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.
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 circle divides each side of a triangle into three equal parts. Prove that this triangle is regular.
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 regular hexagon with sides of length 5 is divided by straight lines, that are parallel to its sides, to form regular triangles with sides of length 1 $($see the figure$)$.
We call the vertices of all such triangles, nodes. It is known that more than half of the nodes are marked. Prove that there are five marked nodes lying on one circumference.
The total number of nodes is $2 \times (6 + 7 + 8 + 9 + 10)$ = 91. Each node, with the exception of the central node, belongs to one of 11 concentric circles centred at the centre of the hexagon $($see the figure$)$.
Suppose that there are no five marked nodes lying on the same circumference. Then the total number of marked nodes is not more than $11 \times 4$ + 1 = 45, that is, not more than half. This is a contradiction.
