3
$\begingroup$

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 .

  • 1
    The original question consisted of only a ink to that question.2012-11-30

1 Answers 1

6

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.