<s>[INST] The sum of three square roots bounded below by $\sqrt{82}$ 
Let $a,b,c >0$ and $a+b+c \le  1$. Prove: $$\sqrt{a^2+1/a^2}+\sqrt{b^2+1/b^2}+\sqrt{c^2+1/c^2} \ge \sqrt{82}$$

Progress: I tried to have 3 vectors $(a,1/a)$, $(b,1/b)$ and $(c,1/c)$, played around with the vector inequalities but no success.
 [/INST] In the first proof only condition $a+b+c=1$ is used. What if $a+b+c<1$? Also a simpler alternative way to show $\frac{1}{a}+ \frac{1}{b}+\frac{1}{c}\geq 9$ when $a+b+c\leq 1$ is the Cauchy-Schwarz inequality:
$9=\|( \sqrt{a}, \sqrt{b}, \sqrt{c})\cdot(\frac{1}{\sqrt{c}}, \frac{1}{\sqrt{b}}, \frac{1}{\sqrt{c}})\|^2\leq (a+b+c)(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})$.
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