Problems

In a football team (made up of 11 people), a captain and his deputy need to be chosen. How many ways can this be done?

If a salary is first increased by 20%, and then reduced by 20%, will the salary paid increase or decrease as a result?

Arrange in a row the numbers from 1 to 100 so that any two neighbouring ones differ by at least 50.

The numbers from 1 to 9999 are written out in a row. How can I remove 100 digits from this row so that the remaining number is a) maximal b) minimal?

A group of 20 tourists go on a trip. The oldest member of the group is 35, the youngest is 20. Is it true that there are members of the group that are the same age?

Two grandmasters in turn put rooks on a chessboard (one turn – one rook) so that they cannot capture each other. The person who cannot put a rook on the chessboard loses. Who will win with the game – the first or second grandmaster?

In each square of a rectangular table of size \(M \times K\), a number is written. The sum of the numbers in each row and in each column, is 1. Prove that \(M = K\).

Is it possible to draw this picture (see the figure), without taking your pencil off the paper and going along each line only once?

One of five brothers baked a cake for their Mum. Alex said: “This was Vernon or Tom.” Vernon said: “It was not I and not Will who did it.” Tom said: “You’re both lying.” David said: “No, one of them told the truth, and the other was lying.” Will said: “No David, you’re wrong.” Mum knows that three of her sons always tell the truth. Who made the cake?

During the election for the government of the planet of Liars and Truth-Tellers, \(12\) candidates each gave a short speech about themselves.
After everyone had spoken, one alien said: “So far, only one lie has been told today.”
Then another said: “And now two have been said so far.”
The third said: “And now three lies have been told so far,” and so on — until the twelfth alien said: “And now twelve lies have been told so far.”
It turned out that at least one candidate had correctly counted how many lies had been told before their own statement.

How many lies were said that day in total?