Đề sai toàn tập, dấu "=" rồi còn tính gì nữa ????

Đề bài vẫn chưa đúng nhé, đúng ra phải là \(M=\frac{1}{ab+a+1}+\frac{1}{bc+b+1}+\frac{1}{abc+ac+c}\)

Ta có : \(M=\frac{1}{ab+a+1}+\frac{1}{bc+b+1}+\frac{1}{abc+ac+c}\)

\(=\frac{1}{ab+a+1}+\frac{a}{abc+ab+a}+\frac{ab}{a^2b^2c+a^2bc+abc}\)

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

Vì a.b.c = 1

Ta có :

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

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

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

\(\Rightarrow M=\frac{c}{1+ac+c}+\frac{ca}{c+ca+1}+\frac{ac}{ac+c+1}\)

\(\Rightarrow M=\frac{c+2ac}{1+ac+c}\)

\(\Rightarrow M=\frac{bc+2}{b+1+bc}\)

\(\Rightarrow M=\frac{bc++1+abc}{b+1+bc}\)

-_-

Năm ngoái a lm ko ra thế này đâu

Đề bài sai nhé, chỗ \(\frac{1}{b.c+b+1}\) phải là \(\frac{b}{b.c+b+1}\) ms đúng

Ta có:

\(\frac{1}{a.b+a+1}+\frac{b}{b.c+b+1}+\frac{1}{a.b.c+b.c+b}=\frac{a.b.c}{a.b+a+a.b.c}+\frac{b}{b.c+b+1}+\frac{1}{1+b.c+b}\)

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

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

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

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

\(=\frac{abc}{ab\left(c+ac+1\right)}+\frac{abc}{b\left(ac+1+c\right)}+\frac{1}{1+c+ac}\)

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

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

= 1

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

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

Theo bài ra ta có: a.b.c = 1

    =>  a=1;b=1;c=1

Ta có: A = \(\frac{1}{a.b+a+1}\)\(+\frac{1}{b.c+b+1}+\frac{1}{c.a+c+1}\)\(=\frac{1}{1.1+1+1}+\frac{1}{1.1+1+1}\)\(+\frac{1}{1.1+1+1}\)

             \(=\frac{1}{1+1+1}+\frac{1}{1+1+1}+\frac{1}{1+1+1}\)\(=\frac{1}{3}+\frac{1}{3}+\frac{1}{3}=\frac{3}{3}=1\)

Vậy A = 1