$\begin{array}{l} \sqrt x + \sqrt {y - 1} + \sqrt {z - 2} = \frac{1}{2}\left( {x + y + z} \right)\\ \Leftrightarrow 2\sqrt x + 2\sqrt {y - 1} + 2\sqrt {z - 2} = x + y + z\\ \Leftrightarrow x - 2\sqrt x + 1 + y - 1 - 2\sqrt {y - 1} + 1 + z - 2 + 2\sqrt {z - 2} + 1 = 0\\ \Leftrightarrow {\left( {\sqrt x - 1} \right)^2} + \left( {\sqrt {y - 1} - 1} \right) + {\left( {\sqrt {z - 2} - 1} \right)^2} = 0\\ \Leftrightarrow \left\{ \begin{array}{l} \sqrt x - 1 = 0\\ \sqrt {y - 1} - 1 = 0\\ \sqrt {z - 2} - 1 = 0 \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} x = 1\\ y = 2\\ z = 3 \end{array} \right. \end{array}$
