MathDB

IMO LongList 1967, Romania 3

Source: IMO LongList 1967, Romania 3

December 16, 2004

algebrapolynomialSummationequationDiophantine equationIMO ShortlistIMO Longlist

Problem Statement

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.