4
Part of 2012 Bundeswettbewerb Mathematik
Problems(2)
7 vertices of a 27-gon, among 3 form isosceles or 4 isosceles trapezoid
Source: Germany Federal - Bundeswettbewerb Mathematik 2012, round 1, p4
4/11/2020
From the vertices of a regular 27-gon, seven are chosen arbitrarily. Prove that among these seven points there are three points that form an isosceles triangle or four points that form an isosceles trapezoid.
combinatorial geometryregular polygonisoscelestrapezoidIsosceles Trianglecombinatoricsgeometry
rectangle with dimensions a,b such no lattice point interior or on it's edge
Source: Germany Federal - Bundeswettbewerb Mathematik 2012, round 2, p4
4/15/2020
A rectangle with the side lengths and with should be placed in a right-angled coordinate system so that there is no point with integer coordinates in its interior or on its edge.
Under what necessary and at the same time sufficient conditions for and is this possible?
rectanglecombinatoricscombinatorial geometrylattice pointslattice