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Which integral values can the sum take?

Source: Austrian Mathematical Olympiad 2004, Part 1, P4

June 18, 2011
calculusintegrationalgebra unsolvedalgebra

Problem Statement

Each of the 2N=20042N = 2004 real numbers x1,x2,…,x2004x_1, x_2, \ldots , x_{2004} equals either 2−1\sqrt 2 -1 or 2+1\sqrt 2 +1. Can the sum ∑k=1Nx2k−1x2k\sum_{k=1}^N x_{2k-1}x_2k take the value 20042004? Which integral values can this sum take?
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