Question

Cho 3 số a;b;c thỏa mãn a.b.c=1.CMR :\(\dfrac{a}{ab+a+1}\)+\(\dfrac{b}{bc+b+1}\)+\(\dfrac{c}{ac+c+1}\)=1

Answer 1

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

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

\(=\dfrac{a}{a\left(b+1+bc\right)}+\dfrac{b}{b\left(c+1+ac\right)}+\dfrac{c}{ac+c+1}\)

\(=\dfrac{1}{b+1+bc}+\dfrac{1}{c+1+ac}+\dfrac{c}{ac+c+1}\)

\(=\dfrac{ac}{abc+ac+abc.c}+\dfrac{1}{ac+c+1}+\dfrac{c}{ac+c+1}\)

\(=\dfrac{ac}{1+ac+c}+\dfrac{1}{ac+c+c}+\dfrac{c}{ac+c+1}\)

\(=\dfrac{ac+1+c}{ac+c+1}=1\) (đpcm)