Bạn tham khảo nhé!

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Lời giải cho bài của bạn ở đây nhé!  http://olm.vn/hoi-dap/question/479780.html

\(\dfrac{1}{x}+\dfrac{2}{y}+\dfrac{3}{z}=0\)

=>\(\dfrac{yz+2xz+3xy}{xyz}=0\)

=>yz+2xz+3xy=0

=>\(xy+\dfrac{2}{3}xz+\dfrac{1}{3}yz=0\)

\(x+\dfrac{y}{2}+\dfrac{z}{3}=1\)

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

=>\(x^2+\dfrac{y^2}{4}+\dfrac{z^2}{9}+2\left(x\cdot\dfrac{y}{2}+x\cdot\dfrac{z}{3}+\dfrac{y}{2}\cdot\dfrac{z}{3}\right)=1\)

=>\(A+2\left(\dfrac{xy}{2}+\dfrac{xz}{3}+\dfrac{yz}{6}\right)=1\)

=>A+xy+2/3xz+1/3yz=1

=>A=1

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Bạn tham khảo ở đây.

\(x+y+z=9\Leftrightarrow\left(x+y+z\right)^2=81\\ \Leftrightarrow x^2+y^2+z^2+2\left(xy+yz+xz\right)=81\\ \Leftrightarrow xy+yz+xz=\dfrac{81-27}{2}=27\\ \Leftrightarrow x^2+y^2+z^2=xy+yz+xz\\ \Leftrightarrow2x^2+2y^2+2z^2=2xy+2yz+2xz\\ \Leftrightarrow\left(x^2-2xy+y^2\right)+\left(y^2-2yz+z^2\right)+\left(z^2-2xz+x^2\right)=0\\ \Leftrightarrow\left(x-y\right)^2+\left(y-z\right)^2+\left(z-x\right)^2=0\\ \Leftrightarrow\left\{{}\begin{matrix}x-y=0\\y-z=0\\z-x=0\end{matrix}\right.\Leftrightarrow x=y=z=\dfrac{9}{3}=3\left(x+y+z=9\right)\)

\(\Leftrightarrow\left(x-4\right)^{2018}+\left(y-4\right)^{2019}+\left(z-4\right)^{2020}\\ =\left(-1\right)^{2018}+\left(-1\right)^{2019}+\left(-1\right)^{2020}=1-1+1=1\)

Ta có:      \(x+y+z=0\)

\(\Leftrightarrow\)  \(\left(x+y+z\right)^2=0\)

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

\(\Leftrightarrow\)\(x^2+y^2+z^2=0\)   (vì  xy + yz + xz =0)

\(\Leftrightarrow\)\(x=y=z=0\)

Vậy      \(S=\left(0-1\right)^{1999}+0^{2003}+\left(0+1\right)^{2006}=0\)