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. analytic geometrycombinatorics unsolvedcombinatorics