Cho x,y,z khác 0 và x khác y. Tính M=|x|/x + |y|/y + |z|/z + |xyz|/xyz
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![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
Vì x+y+z=0
=> \(\hept{\begin{cases}x+y=-z\\y+z=-x\\x+z=-y\end{cases}}\)
Ta có \(A=\frac{x}{y+z-x}+\frac{y}{x+z-y}+\frac{z}{x+y-z}\)
\(=\frac{x}{-x-x}+\frac{y}{-y-y}+\frac{z}{-z-z}=\frac{x}{-2x}+\frac{y}{-2y}+\frac{z}{-2z}\)
\(=\frac{-1}{2}+\frac{-1}{2}+\frac{-1}{2}=\frac{-3}{2}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
Xét (1/x+1/y+1/z)^2=1/x^2+1/y^2+1/z^2+2/xy+2/yz+2/xz
=P+2/xy+2/yz+2/xz=P+(2z+2x+2y)/xyz=P+2(x+y+z)/x+y+z=P+2
mà (1/x+1/y+1/z)^2=3
=>p=3-2=1
![](https://rs.olm.vn/images/avt/0.png?1311)
\(x+\frac{1}{y}=y+\frac{1}{z}=z+\frac{1}{x}\)
\(\Rightarrow\hept{\begin{cases}x-y=\frac{1}{z}-\frac{1}{y}=\frac{y-z}{yz}\\x-z=\frac{1}{x}-\frac{1}{y}=\frac{y-x}{xy}\\y-z=\frac{1}{x}-\frac{1}{z}=\frac{z-x}{xz}\end{cases}}\)
\(\Rightarrow\left(x-y\right)\left(x-z\right)\left(y-z\right)=\frac{\left(y-z\right)\left(y-x\right)\left(z-x\right)}{\left(xyz\right)^2}\)
\(\Rightarrow\left(xyz\right)^2=1\Leftrightarrow\orbr{\begin{cases}xyz=1\\xyz=-1\end{cases}}\).
![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
Áp dụng tính chất dãy tỉ số bằng nhau ta có:
\(\frac{x-y-z}{x}=\frac{-x+y-z}{y}=\frac{-x-y+z}{z}=\frac{x-y-z-x+y-z-x-y+z}{x+y+z}\)\(=\frac{-\left(x+y+z\right)}{x+y+z}\)
Nếu \(x+y+z=0\)thì \(\hept{\begin{cases}x+y=-z\\y+z=-x\\z+x=-y\end{cases}}\)
\(A=\left(1+\frac{y}{x}\right)\left(1+\frac{z}{y}\right)\left(1+\frac{x}{z}\right)\)
\(=\frac{x+y}{x}.\frac{y+z}{y}.\frac{z+x}{z}\)
\(=\frac{-z}{x}.\frac{-x}{y}.\frac{-y}{z}=-1\)
Nếu \(x+y+z\ne0\)thì \(\frac{x-y-z}{x}=\frac{-x+y-z}{y}=\frac{-x-y+z}{z}=-1\)
suy ra: \(\frac{x-y-z}{x}=-1\) \(\Rightarrow\) \(x-y-z=-x\) \(\Rightarrow\) \(y+z=2x\)
\(\frac{-x+y-z}{y}=-1\) \(-x+y-z=-y\) \(x+z=2y\)
\(\frac{-x-y+z}{z}=-1\) \(-x-y+z=-z\) \(x+y=2z\)
\(A=\left(1+\frac{y}{x}\right)\left(1+\frac{z}{y}\right)\left(1+\frac{x}{z}\right)\)
\(=\frac{x+y}{x}.\frac{y+z}{y}.\frac{x+z}{z}\)
\(=\frac{2z}{x}.\frac{2x}{y}.\frac{2y}{z}=8\)