Ta có: \(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=0\Rightarrow\dfrac{xy+yz+xz}{xyz}=0\Rightarrow xy+yz+xz=0\)
\(\Rightarrow xy=-yz-xz;yz=-xy-xz;xz=-xy-yz\)
Ta lại có: \(A=\dfrac{x+y}{z}+\dfrac{x+z}{y}+\dfrac{y+z}{x}=\dfrac{x^2+xy}{xz}+\dfrac{z^2+xz}{yz}+\dfrac{y^2+yz}{xy}\)
\(=\dfrac{x^2-yz-xz}{xz}+\dfrac{z^2-xy-yz}{yz}+\dfrac{y^2-xy-xz}{xy}\)
\(=\dfrac{x\left(x-z\right)}{xz}-\dfrac{yz}{xz}+\dfrac{z\left(z-y\right)}{yz}-\dfrac{xy}{yz}+\dfrac{y\left(y-x\right)}{xy}-\dfrac{xz}{xy}\)
\(=\dfrac{x-z}{z}-\dfrac{y}{x}+\dfrac{z-y}{y}-\dfrac{x}{z}+\dfrac{y-x}{x}-\dfrac{z}{y}\)
\(=\dfrac{x-z-x}{z}+\dfrac{z-y-z}{y}+\dfrac{y-x-y}{x}=\dfrac{-z}{z}+\dfrac{-y}{y}+\dfrac{-x}{x}\)
\(=-1-1-1=-3\). Vậy A=-3
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