MathDB

W. 41

Part of 2019 Jozsef Wildt International Math Competition

Problems(1)

Show that number of points inside the tetrahedron

Source: 2019 Jozsef Wildt International Math Competition

5/19/2020
For n∈Nn \in \mathbb{N}, consider in R3\mathbb{R}^3 the regular tetrahedron with vertices O(0,0,0)O(0, 0, 0), A(n,9n,4n)A(n, 9n, 4n), B(9n,4n,n)B(9n, 4n, n) and C(4n,n,9n)C(4n, n, 9n). Show that the number NN of points (x,y,z)(x, y, z), [x,y,z∈Z][x, y, z \in \mathbb{Z}] inside or on the boundary of the tetrahedron OABCOABC is given byN=343n33+35n22+7n6+1N=\frac{343n^3}{3}+\frac{35n^2}{2}+\frac{7n}{6}+1
3D geometrycoordinate geometrygeometrytetrahedron