Đặt \(\left\{{}\begin{matrix}xy=a\\yz=b\\zx=c\end{matrix}\right.\)

Giả thiết \(\Leftrightarrow a^3+b^3+c^3=3abc\)

\(\Leftrightarrow a^3+b^3+3a^2b+3ab^2+c^3-3abc-3a^2b-3ab^2=0\)

\(\Leftrightarrow\left(a+b\right)^3+c^3-3ab\left(a+b+c\right)=0\)

\(\Leftrightarrow\left(a+b+c\right)\left(a^2+2ab+b^2-bc-ca\right)-3ab\left(a+b+c\right)=0\)

\(\Leftrightarrow\left(a+b+c\right)\left(a^2+b^2+c^2-ab-bc-ca\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}a+b+c=0\\a^2+b^2+c^2-ab-bc-ca=0\end{matrix}\right.\)

+) TH1: \(a+b+c=0\Leftrightarrow xy+yz+zx=0\)

Biến đổi linh tinh P chắc là ra :D

+) TH2: \(a=b=c\Leftrightarrow xy=yz=zx\Leftrightarrow x=y=z\)

\(P=\frac{x+y}{y}\cdot\frac{y+z}{z}\cdot\frac{z+x}{x}=\frac{2y}{y}\cdot\frac{2z}{z}\cdot\frac{2x}{x}=2\cdot2\cdot2=8\)

Vậy....

\(\left(x+y+z\right)^2=x^2+y^2+z^2\\ \Leftrightarrow xy+yz+xz=0\\ \Leftrightarrow\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=0\)

Đặt

\(\dfrac{1}{x}=a;\dfrac{1}{y}=b;\dfrac{1}{z}=c\\ vìa+b+c=0\\ \Rightarrow a^3+b^3+c^3=3abc\\ \Rightarrow\left(\dfrac{1}{x}\right)^3+\left(\dfrac{1}{y}\right)^3+\left(\dfrac{1}{z}\right)^3=\dfrac{3}{xyz}\)

\(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\sqrt{3}\)

\(\Leftrightarrow\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)^2=3\)

\(\Leftrightarrow\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}+2\left(\frac{1}{xy}+\frac{1}{yz}+\frac{1}{zx}\right)=3\)

\(\Leftrightarrow\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}+\frac{2\left(x+y+z\right)}{xyz}=3\)

\(\Leftrightarrow\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}+\frac{2xyz}{xyz}=3\)

\(\Leftrightarrow\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}+2=3\)

\(\Leftrightarrow\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=1\)