Problem

Source: 2023 Olympic Revenge Problem 4

Tags: geometry, 3D geometry, sphere, combinatorial geometry, combinatorics, lattice points



Let $S=\{(x,y,z)\in \mathbb{Z}^3\}$ the set of points with integer coordinates in the space. Gugu has infinitely many solid spheres. All with radii $\ge (\frac{\pi}2)^3$. Is it possible for Gugu to cover all points of $S$ with his spheres?