<s>[INST] If $abc=1$, does $(a^2+b^2+c^2)^2 \geq a+b+c$? This question comes in mind while solving another question.

If $abc=1$, does $(a^2+b^2+c^2)^2 \geq a+b+c$ ?

I wonder if this question (if $abc=1$, then $a^2+b^2+c^2\ge a+b+c$) helps?
I wondered if AM-GM could help, but the extra square keeps bothering me while solving it.
Another thought: If $(a^2+b^2+c^2)^2 \geq 1$, then this statement will be true. But how can I prove it?
 [/INST] Yes, that question helps a lot! Note that by AM-GM inequality $a^2+b^2+c^2\ge 3(a^2b^2c^2)^{1/3}=3$. Since $x^2\geq x$ for all $x\geq 1$, it follows that
$$(a^2+b^2+c^2)^2\geq a^2+b^2+c^2\geq a+b+c.$$
</s>