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prove that |a_i| < 1 for some i, sequence where a_i +a_j has min abs. value

Source: Tuymaada 2019 p1

July 22, 2019
algebraSequenceinequalities

Problem Statement

In a sequence a1,a2,..a_1, a_2, .. of real numbers the product a1a2a_1a_2 is negative, and to define ana_n for n>2n > 2 one pair (i,j)(i, j) is chosen among all the pairs (i,j),1≤i<j<n(i, j), 1 \le i < j < n, not chosen before, so that ai+aja_i +a_j has minimum absolute value, and then ana_n is set equal to ai+aja_i + a_j . Prove that ∣ai∣<1|a_i| < 1 for some ii.
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