Suppose that
![p](/media/m/1/c/8/1c85c88d10b11745150467bf9935f7de.png)
and
![q](/media/m/c/1/d/c1db9b1124cc69b01f9a33595637de69.png)
are two different positive integers and
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
is a real number. Form the product
![(x+p)(x+q).](/media/m/2/8/3/283d81e59c41fa61bdc533c2b9a448a4.png)
Find the sum
![S(x,n) = \sum (x+p)(x+q),](/media/m/1/e/a/1ea31c4fd90fd77451e16c69696cb3b5.png)
where
![p](/media/m/1/c/8/1c85c88d10b11745150467bf9935f7de.png)
and
![q](/media/m/c/1/d/c1db9b1124cc69b01f9a33595637de69.png)
take values from 1 to
![n.](/media/m/4/7/8/478d25bfe04800537ae6e85be9d11ea2.png)
Does there exist integer values of
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
for which
%V0
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 $S(x,n) = 0.$