Đáp án:
\((x;y;z)=(30;10;1020)\)
Giải thích các bước giải:
\(\begin{array}{l}
\quad \sqrt{x-29} + 2\sqrt{y-6} + 3\sqrt{z-2011} + 1016 = \dfrac12(x+y+z)\qquad (*)\\
ĐK:\begin{cases}x \geqslant 29\\y \geqslant 6\\z \geqslant 1011\end{cases}\\
(*)\Leftrightarrow x+y+z = 2\sqrt{x-29} + 4\sqrt{y- 6} + 6\sqrt{z-2011} + 2032\\
\Leftrightarrow \left(x - 29 - 2\sqrt{x-29} + 1\right) + \left(y - 6 - 4\sqrt{y-6} + 4\right) + \left(z - 2011 - 6\sqrt{z-1011} + 9\right) = 0\\
\Leftrightarrow \left(\sqrt{x-29} - 1\right)^2 + \left(\sqrt{y-6} - 2\right)^2 + \left(\sqrt{z-1011} - 3\right)^2 = 0\qquad (**)\\
\text{Ta có:}\\
\begin{cases}\left(\sqrt{x-29} - 1\right)^2\geqslant 0\quad \forall x\geqslant 29\\\left(\sqrt{y-6} - 2\right)^2\geqslant 0\quad \forall y\geqslant 6\\\left(\sqrt{z-1011} - 3\right)^2 \geqslant 0\quad \forall z\geqslant 1011\end{cases}\\
\text{Do đó:}\\
(**)\Leftrightarrow \begin{cases}\left(\sqrt{x-29} - 1\right)^2=0\\\left(\sqrt{y-6} - 2\right)^2=0\\\left(\sqrt{z-1011} - 3\right)^2=0\end{cases}\\
\Leftrightarrow \begin{cases}\sqrt{x-29} - 1 =0\\\sqrt{y-6} -2 =0\\\sqrt{z-1011}-3 =0\end{cases}\\
\Leftrightarrow \begin{cases}\sqrt{x-29} =1\\\sqrt{y-6} =2\\\sqrt{z-1011}=3\end{cases}\\
\Rightarrow \begin{cases}x - 29 =1\\y - 6= 4\\z - 1011 = 9\end{cases}
\Leftrightarrow \begin{cases}X = 30\\y = 10\\z = 1020\end{cases}\quad (nhận)\\
\text{Vậy}\ (x;y;z)=(30;10;1020)
\end{array}\)
