# Inequality Question

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 .

• 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. – Simon Hayward Nov 29 '12 at 20:28
• At least post what the question is. It doesn't have to be perfect, some else can edit it. – Simon Hayward Nov 29 '12 at 20:36
• Right. I didn't -1 it. But I will now. – Simon Hayward Nov 29 '12 at 20:38
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.