Problems

Prove that if 21 people collected 200 nuts between them, there are two people in the group who collected the same number of nuts.

There are \(n\) integers. Prove that among them either there are several numbers whose sum is divisible by \(n\) or there is one number divisible by \(n\) itself.

In Mongolia there are in circulation coins of 3 and 5 tugriks. An entrance ticket to the central park costs 4 tugriks. One day before the opening of the park, a line of 200 visitors queued up in front of the ticket booth. Each of them, as well as the cashier, has exactly 22 tugriks. Prove that all of the visitors will be able to buy a ticket in the order of the queue.

There is a rectangular table. Two players start in turn to place on it one pound coin each, so that these coins do not overlap one another. The player who cannot make a move loses. Who will win with the correct strategy?

a) Two players play in the following game: on the table there are 7 two pound coins and 7 one pound coins. In a turn it is allowed to take coins worth no more than three pounds. The one who takes the last coin wins. Who will win with the correct strategy?

b) The same question, if there are 12 one pound and 12 two pound coins.

a) Prove that in any football team there are two players who were born on the same day of the week.

b) Prove that in the population of London, which is almost 9 million, there will be ten thousand people who celebrate their birthday on the same day.

Kai has a piece of ice in the shape of a “corner” (see the figure). The Snow Queen demanded that Kai cut it into four equal parts. How can he do this?

The king made a test for the future groom of his daughter. He put the princess in one of three rooms, a tiger in the other, and left the last room empty. It is known that the sign on the door where the princess is sitting is true, where the tiger is – it is false, and nothing is known about the sign on the third room. The tablets are as follows:

1 – room 3 is empty

2 – the tiger is in room 1

3 – this room is empty

Can the prince correctly guess the room with the princess?

On Easter Island, people ask each other questions, to which only “yes” or “no” can be answered. In this case, each of them belongs exactly to one of the tribes either A or B. People from tribe A ask only those questions to which the correct answer is “yes”, and from tribe B – those questions to which the correct answer is “no.” In one house lived a couple Ethan and Violet Russell. When Inspector Krugg approached the house, the owner met him on the doorstep with the words: “Tell me, do Violet and I belong to tribe B?”. The inspector thought and gave the right answer. What was the right answer?

a) A 1 or a 0 is placed on each vertex of a cube. The sum of the 4 adjacent vertices is written on each face of the cube. Is it possible for each of the numbers written on the faces to be different?

b) The same question, but if 1 and \(-1\) are used instead.