MathDB

2

Part of 2006 Spain Mathematical Olympiad

Problems(2)

finite number of octahedra cutting a painted wooden octahedron

Source: 2006 Spanish Mathematical Olympiad P2

7/20/2018
The dimensions of a wooden octahedron are natural numbers. We painted all its surface (the six faces), cut it by planes parallel to the cubed faces of an edge unit and observed that exactly half of the cubes did not have any painted faces. Prove that the number of octahedra with such property is finite.
(It may be useful to keep in mind that 123=1,79...<1,8\sqrt[3]{\frac{1}{2}}=1,79 ... <1,8).
[hide=original wording] Las dimensiones de un ortoedro de madera son enteras. Pintamos toda su superficie (las seis caras), lo cortamos mediante planos paralelos a las caras en cubos de una unidad de arista y observamos que exactamente la mitad de los cubos no tienen ninguna cara pintada. Probar que el número de ortoedros con tal propiedad es finito
octahedroncombinatorial geometryColoringgeometry3D geometry
product of 4 consecutive naturals not square nor cube

Source: 2006 Spanish Mathematical Olympiad P5

7/20/2018
Prove that the product of four consecutive natural numbers can not be neither square nor perfect cube.
number theoryProductconsecutivePerfect Squareperfect cube