On a board of size \(8 \times 8\), two in turn colour the cells so that there are no corners of three coloured squares. The player who can’t make a move loses. Who wins with the right strategy?
Find the number of zeros in which the number \(11^{100} - 1\) ends.
On a plane there are 100 sheep-points and one wolf-point. In one move, the wolf moves by no more than 1, after which one of the sheep moves by a distance of no more than 1, after that the wolf again moves, etc. At any initial location of the points, will a wolf be able to catch one of the sheep?
Every evening Ross arrives at a random time to the bus stop. Two bus routes stop at this bus stop. One of the routes takes Ross home, and the other takes him to visit his friend Rachel. Ross is waiting for the first bus and depending on which bus arrives, he goes either home or to his friend’s house. After a while, Ross noticed that he is twice as likely to visit Rachel than to be at home. Based on this, Ross concludes that one of the buses runs twice as often as the other. Is he right? Can buses run at the same frequency when the condition of the task is met? (It is assumed that buses do not run randomly, but on a certain schedule).
\(2n\) 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.