# 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:

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

×