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Points in octahedrons

Source: All russian olympiad 2016,Day1,grade 11,P4

May 5, 2016
coordinate geometrynumber theorygeometry3D geometryoctahedrontetrahedronanalytic geometry

Problem Statement

There is three-dimensional space. For every integer nn we build planes x±y±z=n x \pm y\pm z = n. All space is divided on octahedrons and tetrahedrons. Point (x0,y0,z0)(x_0,y_0,z_0) has rational coordinates but not lies on any plane. Prove, that there is such natural kk , that point (kx0,ky0,kz0)(kx_0,ky_0,kz_0) lies strictly inside the octahedron of partition.
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