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Integer sequence

Source: Brazilian M.O. 2004

October 17, 2004
inductionalgebra unsolvedalgebra

Problem Statement

Let x1,x2,...,x2004x_1, x_2, ..., x_{2004} be a sequence of integer numbers such that xk+3=xk+2+xkxk+1x_{k+3}=x_{k+2}+x_{k}x_{k+1}, ∀1≤k≤2001\forall 1 \le k \le 2001. Is it possible that more than half of the elements are negative?
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