Let $k \le n$ be positive integers and let $m$ be a positive integer. Assume that $x_1, \ldots, x_n$ are non-negative integers and \begin{align} & x_1^2 + x_2^2 + \cdots + x_n^2 - (k-2) m^2=2, \\ & x_1 + \cdots + x_n = k m, \\ & x_1 \ge x_2 \ge \cdots \ge x_n. \end{align} How to show that $x_{k+1}+x_{k+2} + \cdots + x_n < 2m$?

It is easy to see that the result is true for $k=1,2$.

In the case of $k=3$, we have \begin{align} & x_1^2 + x_2^2 + \cdots + x_n^2 = m^2 + 2, \\ & x_1 + \cdots + x_n = 3 m, \\ & x_1 \ge x_2 \ge \cdots \ge x_n. \end{align} We have to estimate the solutions of the above equations. Are there some method to do this? Thank you very much.