\(\frac{x}{y}+\frac{y}{z}+\frac{z}{x}=3\\ \Leftrightarrow\frac{x^2z}{xyz}+\frac{y^2x}{xyz}+\frac{z^2y}{xyz}=3\\ \Leftrightarrow x^2z+y^2x+z^2y=3xyz\\ \Leftrightarrow x^2z+y^2x+z^2y-3xyz=0\\ \Leftrightarrow xz\left(x-y\right)+yx\left(y-z\right)+yz\left(z-x\right)=0\\ \Leftrightarrow\left\{{}\begin{matrix}xz\left(x-y\right)=0\\yx\left(y-z\right)=0\\yz\left(z-x\right)=0\end{matrix}\right.\Leftrightarrow\left(x,y,z>0\Rightarrow xz,yx,yz\ne0\right)\left\{{}\begin{matrix}x-y=0\\y-z=0\\z-x=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=y\\y=z\\z=x\end{matrix}\right.\Leftrightarrow x=y=z\Rightarrow1+1+1=3\left(dpcm\right)\)
Mình bổ sung tí TH2:\(\frac{\sqrt{y}}{x}+\frac{\sqrt{z}}{y}+\frac{\sqrt{x}}{z}=\frac{\sqrt{x}}{x}+\frac{\sqrt{x}}{x}+\frac{\sqrt{x}}{x}=\frac{3}{\sqrt{x}}\le3\)
