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 \((n - 2) \times 180^{\circ}\).