MathDB

Show that number of points inside the tetrahedron

Source: 2019 Jozsef Wildt International Math Competition

May 19, 2020

3D geometrycoordinate geometrygeometrytetrahedron

Problem Statement

For

n \in \mathbb{N}

, consider in

\mathbb{R}^3

the regular tetrahedron with vertices

O(0, 0, 0)

A(n, 9n, 4n)

B(9n, 4n, n)

and

C(4n, n, 9n)

. Show that the number

N

of points

(x, y, z)

[x, y, z \in \mathbb{Z}]

inside or on the boundary of the tetrahedron

OABC

is given by

N=\frac{343n^3}{3}+\frac{35n^2}{2}+\frac{7n}{6}+1