ĐKXĐ:\(x\ge0;y\ge1;z\ge2\)
\(\sqrt{x}+\sqrt{y-1}+\sqrt{z-2}=\frac{x+y+z}{2}\)
\(\Leftrightarrow2\sqrt{x}+2\sqrt{y-1}+2\sqrt{z-2}=x+y+z\)
\(\Leftrightarrow\left(x-2\sqrt{x}+1\right)+\left(y-1+2\sqrt{y-1}+1\right)+\left(z-2+2\sqrt{z-2}+1\right)=0\)
\(\Leftrightarrow\left(\sqrt{x}-1\right)^2+\left(\sqrt{y-1}-1\right)^2+\left(\sqrt{z-2}-2\right)^2=0\)
Mà \(\left\{\begin{matrix}\left(\sqrt{x-1}-1\right)^2\ge0\\\left(\sqrt{y-1}-1\right)^2\ge0\\\left(\sqrt{z-2}-2\right)^2\ge0\end{matrix}\right.\)\(\forall x;y;z\)
\(\Rightarrow\left\{\begin{matrix}\left(\sqrt{x-1}-1\right)^2=0\\\left(\sqrt{y-1}-1\right)^2=0\\\left(\sqrt{z-2}-2\right)^2=0\end{matrix}\right.\)\(\Leftrightarrow\left\{\begin{matrix}\sqrt{x-1}-1=0\\\sqrt{y-1}-1=0\\\sqrt{z-2}-2=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{\begin{matrix}\sqrt{x-1}=1\\\sqrt{y-1}=1\\\sqrt{z-2}=2\end{matrix}\right.\)\(\Leftrightarrow\left\{\begin{matrix}x-1=1\\y-1=1\\z-2=4\end{matrix}\right.\)\(\Leftrightarrow\left\{\begin{matrix}x=2\\y=2\\z=6\end{matrix}\right.\)
=> x02 + y02 + z02 = 22 + 22 + 62 = 44
