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.
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.
Find the number of zeros in which the number \(11^{100} - 1\) ends.
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?
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.
In the TV series “The Secret of Santa Barbara” there are 20 characters. Each episode contains one of the events: some character discovers the Mystery, some character discovers that someone knows the Mystery, some character discovers that someone does not know the Mystery. What is the maximum number of episodes that this tv series can last?
A traveller who came to the planet hired a local as a guide. They went for a walk and saw another alien. The traveller sent the guide to find out to whether this native is a liar or truth teller. The guide returned and said: “The native says that they are a truth teller.” Who was the guide: a liar or a truth teller?