Question

[Nguyễn Bá Tùng]

\(Cho:a+b+c=2016;\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}=\frac{1}{2016}\)

Tính:\(S=\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}\)

Answer 1

[Trịnh Thành Công]

         Vì \(a+b+c=2016\Rightarrow a=2016-\left(b+c\right);b=2016-\left(a+c\right);c=2016-\left(a+b\right)\)

Ta có:\(S=\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}\) 

           \(S=\frac{2016-\left(b+c\right)}{b+c}+\frac{2016-\left(a+c\right)}{a+c}+\frac{2016-\left(a+b\right)}{a+b}\)

           \(S=\frac{2016}{b+c}-1+\frac{2016}{a+c}-1+\frac{2016}{a+b}-1\)

           \(S=2016.\left(\frac{1}{b+c}+\frac{1}{a+c}+\frac{1}{a+b}\right)-3\)

           \(S=2016.\frac{1}{2016}-3\)

          \(S=-2\)