Đk: \(x\ge0,y\ge1,z\ge2\)
\(\sqrt{x}+\sqrt{y-1}+\sqrt{z-2}=\dfrac{1}{2}\left(x+y+z\right)\)
\(\Leftrightarrow\dfrac{1}{2}x-\sqrt{x}+\dfrac{1}{2}y-\sqrt{y-1}+\dfrac{1}{2}z-\sqrt{z-2}=0\)
\(\Leftrightarrow\dfrac{1}{2}\left(x-2\sqrt{x}\right)+\dfrac{1}{2}\left(y-2\sqrt{y-1}\right)+\dfrac{1}{2}\left(z-2\sqrt{z-2}\right)=0\)
\(\Leftrightarrow\dfrac{1}{2}\left(x-2\sqrt{x}+1-1\right)+\dfrac{1}{2}\left[\left(y-1\right)-2\sqrt{y-1}+1\right]+\dfrac{1}{2}\left[\left(z-2\right)-2\sqrt{z-2}+1+1\right]=0\)
\(\Leftrightarrow\dfrac{1}{2}\left(\sqrt{x}-1\right)^2-\dfrac{1}{2}+\dfrac{1}{2}\left(\sqrt{y-1}-1\right)^2+\dfrac{1}{2}\left(\sqrt{z-2}-1\right)^2+\dfrac{1}{2}=0\)
\(\Leftrightarrow\left(\sqrt{x}-1\right)^2+\left(\sqrt{y-1}-1\right)^2+\left(\sqrt{z-2}-1\right)^2=0\)
\(\Leftrightarrow\left\{{}\begin{matrix}\left(\sqrt{x}-1\right)^2=0\\\left(\sqrt{y-1}-1\right)^2=0\\\left(\sqrt{z-2}-1\right)^2=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=1\\y=2\\z=3\end{matrix}\right.\)
