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Assume that $a_1, \dots,a_n $ and $b_1, \dots,b_n$ are $2n$ non-negative real numbers.

We have $$\sum_{i=1}^na_i = \sum_{i=1}^nb_i$$

We're to prove that $$\sqrt2 \sum_{i=1}^n (\sqrt{a_i}-\sqrt {b_i})^2 \ge \sum_{i=1}^n|a_i-b_i|.$$ Can anyone help!

I encountered it while i was surfing in olympiad section of artofproblemsolving and found it interesting , since my olympiads are very near so I tried to solve this inequality but failed to do so. I tried to apply AM-GM-HM Inequality but it doesnt works here & also tried Cauchy-Schwarz & Tchebycheff's Inequality too but with no success . I just cant figure out what to keep as variables in the formulae stated above .

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    Are you kidding me? You are posting a link to your actual question? Please improve this or it's going to end up being deleted as a low quality post.2012-11-29
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    But dude , i don't know how to add readibility to my question in math exchange here , so i had to post a link , and its question which is important not just link or unlinked post . @SimonHayward2012-11-29
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    At least post what the question is. It doesn't have to be perfect, some else can edit it.2012-11-29
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    @SimonHayward : Okay , i will edit it right now but plus undo your -1 :P2012-11-29
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    Right. I didn't -1 it. But I will now.2012-11-29
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    @SimonHayward : wow , thnx anyways for the tip .2012-11-29
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    Undid it and voted up. Thanks for making it easy for people to see what your question is and answer it.2012-11-29
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    Now that the question is readable... Welcome to math.SE: since you are new, you might want to know that, in order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level. Please consider rewriting your post.2012-11-29
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    @SimonHayward : thnxks :)2012-11-30
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    I think it would be nice to mention the it is posted at AoPS, too: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=51&t=5090322012-11-30
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    The original question consisted of only a ink to that question.2012-11-30

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If $n=2$ and $a_1=b_2=100, a_2=b_1=121$, then the inequality becomes $2\sqrt{2}\ge 42$, which is false. So the inequality does not actually hold.