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?
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?
A square area of size \(100\times 100\) is covered in tiles of size \(1\times 1\) in 4 different colours – white, red, black, and grey. No two tiles of the same colour touch one another, that is share a side or a corner. How many red tiles can there be?
One corner square was cut from a chessboard. What is the smallest number of equal triangles that can be cut into this shape?
It is known that a certain polynomial at rational points takes rational values. Prove that all its coefficients are rational.
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?
The order of books on a shelf is called wrong if no three adjacent books are arranged in order of height (either increasing or decreasing). How many wrong orders is it possible to construct from \(n\) books of different heights, if: a) \(n = 4\); b) \(n = 5\)?
101 points are marked on a plane; not all of the points lie on the same straight line. A red pencil is used to draw a straight line passing through each possible pair of points. Prove that there will always be a marked point on the plane through which at least 11 red lines pass.
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?
An adventurer is travelling to the planet of liars and truth tellers with an official guide and is introduced to a local. “Are you a truth teller?” asked the adventurer. The alien answers “Yrrg,” which means either “yes” or “no”. The adventurer asks the guide for a translation. The guide says “"yrrg" means "yes". I will add that the local is actully a liar.” Is the local alien liar or truth teller?