A pentagon is inscribed in a circle of radius 1. Prove that the sum of the lengths of its sides and diagonals is less than 17.
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.
On a plane, there are given 2004 points. The distances between every pair of points is noted. Prove that among these noted distances at least thirty numbers are different.
On the planet Tau Ceti, the landmass takes up more than half the surface area. Prove that the Tau Cetians can drill a hole through the centre of their planet that connects land to land.
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.
An area of airspace contains clouds. It turns out that the area can be divided by 10 aeroplanes into regions such that each region contains no more than one cloud. What is the largest number of clouds an aircraft can fly through whilst holding a straight line course.
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 there is no polyhedron that has exactly seven edges.
Prove that no straight line can cross all three sides of a triangle, at points away from the vertices.