A carpet of size 4 m by 4 m has had 15 holes made in it by a moth. Is it always possible to cut out a 1 m \(\times\) 1 m area of carpet that doesn’t contain any holes? The holes are considered to be points.
The natural number \(a\) was increased by 1, and its square increased by 1001. What is \(a\)?
In a basket, there are 30 red and green apples. Among any 12 apples there is at least one red one, and among any 20 apples there is at least one green one. How many red apples and how many green apples are there in the basket?
In the numbers of MEXAILO and LOMONOSOV, each letter denotes a number (different letters correspond to different numbers). It is known that the products of the numbers of these two words are equal. Can both numbers be odd?
A game with 25 coins. In a row there are 25 coins. For a turn it is allowed to take one or two neighbouring coins. The player who has nothing to take loses.
There are three piles of rocks: in the first pile there are 10 rocks, 15 in the second pile and 20 in the third pile. In this game (with two players), in one turn a player is allowed to divide one of the piles into two smaller piles. The loser is the one who cannot make a move. Which player would be the winner?
In the first pile there are 100 sweets and in the second there are 200. Consider the game with two players where: in one turn a player can take any amount of sweets from one of the piles. The winner is the one who takes the last sweet. Which player would win by using the correct strategy?
The grandad is twice as strong as the grandma, the grandma is three times stronger than the granddaughter, the granddaughter is four times stronger than the dog, the dog is five times stronger than the cat and the cat is six times stronger than the mouse. The grandad, the grandma, the granddaughter, the dog and the cat together with the mouse can pull out the pumpkin from the ground, which they cannot do without the mouse. How many mice should be summoned so that they can pull out the pumpkin themselves?
Is it possible to cut out such a hole in a sheet of paper through which a person could climb through?
There are two hourglasses – one for 7 minutes and another for 11 minutes. An egg is boiled for 15 minutes. How can this time be measured with the help of the available hourglasses?