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A sequence (a_n) and a polynomial P such that a_m+a_n=P(mn)?

Source: Tuymaada 2004, day 1, problem 1. - Author : A. Golovanov.

May 25, 2007

algebrapolynomialcalculusintegrationalgebra proposed

Problem Statement

Do there exist a sequence

a_{1}, a_{2}, a_{3}, \ldots

of real numbers and a non-constant polynomial

P(x)

such that

a_{m}+a_{n}=P(mn)

for every positive integral

m

and

n?

Proposed by A. Golovanov