A Cartesian plane is coloured in in two colours. Prove that there will be two points on the plane that are a distance of 1 apart and are the same colour.
A circle divides each side of a triangle into three equal parts. Prove that this triangle is regular.
There are 5 points inside an equilateral triangle with side of length 1. Prove that the distance between some two of them is less than 0.5.
A white plane is arbitrarily sprinkled with black ink. Prove that for any positive \(l\) there exists a line segment of length \(l\) with both ends of the same colour.