20 unknowns in one equation?

Algebra Level 5

Let \(x_1, x_2,\ldots,x_{20} \) be real numbers satisfying

\[ \large \sum_{m=1}^{20} m \sqrt{x_m - m^2} = \frac12 \sum_{m=1}^{20} x_m. \]

It is given that there is only one unique solution to the 20-tuple, \((x_1,x_2,\ldots,x_{20})\). Find the value of \(x_1 + x_2 + \cdots + x_{20} \).

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