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.
Find the first 99 decimal places in the number expansion of \((\sqrt{26} + 5)^{99}\).
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.
A convex figure and point \(A\) inside it are given. Prove that there is a chord (that is, a segment joining two boundary points of a convex figure) passing through point \(A\) and dividing it in half at point \(A\).
In a communication system consisting of 2001 subscribers, each subscriber is connected with exactly \(n\) others. Determine all the possible values of \(n\).
A raisin bag contains 2001 raisins with a total weight of 1001 g, and no raisin weighs more than 1.002 g.
Prove that all the raisins can be divided onto two scales so that they show a difference in weight not exceeding 1 g.
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 20 students in a class, and each one is friends with at least 14 others. Can you prove that there are four students in this class who are all friends?
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.