Problem #PRU-65760

Problem

Author: A. Glazyrin

In the coordinate space, all planes with the equations $x\pm y\pm z=n$ (for all integers $n$) were carried out. They divided the space into tetrahedra and octahedra. Suppose that the point $(x_0,y_0,z_0)$ with rational coordinates does not lie in any plane. Prove that there is a positive integer $k$ such that the point $(k{x}_0,k{y}_0,k{z}_0)$ lies strictly inside some octahedron from the partition.