Problems

This time Alex is colouring a circle. They once again used only red and blue. Show that there are three points on the circle which are all the same colour and form an isosceles triangle.

Alex coloured the plane with red and green crayons. Show that there is an equilateral triangle with all vertices of the same colour somewhere on the plane.

Alex used two colours to colour the plane, red and green. Show that there is a segment of length \(1\) whose endpoints are the same colour.

Find the minimum number of colours Alex needs to paint the following figure in such a way that no connected points are of the same colour.

On the green and red coloured plane, show that there is always a rectangle whose vertices all have the same colour.

Show that if Alex uses only \(7\) colours to colour the entire plane, it is always possible to find two points of different colours with distance \(1\) from each other.

Alex now wants each straight line on the plane to only consist of points of two or fewer colours. How many colours in total can they use to colour the plane?

Alex got tired of colouring the entire plane for now, and decided to only colour grid points - points where lines meet on a sheet of square grid paper. They used red and blue crayons, each at least once. The distance between adjacent grid points is \(1\). Show that there is always a segment of length \(5\) whose ends are different colours.

Alex now decided to use three colours to colour the entire plane again, red, green and gold. Show that there are two points, distance one apart, which are the same colour.
It is also possible to show the same to be true if she used \(4\) colours, including grey for example, but that is significantly more difficult;

The statement of the problem 9 can be generalised for higher dimensions, in particular for the \(3\)-dimensional space. Prove that in order to colour the entire \(3\)-dimensional space in such a way that no two points distance \(1\) apart are of the same colour Alex needs at least \(5\) colours.