<s>[INST] Inequality. $a^2+b^2+c^2 \geq a+b+c$ Let $a,b,c$ be positive real numbers such that $abc=1$. Prove that 
$a^2+b^2+c^2 \geq a+b+c$. 
Thanks
 [/INST] We have:
$$\sqrt{\frac{a^2+b^2+c^2}{3}}\underset{Q\geq G}{\geq} \sqrt[3]{abc} = 1$$
But $\sqrt{x} \geq 1 \iff x \geq 1 \iff x\geq \sqrt{x}$ so:
$$\frac{a^2+b^2+c^2}{3} \geq\sqrt{\frac{a^2+b^2+c^2}{3}}\underset{Q\geq A}{\geq}\frac{a+b+c}{3}$$
$$a^2+b^2+c^2 \geq a+b+c$$
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