Cho tam giác $ABC$ thỏa mãn điều kiện :
                    $\frac{1}{{2 - cos2A}} + \frac{1}{{2 + cos2B}} + \frac{1}{{2 + cos2C}} \ge \frac{6}{5}$
             CMR tam giác $ABC$ cân với góc ở đỉnh $A = {120^0}$


Áp dụng bất đẳng thức Côsi ta có
($\frac{1}{{2 - c{\rm{os}}2A}} + \frac{1}{{2 + c{\rm{os}}2B}} + \frac{1}{{2 + c{\rm{os}}2C}}$)$[(2-cos2A+(2+cos2B)+(2+cos2C))]\geq 9$
$ \Rightarrow \frac{1}{{2 - c{\rm{os}}2A}} + \frac{1}{{2 + c{\rm{os}}2B}} + \frac{1}{{2 + c{\rm{os}}2C}} \ge \frac{9}{{6 + c{\rm{os}}2B + c{\rm{os}}2C - c{\rm{os}}2A}}           (1)$
Dấu “$=$” xảy ra ở $(1)$ khi  $cos2B=cos2C=-cos2A                                              (2)$
Ta có :
 $Cos2B+cos2C-cos2A=2cos(B+C)cos(B-C)-2cos2A+1$
                                    $=-2cos2A-2cosAcos(B-C)+1$
                                    $\begin{array}{l}
 =  - 2\left[ {c{\rm{os}}2A - \cos Ac{\rm{os}}(B - C)} \right] + 1\\
 =  - 2{\left[ {\cos A + \frac{1}{2}c{\rm{os}}(B - C)} \right]^2} + \frac{1}{2}c{\rm{o}}{{\rm{s}}^2}(B - C) + 1
\end{array}$
$ \Rightarrow c{\rm{os}}2B + c{\rm{os}}2C - c{\rm{os}}2A \le \frac{3}{2}$
$\Rightarrow 6 + c{\rm{os}}2B + c{\rm{os}}2C - c{\rm{os}}2A \le \frac{{15}}{2}                                              (3)$
Dấu “$=$” trong $(3)$ xảy ra khi
                                         $\left\{ \begin{array}{l}
c{\rm{os}}(B - C) = 1\\
\cos A + \frac{1}{2}c{\rm{os}}(B - C) = 0
\end{array} \right.$                                      (4)
Từ $(1)(3)$ suy ra :  $\frac{1}{{2 - c{\rm{os}}2A}} + \frac{1}{{2 + c{\rm{os}}2B}} + \frac{1}{{2 + c{\rm{os}}2C}} \ge \frac{6}{5}                           (5)$
Dấu “$=$” trong ($5$) xảy ra khi có dấu “$=$” ở $(1)$ và $(3)$,tương đương có $(2)(4)$
Khi đó
               $\left\{ \begin{array}{l}
c{\rm{os}}(B - C) = 1\\
c{\rm{os}}2B = c{\rm{os}}2C =  - c{\rm{os}}2A\\
\cos A = \frac{{ - 1}}{2}c{\rm{os}}(B - C)
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
B = C\\
A = {120^0}
\end{array} \right.$
Từ giả thiết suy ra trong $(5$) có dấu “$=$”.Vậy theo lập luận trên ta có (đpcm)
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