A raisin bag contains 2001 raisins with a total weight of 1001 g, and no raisin weighs more than 1.002 g.
Prove that all the raisins can be divided onto two scales so that they show a difference in weight not exceeding 1 g.
10 numbers are written around the circle, the sum of which is equal to 100. It is known that the sum of every three numbers standing side by side is not less than 29.
Specify the smallest number \(A\) such that in any such set of numbers each of the numbers does not exceed \(A\).
A daisy has a) 12 petals; b) 11 petals. Consider the game with two players where: in one turn a player is allowed to remove either exactly one petal or two petals which are next to each other. The loser is the one who cannot make a turn. How should the second player act, in cases a) and b), in order to win the game regardless of the moves of the first player?
On the board the number 1 is written. Two players in turn add any number from 1 to 5 to the number on the board and write down the total instead. The player who first makes the number thirty on the board wins. Specify a winning strategy for the second player.
There are two stacks of coins on a table: in one of them there are 30 coins, and in the other – 20. You can take any number of coins from one stack per move. The player who cannot make a move is the one that loses. Which player wins with the correct strategy?
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.
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.)
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?