Question

CMR: \(\frac{a}{ab+a+1}=\frac{b}{bc+b+1}=\frac{c}{ca+c+1}=1\) biết abc=1

Answer 1

\(A=\frac{a}{ab+a+1}+\frac{b}{bc+b+1}+\frac{c}{ca+c+1}=1\)

Từ \(abc=1\Rightarrow a=\frac{1}{bc}\) thay vào ta có:

\(A=\frac{\frac{1}{bc}}{\frac{1}{bc}\cdot b+\frac{1}{bc}+1}+\frac{b}{bc+b+1}+\frac{c}{c\cdot\frac{1}{bc}+c+1}\)

\(=\frac{\frac{1}{bc}}{\frac{1}{c}+\frac{1}{bc}+1}+\frac{b}{bc+b+1}+\frac{c}{\frac{1}{b}+c+1}\)

\(=\frac{1}{bc\left(\frac{1}{bc}+\frac{1}{c}+1\right)}+\frac{b}{bc+b+1}+\frac{bc}{bc+b+1}\)

\(=\frac{1}{bc+b+1}+\frac{b}{bc+b+1}+\frac{bc}{bc+b+1}\)

\(=\frac{1+b+bc}{bc+b+1}=1\)