is the sum of three positive terms, it is surely greater than or equal to zero, which completes the proof.
</s>