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Let $c_1, \ldots, c_n, b_1, \ldots, b_n(n \geq 2)$ be positive real numbers. Prove that the equation
$\sum_{i=1}^n c_i \sqrt{x_i-b_i}=\frac{1}{2} \sum_{i=1}^n x_i$
has a unique solution $\left(x_1, \ldots, x_n\right)$ if and only if $\sum_{i=1}^n c_i^2=\sum_{i=1}^n b_i$.
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