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#### Central angle. Arc length and circumference , Chords and secants (other) , Isosceles, inscribed, and circumscribed trapeziums , Pigeonhole principle (finite number of poits, lines etc.) , Regular polygons , Two tangent lines to a circle, intersecting at a particular point

In a regular shape with 25 vertices, all the diagonals are drawn.
Prove that there are no nine diagonals passing through one interior point of the shape.

#### Discrete geometry (other) , Pigeonhole principle (finite number of poits, lines etc.) , Regular polygons , Trapeziums (other)

A regular 1981-gon has 64 vertices. Prove that there exists a trapezium with vertices at the marked points.

#### Incirlce and circumcircle of a triangle , Pentagons , Pigeonhole principle (finite number of poits, lines etc.) , Regular polygons

All the points on the edge of a circle are coloured in two different colours at random. Prove that there will be an equilateral triangle with vertices of the same colour inside the circle – the vertices are points on the circumference of the circle.

#### Partitions into pairs and groups bijections , Pigeonhole principle (finite number of poits, lines etc.) , Regular polygons

2001 vertices of a regular 5000-gon are painted. Prove that there are three coloured vertices lying on the vertices of an isosceles triangle.

#### A segment inside the triangle is smaller than the largest side , Hexagons , Pigeonhole principle (angles and lengths) , Regular polygons

There are 7 points placed inside a regular hexagon of side length 1 unit. Prove that among the points there are two which are less than 1 unit apart.

#### 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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