Bài này phân tích hết ra.
Đặt \(A=\frac{1}{x+xy+1}+\frac{1}{y+yz+1}+\frac{1}{z+xz+1}\)
\(=\frac{\left(x+xy+1\right)\left(y+yz+1\right)+\left(x+xy+1\right)\left(z+xz+1\right)+\left(y+yz+1\right)\left(z+xz+1\right)}{\left(x+xy+1\right)\left(y+yz+1\right)\left(z+xz+1\right)}\)
Đặt \(M=\left(x+xy+1\right)\left(y+yz+1\right)+\left(x+xy+1\right)\left(z+xz+1\right)+\left(y+yz+1\right)\left(z+xz+1\right)\)
\(=\left(x+y+2xy+yz+xy^2+xyz+xy^2z+1\right)+\left(2xz+x+z+xyz+x^2yz+x^2z+xy+1\right)+\left(y+z+2yz+yz^2+xz+xyz+xyz^2+1\right)\)
Thay \(xyz=1;\)có :
\(M=\left(x+y+2xy+yz+xy^2+1+y.1+1\right)+\left(2xz+x+z+1+x.1+x^2z+xy+1\right)+\left(y+z+2yz+yz^2+xz+1+z.1+1\right)\)
\(=3x+3y+3z+3xy+3yz+3xz+xy^2+x^2z+yz^2+6\)
Đặt \(N=\left(x+xy+1\right)\left(y+yz+1\right)\left(z+xz+1\right)\)
\(=\left(x+y+2xy+yz+xy^2+1+y.1+1\right)\left(z+xz+1\right)\)
\(=\left(x+2y+2xy+yz+xy^2+2\right)\left(z+xz+1\right)\)
\(=xz+x^2z+x+3yz+2xy+2y+4xyz+2x^2yz+2xy+yz^2+xyz^2+xy^2z+x^2y^2z+xy^2+2z+2\)
Lần lượt thay \(xyz=1\); cuối cùng có :
\(N=3x+3y+3z+3xy+3yz+3xz+xy^2+x^2z+yz^2+6\)
\(\Rightarrow M=N\)
\(\Rightarrow A=\frac{M}{N}=1\)
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