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?
A raisin bag contains 2001 raisins with a total weight of 1001 g, and no raisin weighs more than 1.002 g.
Prove that all the raisins can be divided onto two scales so that they show a difference in weight not exceeding 1 g.
Three people are talking at dinner: Greyson, Blackburne and Reddick. The black-haired person told Greyson: “It is curious that one of us is grey-haired, the other is black-haired, and the third is red-haired, but no one has hair colour that matches their surname.” What colour hair does each of the men chatting have?
In a group of friends, each two people have exactly five common acquaintances. Prove that the number of pairs of friends is divisible by 3.
Of 11 balls, 2 are radioactive. For any set of balls in one check, you can find out if there is at least one radioactive ball in it (but you cannot tell how many of them are radioactive). Is it possible to find both radioactive balls in 7 checks?
One and a half diggers dig for a half hour and end up having dug half a pit. How many pits will two diggers dig in two hours?