1. Equation
$$
\begin{array}{l}
\sqrt{x_{1}-1}+2 \sqrt{x_{2}-4}+3 \sqrt{x_{3}-9} \\
=\frac{1}{2}\left(x_{1}+x_{2}+x_{3}\right)
\end{array}
$$

The real solution $\left(x_{1}, x_{2}, x_{3}\right)=$
-1. $(2,8,18)$.
Multiplying both sides of the equation by 2 and rearranging, we get
$$
\begin{array}{l}
x_{1}+x_{2}+x_{3}-2 \sqrt{x_{1}-1}-4 \sqrt{x_{2}-4}- \\
6 \sqrt{x_{3}-9}=0 .
\end{array}
$$

Completing the square, we have
$$
\begin{array}{l}
\left(\sqrt{x_{1}-1}-1\right)^{2}+\left(\sqrt{x_{2}-4}-2\right)^{2}+ \\
\quad\left(\sqrt{x_{3}-9}-3\right)^{2}=0 \\
\Rightarrow \sqrt{x_{1}-1}-1=0, \sqrt{x_{2}-4}-2=0, \\
\sqrt{x_{3}-9}-3=0 .
\end{array}
$$

Solving, we get $x_{1}=2, x_{2}=8, x_{3}=18$.