Question

CMR \(x\ne0,y\ne0,z\ne0\) và \(x+\frac{1}{y}=y+\frac{1}{z}=z+\frac{1}{x}\)thì x=y=z và \(zyx=\pm1\)

Answer 1

\(x+\frac{1}{y}=y+\frac{1}{z}=z+\frac{1}{x}\). do đó :

\(x-y=\frac{1}{z}-\frac{1}{y}=\frac{y-z}{yz},y-z=\frac{1}{x}-\frac{1}{z}=\frac{z-x}{xz},z-x=\frac{1}{y}-\frac{1}{x}=\frac{x-y}{xy}\)

suy ra : ( x - y ) ( y - z ) ( z - x ) = \(\frac{\left(x-y\right)\left(y-z\right)\left(z-x\right)}{x^2y^2z^2}\)

\(\Rightarrow\left(x-y\right)\left(y-z\right)\left(z-x\right)\left(x^2y^2z^2-1\right)=0\)

\(\Rightarrow\orbr{\begin{cases}x=y=z\\x^2y^2z^2=1\Rightarrow xyz=\mp1\end{cases}}\)