Uzumaki's impossible equality

Algebra Level 5

If real numbers \(x_1, x_2, \ldots x_{13}\) satisfy

\[ \sum_{i=1}^{13} x_{i} = 2 \sum_{i=1}^{13} i \sqrt{x_{i} -i^{2}} ,\]

what is the value of \( x_{13}\)?

This problem is posed by Uzumaki N.

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