Each whole number is painted either blue or yellow, with the following rules:
The sum of two numbers painted in different colours is painted yellow.
The product of two numbers painted in different colours is painted blue.
There is at least one number of each colour. What possible colours can the product of two blue numbers have?
Every point in the plane is coloured red or blue. Show that there is a colour such that for any distance \(d\), there is a pair of points of that colour that are exactly distance \(d\) apart.