Problems

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 a tournament by the Olympic system (the loser is eliminated), 50 boxers participate. What is the minimum number of matches needed to be conducted in order to identify the winner?

Four aliens – Dopey, Sleepy, Happy, Moody from the planet of liars and truth tellers had a conversation: Dopey to Sleepy: “you are a liar”; Happy to Sleepy: “you are a liar”; Moody to Happy: “Yes, they are both liars,” (after a moment’s thought), “however, so are you.” Which of them is telling the truth?

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?

What is the maximum number of rooks – also known as castles – you could place on an 8 by 8 chess board such that no two could take one another? Rooks can attack any number of squares horizontally and vertically, but not diagonally.

Lessons at the Evening Mathematical School take place in nine auditoriums. Amongst the class were 19 students from the same school.

a) Prove that no matter how these students are arranged at least one auditorium will contain no fewer than 3 of these students.

b) Is it true that one of the auditoriums must contain exactly 3 of these students?

12 straight lines passing through the origin are drawn on a plane. Prove that it is possible to choose two of these lines such that the angle between them is less than 17 degrees.

It is known that among the members of the government of the Planet of Liars and truth tellers, consisting of 20 members, there is at least one honest one, and also that from any two at least one is a bribe taker. How many bribe takers are there in the government?