Olympiad problems

2013 ELMO Shortlist, 2

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 $P(n)$? [i]Proposed by Andre Arslan[/i]

2013 ELMO Problems, 5

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 $P(n)$? [i]Proposed by Andre Arslan[/i]

2013 ELMO Shortlist, 2

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 $P(n)$? [i]Proposed by Andre Arslan[/i]