CAPS 2001 Problem 6

Points with integer coordinates in cartesian space are called lattice points. We color $2000$ lattice points blue and $2000$ other lattice points red in such a way that no two blue-red segments have a common interior point (a segment is blue-red if its two endpoints are colored blue and red). Consider the smallest rectangular parallelepiped that covers all the colored points. (a) Prove that this rectangular parallelepiped covers at least $500,000$ lattice points. (b) Give an example of a coloring for which the considered rectangular paralellepiped covers at most $8,000,000$ lattice points.