Question

Cmr: Nếu 1/a + 1/b + 1/c = 1/a+b+c thì (a+b) *(a+c) *(b+c) =0

Answer 1

\(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=\dfrac{1}{a+b+c}\)

\(\Leftrightarrow\dfrac{bc+ac+ab}{abc}=\dfrac{1}{a+b+c}\)

\(\Leftrightarrow\left(a+b+c\right)\left(ab+bc+ac\right)=abc\)

\(\Leftrightarrow a^2b+abc+a^2c+ab^2+b^2c+abc+bc^2+ac^2=0\)

\(\Leftrightarrow ab\left(a+b+c\right)+bc\left(a+b+c\right)+ac\left(a+c\right)=0\)

\(\Leftrightarrow\left(a+b+c\right)b\left(a+c\right)+ac\left(a+c\right)=0\)

\(\Leftrightarrow\left(a+c\right)\left(ab+b^2+bc+ac\right)=0\)

\(\Leftrightarrow\left(a+c\right)\left(a+b\right)\left(b+c\right)=0\)