\(1+x=x+\sqrt{xy}+\sqrt{yz}+\sqrt{xz}=\sqrt{x}\left(\sqrt{x}+\sqrt{y}\right)+\sqrt{z}\left(\sqrt{x}+\sqrt{y}\right)=\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}+\sqrt{z}\right)\)
Tương tự, ta có:
\(1+y=\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{y}+\sqrt{z}\right)\)
\(1+z=\left(\sqrt{x}+\sqrt{z}\right)\left(\sqrt{y}+\sqrt{z}\right)\)
\(\Rightarrow P=\sqrt{\left(\sqrt{x}+\sqrt{y}\right)^2\left(\sqrt{x}+\sqrt{z}\right)^2\left(\sqrt{y}+\sqrt{z}\right)^2}\left(\dfrac{\sqrt{x}}{\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}+\sqrt{z}\right)}+\dfrac{\sqrt{y}}{\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{y}+\sqrt{z}\right)}+\dfrac{\sqrt{z}}{\left(\sqrt{x}+\sqrt{z}\right)\left(\sqrt{y}+\sqrt{z}\right)}\right)\)
\(P=\sqrt{x}\left(\sqrt{y}+\sqrt{z}\right)+\sqrt{y}\left(\sqrt{x}+\sqrt{z}\right)+\sqrt{z}\left(\sqrt{x}+\sqrt{y}\right)\)
\(P=2\left(\sqrt{xy}+\sqrt{xz}+\sqrt{yz}\right)=2\)
