MathDB

4

Part of 2021 Bundeswettbewerb Mathematik

Problems(2)

Colouring line segments in a pyramid

Source: Bundeswettbewerb Mathematik 2021, Round 1 - Problem 4

4/12/2021
Consider a pyramid with a regular nn-gon as its base. We colour all the segments connecting two of the vertices of the pyramid except for the sides of the base either red or blue. Show that if n=9n=9 then for each such colouring there are three vertices of the pyramid connecting by three segments of the same colour, and that this is not necessarily the case if n=8n=8.
combinatoricsgraph theoryColoringGraph coloringgeometryedge coloring
Taming a wild rectangle

Source: Bundeswettbewerb Mathematik 2021, Round 2 - Problem 4

9/22/2021
In the Cartesian plane, a line segment is called tame if it lies parallel to one of the coordinate axes and its distance to this axis is an integer. Otherwise it is called wild.
Let mm and nn be odd positive integers. The rectangle with vertices (0,0),(m,0),(m,n)(0,0),(m,0),(m,n) and (0,n)(0,n) is partitioned into finitely many triangles. Let MM be the set of these triangles. Assume that
(1) Each triangle from MM has at least one tame side. (2) For each tame side of a triangle from MM, the corresponding altitude has length 11. (3) Each wild side of a triangle from MM is a common side of exactly two triangles from MM.
Show that at least two triangles from MM have two tame sides each.
analytic geometrygeometryrectangle