Suppose that $p$ and $q$ are two different positive integers and $x$ is a real number. Form the product $(x+p)(x+q).$ Find the sum $S(x,n) = \sum (x+p)(x+q),$ where $p$ and $q$ take values from 1 to $n.$ Does there exist integer values of $x$ for which $S(x,n) = 0.$
Problem
Source: IMO LongList 1967, Romania 3
Tags: algebra, polynomial, Summation, equation, Diophantine equation, IMO Shortlist, IMO Longlist