\(x+y+z=0\Rightarrow\left\{{}\begin{matrix}x=-\left(y+z\right)\\y=-\left(x+z\right)\\z=-\left(x+y\right)\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}-x=y+z\\-y=x+z\\-z=x+y\end{matrix}\right.\)
\(C=\left(1+\dfrac{x}{y}\right)\left(1+\dfrac{x}{z}\right)\left(1+\dfrac{z}{x}\right)\)
\(C=\left(\dfrac{y}{y}+\dfrac{x}{y}\right)\left(\dfrac{z}{z}+\dfrac{x}{z}\right)\left(\dfrac{x}{x}+\dfrac{z}{x}\right)\)
\(C=\dfrac{x+y}{y}.\dfrac{x+z}{z}.\dfrac{x+z}{x}\)
\(C=\dfrac{\left(-z\right).2\left(-y\right)}{xyz}=\dfrac{2yz}{xyz}=\dfrac{2}{x}\)
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