7 natural numbers are written around the edges of a circle. It is known that in each pair of adjacent numbers one is divisible by the other. Prove that there will be another pair of numbers that are not adjacent that share this property.
There are 7 points placed inside a regular hexagon of side length 1 unit. Prove that among the points there are two which are less than 1 unit apart.
Is it possible to transport 50 stone blocks, whose masses are equal to \(370, 372,\dots, 468\) kg, from a quarry on seven 3-tonne trucks?
There is a counter on the chessboard. Two in turn move the counter to an adjacent on one side cell. It is forbidden to put a counter on a cell, which it has already visited. The one who can not make the next turn loses. Who wins with the right strategy?
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?