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Part of 2015 Spain Mathematical Olympiad
Problems(2)
Graph of polynomial with integer coefficients
Source: Spanish Mathematical Olympiad. 2015. Problem 1
4/30/2015
On the graph of a polynomial with integer coefficients, two points are chosen with integer coordinates. Prove that if the distance between them is an integer, then the segment that connects them is parallel to the horizontal axis.
algebrapolynomial
Three colours for each face of a polyhedron
Source: Spanish Mathematical Olympiad, 2015, Problem 4
4/30/2015
All faces of a polyhedron are triangles. Each of the vertices of this polyhedron is assigned independently one of three colors : green, white or black. We say that a face is Extremadura if its three vertices are of different colors, one green, one white and one black. Is it true that regardless of how the vertices's color, the number of Extremadura faces of this polyhedron is always even?
polyhedronGraph coloringcombinatorics