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?
For which \(n > 3\), can a set of weights with masses of \(1, 2, 3, ..., n\) grams be divided into three groups of equal mass?
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?
A family went to the bridge at night. The dad can cross it in 1 minute, the mum in 2 minutes, the child in 5 minutes, and the grandmother in 10 minutes. They have one flashlight. The bridge only withstands two people. How can they cross the bridge in 17 minutes? (If two people cross, then they pass with the lower of the two speeds. They cannot pass along the bridge without a flashlight. They cannot shine the light from afar. They cannot carry anyone in their arms. They cannot throw the flashlight.)
There are \(n\) cities in a country. Between each two cities an air service is established by one of two airlines. Prove that out of these two airlines at least one is such that from any city you can get to any other city whilst traveling on flights only of this airline.
Several stones weigh 10 tons together, each weighing not more than 1 ton.
a) Prove that this load can be taken away in one go on five three-ton trucks.
b) Give an example of a set of stones satisfying the condition for which four three-ton trucks may not be enough to take the load away in one go.
Three people play table tennis, and the player who lost the game gives way to the player who did not participate in it. As a result, it turned out that the first player played 10 games and the second played 21 games. How many games did the third player play?
In the secret service, there are \(n\) agents – 001, 002, ..., 007, ..., \(n\). The first agent monitors the one who monitors the second, the second monitors the one who monitors the third, etc., the nth monitors the one who monitors the first. Prove that \(n\) is an odd number.