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a summation over pairs of integers, polynomial

Source: Yugoslav TST 1968 P5

May 29, 2021
Summationalgebrapolynomial

Problem Statement

Let nn be an integer greater than 11. Let x∈Rx\in\mathbb R.
(a) Evaluate S(x,n)=∑(x+p)(x+q)S(x,n)=\sum(x+p)(x+q), where the summation is over all pairs (p,q)(p,q) of different numbers from {1,2,…,n}\{1,2,\ldots,n\}. (b) Do there exist integers x,nx,n for which S(x,n)=0S(x,n)=0?
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