4

Part of 2018 Bundeswettbewerb Mathematik

Problems(2)

Points in space with distinct distances form certain triangles

Source: Bundeswettbewerb Mathematik 2018, Round 1 - #4

We are given six points in space with distinct distances, no three of them collinear. Consider all triangles with vertices among these points.
Show that among these triangles there is one such that its longest side is the shortest side in one of the other triangles.

geometrycombinatorics3D geometryTriangledistance

An isosceles right triangle in a colouring of lattice points

Source: Bundeswettbewerb Mathematik 2018, Round 2, Problem 4

Determine alle positive integers

n>1

with the following property: For each colouring of the lattice points in the plane with

n

colours, there are three lattice points of the same colour forming an isosceles right triangle with legs parallel to the coordinate axes.

combinatoricscombinatorics unsolvedcombinatorial geometry