IMO 1967 LL ROM44

Suppose p and q are two different positive integers and x is a

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IMO 1967 LL ROM44

Origin: ROM

Problem

Suppose p and q are two different positive integers and x is a real number. Form the product (x + p)(x + q). (a) Find the sum S(x, n) = (x + p)(x + q), where p and q take values from 1 to n. (b) Do there exist integer values of x for which S(x, n) = 0?

Solution

(a) S(x, n) = n(n −1) / x2 + (n + 1)x + (n + 1)(3n + 2)/12 . (b) It is easy to see that the equation S(x, n) = 0 has two roots x1,2 =  −(n + 1) \pm

(n + 1)/3

/2. They are integers if and only if n = 3k2 −1 for some k \inN.