Lời giải:

Ta có:

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

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

Thay \(abc=1\) ta có:

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

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

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

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

\(=\dfrac{a}{ab+a+1}+\dfrac{ab}{1+ba+a}+\dfrac{1}{a+1+ab}=\dfrac{ab+a+1}{ab+a+1}=1\)

Bài toán cơ bản:

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

Bunhiacopxki:

\(\left(a+b+c\right)\left(\dfrac{a}{\left(ab+a+1\right)^2}+\dfrac{b}{\left(bc+b+1\right)^2}+\dfrac{c}{\left(ac+c+1\right)^2}\right)\ge\left(\dfrac{a}{ab+a+1}+\dfrac{b}{bc+b+1}+\dfrac{c}{ac+c+1}\right)^2=1\)

\(\Rightarrow\dfrac{a}{\left(ab+a+1\right)^2}+\dfrac{b}{\left(bc+b+1\right)^2}+\dfrac{c}{\left(ac+c+1\right)^2}\ge\dfrac{1}{a+b+c}\) (đpcm)

Dấu "=" xảy ra khi \(a=b=c\)

Cách 2:

Do \(abc=1\), đặt \(\left(a;b;c\right)=\left(\dfrac{x}{y};\dfrac{y}{z};\dfrac{z}{x}\right)\)

Ta có \(\dfrac{a}{\left(ab+a+1\right)^2}=\dfrac{\dfrac{x}{y}}{\left(\dfrac{x}{z}+\dfrac{x}{y}+1\right)^2}=\dfrac{\dfrac{x}{y}.y^2z^2}{\left(xy+yz+zx\right)^2}=\dfrac{xyz^2}{\left(xy+yz+zx\right)^2}\)...

Từ đó, BĐT cần chứng minh trở thành:

\(\dfrac{xyz^2}{\left(xy+yz+zx\right)^2}+\dfrac{x^2yz}{\left(xy+yz+zx\right)^2}+\dfrac{xy^2z}{\left(xy+yz+zx\right)^2}\ge\dfrac{1}{\dfrac{x}{y}+\dfrac{y}{z}+\dfrac{z}{x}}\)

\(\Leftrightarrow xyz\left(x+y+z\right)\left(\dfrac{x}{y}+\dfrac{y}{z}+\dfrac{z}{x}\right)\ge\left(xy+yz+zx\right)^2\)

\(\Leftrightarrow\left(x+y+z\right)\left(x^2z+y^2x+z^2y\right)\ge\left(xy+yz+zx\right)^2\)

Thật vậy, áp dụng BĐT Bunhiacopxki:

\(\left(z+x+y\right)\left(x^2z+y^2x+z^2y\right)\ge\left(\sqrt{zx^2z}+\sqrt{xy^2x}+\sqrt{yz^2y}\right)^2=\left(xy+yz+zx\right)^2\) (đpcm)

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

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

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

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

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

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

\(=\dfrac{ac+1+c}{ac+c+1}=1\) (đpcm)

Ta có :

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

\(A=\dfrac{a}{ab+a+1}+\dfrac{ab}{abc+ab+a}+\dfrac{abc}{aabc+abc+ab}\)

\(A=\dfrac{a}{ab+a+1}+\dfrac{ab}{1+ab+a}+\dfrac{1}{a+1+ab}\)

\(A=\dfrac{a+ab+1}{ab+a+1}\)

\(\Rightarrow A=1\left(đpcm\right)\)

kiểm tra lại đề đi bạn

\(B=\dfrac{1}{1+a+ab}+\dfrac{1}{1+b+bc}+\dfrac{1}{1+c+ca}=\dfrac{1}{1+a+ab}+\dfrac{a}{a+ab+abc}+\dfrac{ab}{ab+abc+abca}\)

vì abc =1 nên B=\(\dfrac{1}{1+a+ab}+\dfrac{a}{a+ab+1}+\dfrac{ab}{ab+1+a}=\dfrac{1+a+ab}{a+1+ab}=1\)

chúc bạn học tót ^^

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

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

\(A=\dfrac{a^2bc}{ab\left(1+ac+c\right)}+\dfrac{b}{b\left(c+1+ac\right)}+\dfrac{c}{ac+c+1}\)

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

\(A=1\)

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

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

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

\(A=1\)