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For postive integer $a, b, c $ prove the following inequality $$\frac{1}{\sqrt{b+\frac{1}{a}+\frac{1}{2}}} + \frac{1}{\sqrt{c+\frac{1}{b}+\frac{1}{2}}} + \frac{1}{\sqrt{a+\frac{1}{c}+\frac{1}{2}}} > \sqrt{2}$$ How we can prove this inequality?

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    $\begingroup$ The title is missing a piece and the question body too: what are the assumptions on $a,b,c$? $\endgroup$ – Jack D'Aurizio Sep 21 '15 at 13:48
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    $\begingroup$ Please provide the missing parts. $\endgroup$ – Kushal Bhuyan Sep 21 '15 at 13:53
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    $\begingroup$ @AdityaAgarwal Not dealing with limits here. $1/0$ is undefined. $\endgroup$ – Gummy bears Sep 21 '15 at 14:00
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    $\begingroup$ That is what I am saying, there have to be some assumptions. $\endgroup$ – Aditya Agarwal Sep 21 '15 at 14:00
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    $\begingroup$ @AdityaAgarwal He's obviously missin the sum of a, b, c $\endgroup$ – Gummy bears Sep 21 '15 at 14:01
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$$\frac{1}{\sqrt{b+\frac{1}{a}+\frac{1}{2}}} + \frac{1}{\sqrt{c+\frac{1}{b}+\frac{1}{2}}} + \frac{1}{\sqrt{a+\frac{1}{c}+\frac{1}{2}}} > \sqrt{2}$$ By$ AM-GM$ $$\sum_{cyc}\frac{1}{\sqrt{b+\frac{1}{a}+\frac{1}{2}}}=\sum_{cyc}\frac{\sqrt2}{2\cdot\frac{1}{\sqrt2}\sqrt{b+\frac{1}{a}+\frac{1}{2}}}\geq\sum_{cyc}\frac{\sqrt2}{b+\frac{1}{a}+1}=\sqrt2$$

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  • $\begingroup$ How does the last equality hold? ;) $\endgroup$ – Khue Sep 23 '15 at 22:54

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