The city plan is a rectangle of \(5 \times 10\) cells. On the streets, a one-way traffic system is introduced: it is allowed to go only to the right and upwards. How many different routes lead from the bottom left corner to the upper right?
27 coins are given, of which one is a fake, and it is known that a counterfeit coin is lighter than a real one. How can the counterfeit coin be found from 3 weighings on the scales without weights?
Is it possible to arrange 1000 line segments in a plane so that both ends of each line segment rest strictly inside another line segment?
Two people toss a coin: one tosses it 10 times, the other – 11 times. What is the probability that the second person’s coin showed heads more times than the first?
It is known that \[35! = 10333147966386144929 * 66651337523200000000.\] Find the number replaced by an asterisk.
10 people collected a total of 46 mushrooms in a forest. It is known that no two people collected the same number of mushrooms. How many mushrooms did each person collect?
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?
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?
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?