What? Cauchy? AM-GM? Max again?

Algebra Level 5

Find the value of \( \lceil 1000C \rceil \), where \( C \) satisfies the following:

\( C \) is the largest constant such that for all real numbers \( x_1, x_2, \ldots, x_{913} \) that satisfy \( x_i \in (0,1) \) where \( i = 1,2, \ldots, n \) and \( (1-x_i)(1-x_j) \geq \frac{1}{4} \) where \( 1 \leq i < j \leq n \), we have:

\( \sum_{i=1}^n x_i \geq C \sum_{1 \leq i < j \leq n} (2x_ix_j + \sqrt{x_ix_j}) \)

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