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Snow White cut out a big square of cotton fabric and placed it in a chest. The first gnome came, took out the square of fabric from the chest, cut it into four squares, put these back in the chest and left. Later the second gnome came and took out one of the squares and then cut it into four pieces and placed all of these in the chest. Then came the third gnome. He also took out one of the squares and cut it into four squares and put them all back in the chest. The rest of the gnomes also did the same thing. How many squares of fabric were in the chest after the seventh gnome left?

Carpenters were sawing some logs. They made 10 cuts and this produced 16 pieces of wood. How many logs did they saw?

In a 10-storey house, 1 person lives on the first floor, 2 on the second floor, 3 on the third, 4 on the fourth, … 10 on the tenth. On which floor does the elevator stop most often?

Among 4 people there are no three with the same name, the same middle name and the same surname, but any two people have either the same first name, middle name or surname. Can this be so?

Several natives of an island met up $($each either a liar or a knight$)$, and everyone said to everyone else: “You are all liars.” How many knights were there among them?

The population of China is one billion people. It would seem that on a map of China with a scale of 1: 1,000,000 $($1 cm: 10 km$)$, it would be possible to fit a million times fewer people than there is in the whole country. However, in fact, not only 1000, but even 100 people will not be able to be placed on this map. Can you explain this contradiction?

Elephants, rhinoceroses, giraffes. In all zoos where there are elephants and rhinoceroses, there are no giraffes. In all zoos where there are rhinoceroses and there are no giraffes, there are elephants. Finally, in all zoos where there are elephants and giraffes, there are also rhinoceroses. Could there be a zoo in which there are elephants, but there are no giraffes and no rhinoceroses?

20 birds fly into a photographer’s studio – 8 starlings, 7 wagtails and 5 woodpeckers. Each time the photographer presses the shutter to take a photograph, one of the birds flies away and doesn’t come back. How many photographs can the photographer take to be sure that at the end there will be no fewer than 4 birds of one species and no less than 3 of another species remaining in the studio.

The case of Brown, Jones and Smith is being considered. One of them committed a crime. During the investigation, each of them made two statements. Brown: “I did not do it. Jones did not do it. ” Smith: “I did not do it. Brown did it. “Jones:” Brown did not do it. This was done by Smith. “Then it turned out that one of them had told the truth in both statements, another had lied both times, and the third had told the truth once, and he had lied once. Who committed the crime?

So, the mother exclaimed – “It’s a miracle!”, and immediately the mum, dad and the children went to the pet store. “But there are more than fifty bullfinches here, how will we decide?,” the younger brother nearly cried when he saw bullfinches. “Don’t worry,” said the eldest, “there are less than fifty of them”. “The main thing,” said the mother, “is that there is at least one!”. “Yes, it’s funny,” Dad summed up, “of your three phrases, only one corresponds to reality.” Can you say how many bullfinches there was in the store, knowing that they bought the child a bullfinch?

100 cars are parked along the right hand side of a road. Among them there are 30 red, 20 yellow, and 20 pink Mercedes. It is known that no two Mercedes of different colours are parked next to one another. Prove that there must be three Mercedes cars parked next to one another of the same colour somewhere along the road.

In a basket, there are 30 mushrooms. Among any 12 of them there is at least one brown one, and among any 20 mushrooms, there is at least one chanterelle. How many brown mushrooms and how many chanterelles are there in the basket?

Two players in turn paint the sides of an n-gon. The first one can paint the side that borders either zero or two colored sides, the second – the side that borders one painted side. The player who can not make a move loses. At what n can the second player win, no matter how the first player plays?

2022 dollars were placed into some wallets and the wallets were placed in some pockets. It is known that there are more wallets in total than there are dollars in any pocket. Is it true that there are more pockets than there are dollars in one of the wallets? You are not allowed to place wallets one inside the other.

On a table there are 2022 cards with the numbers 1, 2, 3, …, 2022. Two players take one card in turn. After all the cards are taken, the winner is the one who has a greater last digit of the sum of the numbers on the cards taken. Find out which of the players can always win regardless of the opponent’s strategy, and also explain how he should go about playing.

Janine and Zahara each thought of a natural number and said them to Alex. Alex wrote the sum of the thought of numbers onto one sheet of paper, and on the other – their product, after which one of the sheets was hidden, and the other $($ on it was written the number of 2002 $)$ was shown to Janine and Zahara. Seeing this number, Janine said that she did not know what number Zahara had thought of. Hearing this, Zahara said that she did not know what number Janine had thought of. What was the number which Zahara had thought of?

Are there such irrational numbers $a$ and $b$ so that $a > 1,\, b > 1,$ and [$a^m$] is different from [$b^n$] for any natural numbers $m$ and $n$?

In Conrad’s collection there are four royal gold five-pound coins. Conrad was told that some two of them were fake. Conrad wants to check $($ prove or disprove $)$ that among the coins there are exactly two fake ones. Will he be able to do this with the help of two weighings on weighing scales without weights? $($ Counterfeit coins are the same in weight, real ones are also the same in weight, but false ones are lighter than real ones. $)$

Two people are playing. The first player writes out numbers from left to right, randomly alternating between 0 and 1, until there are 2021 numbers in total. Each time after the first one writes out the next digit, the second switches two numbers from the already written row $($ when only one digit is written, the second misses its move $)$. Is the second player always able to ensure that, after his last move, the arrangement of the numbers is symmetrical relative to the middle number?

A game takes place on a squared 9 × 9 piece of checkered paper. Two players play in turns. The first player puts crosses in empty cells, its partner puts noughts. When all the cells are filled, the number of rows and columns in which there are more crosses than zeros is counted, and is denoted by the number K, and the number of rows and columns in which there are more zeros than crosses is denoted by the number H $($ 18 rows in total $)$. The difference B = K – H is considered the winnings of the player who goes first. Find a value of B such that

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1$)$ the first player can secure a win of no less than B, no matter how the second player played;

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2$)$ the second player can always make it so that the first player will receive no more than B, no matter how he plays.