We are given a convex 200-sided polygon in which no three diagonals intersect at the same point. Each of the diagonals is coloured in one of 999 colours. Prove that there is some triangle inside the polygon whose sides lie some of the diagonals, so that all 3 sides are the same colour. The vertices of the triangle do not necessarily have to be the vertices of the polygon.
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.
Arrows are placed on the sides of a polygon. Prove that the number of vertices in which two arrows converge is equal to the number of vertices from which two arrows emerge.
2001 vertices of a regular 5000-gon are painted. Prove that there are three coloured vertices lying on the vertices of an isosceles triangle.
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.
Prove that a convex quadrilateral \(ABCD\) can be drawn inside a circle if and only if \(\angle ABC + \angle CDA = 180^{\circ}\).
a) Prove that the axes of symmetry of a regular polygon intersect at one point.
b) Prove that the regular \(2n\)-gon has a centre of symmetry.
a) The convex \(n\)-gon is cut by diagonals that do not cross to form triangles. Prove that the number of these triangles is equal to \(n - 2\).
b) Prove that the sum of the angles at the vertices of a convex \(n\)-gon is \((n - 2) \times 180^{\circ}\).
Prove that the midpoints of the sides of a regular polygon form a regular polygon.
A unit square contains 51 points. Prove that it is always possible to cover three of them with a circle of radius \(\frac{1}{7}\).