Problem

Source: ELMO 2013/5, by Andre Arslan; also Shortlist N2

Tags: algebra, polynomial, geometry, 3D geometry, modular arithmetic, number theory, hehecombinatoricstoo



For what polynomials $P(n)$ with integer coefficients can a positive integer be assigned to every lattice point in $\mathbb{R}^3$ so that for every integer $n \ge 1$, the sum of the $n^3$ integers assigned to any $n \times n \times n$ grid of lattice points is divisible by $P(n)$? Proposed by Andre Arslan