\(2n\) diplomats sit around a round table. After a break the same \(2n\) diplomats sit around the same table, but this time in a different order.
Prove that there will always be two diplomats with the same number of people sitting between them, both before and after the break.
The planet has \(n\) residents, some are liars and some are truth tellers. Each resident said: “Among the remaining residents of the island, more than half are liars.” How many liars are on the island?
You are given a table of size \(m \times n\) (\(m, n > 1\)). In it, the centers of all cells are marked. What is the largest number of marked centers that can be chosen so that no three of them are the vertices of a right triangle?
A gang contains 50 gangsters. The whole gang has never taken part in a raid together, but every possible pair of gangsters has taken part in a raid together exactly once. Prove that one of the gangsters has taken part in no less than 8 different raids.
A \(99 \times 99\) chequered table is given, each cell of which is painted black or white. It is allowed (at the same time) to repaint all of the cells of a certain column or row in the colour of the majority of cells in that row or column. Is it always possible to have that all of the cells in the table are painted in the same colour?
There are several cities (more than one) in a country; some pairs of cities are connected by roads. It is known that you can get from every city to any other city by driving along several roads. In addition, the roads do not form cycles, that is, if you leave a certain city on some road and then move so as not to pass along one road twice, it is impossible to return to the initial city. Prove that in this country there are at least two cities, each of which is connected by a road with exactly one city.
Two points are placed inside a convex pentagon. Prove that it is always possible to choose a quadrilateral that shares four of the five vertices on the pentagon, such that both of the points lie inside or on the boundary the quadrilateral.
You are given 10 different positive numbers. In which order should they be named \(a_1, a_2, \dots , a_{10}\) such that the sum \(a_1 +2a_2 +3a_3 +\dots +10a_{10}\) is at its maximum?
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.