\(A=\dfrac{ab}{a+c+2b+b+c}+\dfrac{bc}{a+b+2c+c+a}+\dfrac{ca}{c+b+2a+a+b}\)
\(A\le\dfrac{ab}{9}\left(\dfrac{1}{a+c}+\dfrac{1}{2b}+\dfrac{1}{b+c}\right)+\dfrac{bc}{9}\left(\dfrac{1}{a+b}+\dfrac{1}{2c}+\dfrac{1}{c+a}\right)+\dfrac{ca}{9}\left(\dfrac{1}{b+c}+\dfrac{1}{2a}+\dfrac{1}{a+b}\right)\)
\(A\le\dfrac{a+b+c}{18}+\dfrac{1}{9}\left(\dfrac{bc+ca}{a+b}+\dfrac{ab+bc}{c+a}+\dfrac{ab+ca}{b+c}\right)\)
\(A\le\dfrac{a+b+c}{18}+\dfrac{1}{9}\left(a+b+c\right)=1\)
Dấu "=" xảy ra khi \(a=b=c=2\)
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