RU-ALLRUSSIAN 2016 Problem 4

There is three-dimensional space. For every integer $n$ we build planes $x \pm y\pm z = n$ . All space is divided on octahedrons and tetrahedrons. Point $(x_0,y_0,z_0)$ has rational coordinates but not lies on any plane. Prove, that there is such natural $k$ , that point $(kx_0,ky_0,kz_0)$ lies strictly inside the octahedron of partition.