\(A=\frac{1}{a+a+a+a+b+c}+\frac{1}{a+b+b+b+b+c}+\frac{1}{a+b+c+c+c+c}\)
\(\Rightarrow A\le\frac{1}{36}\left(\frac{4}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{a}+\frac{4}{b}+\frac{1}{c}+\frac{1}{a}+\frac{1}{b}+\frac{4}{c}\right)\)
\(\Rightarrow A\le\frac{1}{6}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\le\frac{1}{6}\)
Dấu "=" xảy ra khi \(a=b=c=3\)