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#### Equilateral triangle , Painting problems , Pigeonhole principle (finite number of poits, lines etc.)

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.

#### Equilateral triangle , Pigeonhole principle (angles and lengths) , Pigeonhole principle (finite number of poits, lines etc.)

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.

#### Equilateral triangle

A circle divides each side of a triangle into three equal parts. Prove that this triangle is regular.

#### Equilateral triangle , Painting problems , Pigeonhole principle (finite number of poits, lines etc.)

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.

#### Equilateral triangle , Hexagons , Pigeonhole principle (finite number of poits, lines etc.) , Polygons and polyhedra with vertices in lattice points , Regular polygons

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.

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