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Xét (1/x+1/y+1/z)^2=1/x^2+1/y^2+1/z^2+2/xy+2/yz+2/xz
=P+2/xy+2/yz+2/xz=P+(2z+2x+2y)/xyz=P+2(x+y+z)/x+y+z=P+2
mà (1/x+1/y+1/z)^2=3
=>p=3-2=1
\(x+y+z=xyz\Leftrightarrow\dfrac{1}{xy}+\dfrac{1}{yz}+\dfrac{1}{zx}=1\)
\(\dfrac{1}{x^2}+\dfrac{1}{y^2}+\dfrac{1}{z^2}=\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right)^2-2\left(\dfrac{1}{xy}+\dfrac{1}{yz}+\dfrac{1}{zx}\right)=2^2-2.1=2\) (đpcm)
Ta có \(\dfrac{1}{x^2}+\dfrac{1}{y^2}+\dfrac{1}{z^2}+\dfrac{2}{xyz}=1\)
\(\Leftrightarrow\dfrac{\left(yz\right)^2+\left(xz\right)^2+\left(xy\right)^2+2xyz}{\left(xyz\right)^2}=1\)
<=> (xy)2 + (yz)2 + (zx)2 + 2xyz = (xyz)2
<=> (xy)2 + (yz)2 + (xz)2 + 2xyz(x + y + z) = (xyz)2
<=> (xy + yz + zx)2 = (xyz)2
<=> \(\left[{}\begin{matrix}xy+yz+zx=xyz\\xy+yz+zx=-xyz\end{matrix}\right.\)
+) Khi xy + yz + zx = -xyz
=> \(\dfrac{xy+yz+zx}{xyz}=\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=-1< 0\left(\text{loại}\right)\)
=> xy + yz + zx = xyz
<=> \(xyz\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right)=xyz\Leftrightarrow xyz\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}-1\right)=0\)
<=> \(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=1\)
<=> \(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=\dfrac{1}{x+y+z}\)
<=> \(\dfrac{x+y}{xy}=\dfrac{-\left(x+y\right)}{\left(x+y+z\right)z}\)
<=> \(\left(x+y\right)\left(\dfrac{1}{xz+yz+z^2}+\dfrac{1}{xy}\right)=0\)
<=> \(\dfrac{\left(x+y\right)\left(y+z\right)\left(z+x\right)}{\left(zx+yz+z^2\right)xy}=0\)
<=> \(\left[{}\begin{matrix}x=-y\\y=-z\\z=-x\end{matrix}\right.\)
Khi x = -y => y = 1 => P = 1
Tương tự y = -z ; z = -x được P = 1
Vậy P = 1
\(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\sqrt{3}\)
\(\Leftrightarrow\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)^2=3\)
\(\Leftrightarrow\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}+2\left(\frac{1}{xy}+\frac{1}{yz}+\frac{1}{zx}\right)=3\)
\(\Leftrightarrow\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}+\frac{2\left(x+y+z\right)}{xyz}=3\)
\(\Leftrightarrow\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}+\frac{2xyz}{xyz}=3\)
\(\Leftrightarrow\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}+2=3\)
\(\Leftrightarrow\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=1\)
(x+y+z)^2=x^2+y^2+z^2
=>2(xy+yz+xz)=0
=>xy+xz+yz=0
=>xy/xyz+xz/xyz+yz/xyz=0
=>1/x+1/y+1/z=0
