\(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=0\)
\(\Leftrightarrow yz+zx+xy=0\)
\(\Leftrightarrow\left[{}\begin{matrix}yz=-zx-xy\\zx=-xy-yz\\xy=-yz-zx\end{matrix}\right.\)
\(\Leftrightarrow\dfrac{1}{x^2+2yz}=\dfrac{1}{x^2-xz-xy+yz}=\dfrac{1}{\left(x-y\right)\left(x-z\right)}\)
CMTT\(\Rightarrow\dfrac{1}{y^2+2zx}=\dfrac{1}{\left(y-z\right)\left(y-x\right)}\)
\(\dfrac{1}{z^2+2xy}=\dfrac{1}{\left(z-x\right)\left(z-y\right)}\)
\(\Rightarrow A=\dfrac{1}{\left(x-y\right)\left(x-z\right)}+\dfrac{1}{\left(y-z\right)\left(y-x\right)}+\dfrac{1}{\left(z-x\right)\left(z-y\right)}\)
\(A=\dfrac{y-z}{\left(x-y\right)\left(x-z\right)\left(y-z\right)}+\dfrac{z-x}{\left(x-y\right)\left(x-z\right)\left(y-z\right)}+\dfrac{x-y}{\left(x-y\right)\left(x-z\right)\left(y-z\right)}\)
\(A=\dfrac{y-z+z-x+x-y}{\left(x-y\right)\left(x-z\right)\left(y-z\right)}=0\left(đpcm\right)\)