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Given a positive integer $n$ and a real number $a>0$, consider the equation
$\sum_{i=1}^n\left(x_i^2+\left(a-x_i\right)^2\right)=n a^2$
How many solutions $\left(x_1, \ldots, x_n\right)$ with $0 \leq x_i \leq a$ for each $i$ does the equation have?
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