*Từ abc=1 => a;b;c khác 0

Khi đó : \(\frac{1}{ab+a+1}\) = \(\frac{1}{ab+a+1}\) .\(\frac{bc}{bc}\) = \(\frac{bc}{ab.bc+abc+bc}\) = \(\frac{bc}{abc.b+abc+bc}\) = \(\frac{bc}{bc+b+1}\)

(do abc=1)

*Do abc = 1 => \(\frac{1}{abc+bc+b}\) = \(\frac{1}{bc+b+1}\)

Khi đó : \(\frac{1}{ab+a+1}\) + \(\frac{b}{bc+b+1}\) + \(\frac{1}{abc+bc+b}\)

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

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

Hay \(\frac{1}{ab+a+1}\) + \(\frac{b}{bc+b+1}\) + \(\frac{1}{abc+bc+b}\) = 1 (đpcm).

*Chú ý : Đây là phương pháp thế số bởi chữ !

Lời giải:
Dựa vào điều kiện $abc=1$ ta có:

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

\(=\frac{1}{ab+a+1}+\frac{a}{abc+ab+a}+\frac{ab}{ab+ab.ca+ab.c}\)

\(=\frac{1}{ab+a+1}+\frac{a}{1+ab+a}+\frac{ab}{ab+a+1}=\frac{1+a+ab}{ab+a+1}=1\)

Ta có đpcm.

Ta có: \(a.b.c=1\)

\(=\frac{1}{ab+a+1}+\frac{1}{bc+b+1}+\frac{1}{abc+bc+b}\)

\(=\frac{1}{ab+a+1}+\frac{ab}{abc+ab+a}+\frac{a}{abc.a+abc+ab}\)

\(=\frac{1}{ab+a+1}+\frac{ab}{1+ab+a}+\frac{a}{a+1+ab}\)

\(=\frac{1+ab+a}{1+ab+a}\)

\(=1.\)

\(\Rightarrow\frac{1}{ab+a+1}+\frac{1}{bc+b+1}+\frac{1}{abc+bc+b}=1\left(đpcm\right).\)

Chúc bạn học tốt!

Đặt \(T=\frac{1}{1+a+ab}+\frac{1}{1+b+bc}+\frac{1}{1+c+ac}\) (*)

Ta có: \(abc=1\Rightarrow c=\frac{1}{ab}\).Thay vào (*) ta có:

\(T=\frac{1}{1+a+ab}+\frac{1}{1+b+\frac{1}{a}}+\frac{1}{1+\frac{1}{ab}+\frac{1}{b}}\)

\(=\frac{1}{1+a+ab}+\frac{1}{\frac{a+ab+1}{a}}+\frac{1}{\frac{ab+1+a}{ab}}\)

\(=\frac{1}{1+a+ab}+\frac{a}{a+ab+1}+\frac{ab}{ab+1+a}\)

\(=\frac{1+a+ab}{1+a+ab}=1=VP\) (Đpcm)