4 replies to this topic

[nguyen minh hieu hp]

Cho a,b,c>0 thỏa mãn a+b+c=1.Chứng minh rằng

$\frac{bc}{\sqrt{a+bc}} +\frac{ca}{\sqrt{b+ca}} + \frac{ab}{\sqrt{c+ab}} \leq \frac{1}{2}$

[onpiece123]

Ta có :$\sum \frac{bc}{\sqrt{a+ac}}=\sum \frac{bc}{\sqrt{a^{2}+ab+bc+ac}}$

          =$\sum \frac{bc}{\sqrt{(a+c)(b+a)}}\leq\frac{1}{2} \sum \left ( \frac{bc}{a+b}+\frac{bc}{a+c} \right )$

          $\leq \frac{1}{2}$

Suy ra (đpcm)

Edited by onpiece123, 26-09-2018 - 17:04.

[nguyen minh hieu hp]

Phải là $\sum \frac{bc}{\sqrt{(a+b)(a+c)}} \leq \frac{1}{2} \sum \frac{b^{2}}{a+b}$ + $\frac{c^{2}}{a+c}$ chứ

Edited by nguyen minh hieu hp, 25-09-2018 - 12:49.

[nguyen minh hieu hp]

$\frac{bc}{\sqrt{{a+bc}}}$ mà bạn

[onpiece123]

$\frac{bc}{\sqrt{{a+bc}}}$ mà bạn

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