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Let $a$, $b$ and $c$ be positive real numbers such that $abc = 1$. Prove that $$\displaystyle \sum_{cyc} \sqrt{\frac{a}{a+8}} \geq 1.$$

I have tried using some substitutions but still cannot do it. Please suggest.

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merged by Jyrki Lahtonen Jul 19 '17 at 8:51

This question was merged with Prove the inequality $\sqrt\frac{a}{a+8} + \sqrt\frac{b}{b+8} +\sqrt\frac{c}{c+8} \geq 1$ with the constraint $abc=1$ because it is an exact duplicate of that question.

  • $\begingroup$ I don't think that it's not so duplicate because at least one of my solutions different than solution in the linked topic. $\endgroup$ – Michael Rozenberg Jul 19 '17 at 5:42
  • $\begingroup$ @MichaelRozenberg BTW by using (at)Martin instead of (at)MartinR you pinged me - my guess is that this was not what you intended. I think that you have participated in a few discussions about duplicates - so you already know that closing a post as a duplicated does not delete the answers. $\endgroup$ – Martin Sleziak Jul 19 '17 at 8:13
  • $\begingroup$ @Martin Sleziak See here math.stackexchange.com/questions/2347703 With duplicates sometimes happens the similar thing. I'll get an example. I did not ping you. Why do you speak to me with this tone? $\endgroup$ – Michael Rozenberg Jul 19 '17 at 8:17