Given a board (divided into squares) of the size: a) \(10\times 12\), b) \(9\times 10\), c) \(9\times 11\), consider the game with two players where: in one turn a player is allowed to cross out any row or any column if there is at least one square not crossed out. The loser is the one who cannot make a move. Is there a winning strategy for one of the players?
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.
There are 101 buttons of 11 different colours. Prove that amongst them there are either 11 buttons of the same colour, or 11 buttons of different colours.
Prove that the equation \[a_1 \sin x + b_1 \cos x + a_2 \sin 2x + b_2 \cos 2x + \dots + a_n \sin nx + b_n \cos nx = 0\] has at least one root for any values of \(a_1 , b_1, a_2, b_2, \dots, a_n, b_n\).
A journalist came to a company which had \(N\) people. He knows that this company has a person \(Z\), who knows all the other members of the company, but nobody knows him. A journalist can address each member of the company with the question: “Do you know such and such?” Find the smallest number of questions sufficient to surely find \(Z\). (Everyone answers the questions truthfully. One person can be asked more than one question.)
Let \(f (x)\) be a polynomial about which it is known that the equation \(f (x) = x\) has no roots. Prove that then the equation \(f (f (x)) = x\) does not have any roots.
At a round table, 10 boys and 15 girls were seated. It turned out that there are exactly 5 pairs of boys sitting next to each other.
How many pairs of girls are sitting next to each other?
Prove that there exist numbers, that can be presented in no fewer than 100 ways in the form of a summation of 20001 terms, each of which is the 2000th power of a whole number.
Arrows are placed on the sides of a polygon. Prove that the number of vertices in which two arrows converge is equal to the number of vertices from which two arrows emerge.