10. 2019 circles split a plane into a number of parts whose boundaries are arcs of those circles. How many colors are needed to color this geographic map if any two neighboring parts must be coloured with different colours?
Solution. Each point lies either within an even number of circles (area of type 1), or inside an odd number of them (area of type 2). Crossing an arc we move from area of one type to an area of another type. Therefore, two colors would be enough colouring areas of one type, say with a red colour and areas of another type with a blue colour.