1 equation, n variables

Algebra Level 3

\[ \sqrt{x_{1}-1^{2}} + 2 \sqrt{x_{2}-2^{2}} + \ldots + n \sqrt{x_{n}-n^{2}} = \frac{1}{2} \left [ x_{1} + x_{2}+ \ldots + x_{n} \right ] \]

Then which of the following is true?

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