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Suppose that \[ x + \frac{1}{x} = \frac{1 + \sqrt{5}}{2}. \] Compute \(x^{2000} + x^{-2000}\).

Let \(0 \leq x_k \leq 1\) for all \(k = 1, 2, ..., 2018\). Maximize \[ f(x_1, x_2, ..., x_{2018}) = x_1 + x_2 + ... + x_{2018} - x_1x_2...x_{2018}. \]

Define the function \[ f(x) = \prod_{n= 1}^\infty \cos \frac{x}{2^n}. \] Compute \(f(1)\).

Let \(P(x)\) be a real polynomial with degree at most 9 such that \(P(2n) = F_{2n+1}\) for \(1 \leq n \leq 9\). What is the value of ...

Consider the \(2^{1013}\) numbers \(\pm 1 \pm 2 \pm \cdots \pm 1013\). How many of these numbers are \(2017\) in modulo \(2027?\) Express your answer in modulo \(1000.\)

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