NL-NWO 2016 Problem 2

For an integer $n \ge 1$ we consider sequences of $2n$ numbers, each equal to $0, -1$ or $1$ . 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 $n = 2$ and the sequence $0,1, 1, -1$ , then we find the products $0\cdot 1, 0\cdot 1, 0\cdot -1, 1\cdot 1, 1\cdot -1, 1\cdot -1$ . Adding these six results gives the sum product value of this sequence: $0+0+0+1+(-1)+(-1) = -1$ . The sum product value of this sequence is therefore smaller than the sum product value of the sequence $0, 0, 0, 0$ , which equals $0$ . Determine for each integer $n \ge 1$ the smallest sum product value that such a sequence of $2n$ numbers could have.

Attention: you are required to prove that a smaller sum product value is impossible.