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smallest sum product value of a sequence of 2n numbers

Source: Dutch NMO 2016 p2

September 7, 2019
algebraSequenceSumProductminimum

Problem Statement

For an integer n≥1n \ge 1 we consider sequences of 2n2n numbers, each equal to 0,−10, -1 or 11. The sum product value of such a sequence is calculated by first multiplying each pair of numbers from the sequence, and then adding all the results together. For example, if we take n=2n = 2 and the sequence 0,1,1,−10,1, 1, -1, then we find the products 0⋅1,0⋅1,0⋅−1,1⋅1,1⋅−1,1⋅−10\cdot 1, 0\cdot 1, 0\cdot -1, 1\cdot 1, 1\cdot -1, 1\cdot -1. Adding these six results gives the sum product value of this sequence: 0+0+0+1+(−1)+(−1)=−10+0+0+1+(-1)+(-1) = -1. The sum product value of this sequence is therefore smaller than the sum product value of the sequence 0,0,0,00, 0, 0, 0, which equals 00. Determine for each integer n≥1n \ge 1 the smallest sum product value that such a sequence of 2n2n numbers could have.
Attention: you are required to prove that a smaller sum product value is impossible.
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