Messy quadratic

Algebra Level 5

This problem is from OMO.

For reals \(x \ge 3\), let \(f(x)\) denote the function \(f(x) = \frac{-x + x\sqrt{4x - 3}}{2}\). Suppose \(a_1, a_2, ..., a_{2013}\) is a sequence of real numbers such that \(a_1 > 3\), \(a_{2013} = 2013\), and for \(n = 1, 2, ..., 2012\), \(a_{n + 1} = f(a_n)\). Determine the value of \(a_1 + \displaystyle\sum_{i = 1}^{2012}\frac{a_{i + 1}^3}{a_i^2 + a_ia_{i + 1} + a_{i + 1}^2}\)

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