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[Clwn]

Cho a,b,c là 3 cạnh tam giác. Tìm max : 

$\sum \sqrt{1-\frac{a}{b+c}}$

 

Edited by Clwn, 03-06-2017 - 21:33.

[tuaneee111]

Có bổ đề $Nesbit$: $$\frac{a}{{b + c}} + \frac{b}{{c + a}} + \frac{c}{{a + b}} \geqslant \frac{3}{2} \Leftrightarrow \frac{1}{2}\sum\limits_{cyc} {\frac{{{{\left( {a - b} \right)}^2}}}{{\left( {a + c} \right)\left( {b + c} \right)}}}  \geqslant 0$$Theo $Cauchy - Schwarz$ ta có: $$\sum\limits_{cyc} {\sqrt {1 - \frac{a}{{b + c}}} }  \leqslant \sqrt {3\left( {3 - \sum\limits_{cyc} {\frac{a}{{b + c}}} } \right)}  \leqslant \frac{{3\sqrt 2 }}{2}$$