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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.

a) The vertices (corners) in a regular polygon with 10 sides are colored black and white in an alternating fashion (i.e. one vertex is black, the next is white, etc). Two people play the following game. Each player in turn draws a line connecting two vertices of the same color. These lines must not have common vertices (i.e. must not begin or end on the same dot as another line) with the lines already drawn. The winner of the game is the player who made the final move. Which player, the first or the second, would win if the right strategy is used?

b) The same problem, but for a regular polygon with 12 sides.

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

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.

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

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.

What is the maximum number of pairwise non-parallel segments with endpoints at the vertices of a regular $n$-gon?

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.