a) ĐK: \(x\ge2;y\ge-2009;z\ge2010\)
Ta có: \(\sqrt{x-2}+\sqrt{y+2009}+\sqrt{z-2010}=\dfrac{1}{2}\left(x+y+z\right)\)
<=> \(2\sqrt{x-2}+2\sqrt{y+2009}+2\sqrt{z-2010}=x+y+z\)
<=> \(\left(x-2+2\sqrt{x-2}+1\right)+\left(y+2009-2\sqrt{y+2009}+1\right)\)
\(+\left(z-2010-2\sqrt{z-2010}+1\right)=0\)
<=> \(\left(\sqrt{x-2}-1\right)^2+\left(\sqrt{y+2009}-1\right)^2+\left(\sqrt{z-2010}+1\right)^2=0\)
<=> \(\left\{{}\begin{matrix}\sqrt{x-2}-1=0\\\sqrt{y+2009}-1=0\\\sqrt{z-2010}-1=0\end{matrix}\right.\) => \(\left\{{}\begin{matrix}\sqrt{x-2}=1\\\sqrt{y+2009}=1\\\sqrt{z-2010}=1\end{matrix}\right.\)
<=> \(\left\{{}\begin{matrix}x-2=1\\y+2009=1\\z-2010=1\end{matrix}\right.\) <=> \(\left\{{}\begin{matrix}x=3\\y=-2008\\z=2011\end{matrix}\right.\) (TM)
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