(x_i )_(1≤i≤n) n real number positifs wish verfy Σ_(i=1) ^(i=n) x_i =1 prove that Σ_(1≤i≤n) x


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Question Number 26244 by abdo imad last updated on 22/Dec/17

${(x_{i})}_{1 ⩽ i ⩽ n} n r e a l n u m b e r p o s i t i f s w i s h v e r f y \sum_{i = 1}^{i = n} x_{i} = 1$ $p r o v e t h a t \sum_{1 ⩽ i ⩽ n} x_{i}^{2} ⩾ \frac{1}{n} .$
Commented by abdo imad last updated on 28/Dec/17

$f o r a l l s e c o n s e s o f r e a l n u m b e r s {(a_{i})}_{1 ⩽ i ⩽ n} a n d {(b_{i})}_{1 ⩽ i ⩽ n} p o s i t i f s$ $\sum_{i = 1}^{i = n} a_{i} b_{i} ⩽ {(\sum_{i = 1}^{i = n} a_{i}^{2})}^{\frac{1}{2}} . {(\sum_{i = 1}^{i = n} b_{i}^{2})}^{\frac{1}{2}} (h o l d e r i n e q u a l i t y)$ $l e t t a k e b_{i} = 1 \Rightarrow \sum_{i = 1}^{i = n} a_{i} ⩽ \sqrt{n} {(\sum_{i = 1}^{i = n} a_{i}^{2})}^{\frac{1}{2}} a n d f o r a_{i} x_{i}$ $w e o b t a i n {(\sum_{i =}^{i = n} x_{i})}^{2} ⩽ n (\sum_{i = 1}^{i = n} x_{i}^{2})$ $\Leftrightarrow 1 ⩽ n (\sum_{i = 1}^{i = n} x_{i}^{2}) \Rightarrow \sum_{i = 1}^{i = n} x_{i}^{2} ⩾ \frac{1}{n} .$
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