# 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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