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There are 6 locked suitcases and 6 keys for them. It is not known which keys are for which suitcase. What is the smallest number of attempts do you need in order to open all the suitcases? How many attempts would you need if there are 10 suitcases and keys instead of 6?

A traveller rents a room in an inn for a week and offers the innkeeper a chain of seven silver links as payment – one link per day, with the condition that they will be payed everyday. The innkeeper agrees, with the condition that the traveller can only cut one of the links. How did the traveller manage to pay the innkeeper?

With a non-zero number, the following operations are allowed: $x \rightarrow \frac{1+x}{x}, x \rightarrow \frac{1-x}{x}$. Is it true that from every non-zero rational number one can obtain each rational number with the help of a finite number of such operations?

A staircase has 100 steps. Vivian wants to go down the stairs, starting from the top, and she can only do so by jumping down and then up, down and then up, and so on. The jumps can be of three types – six steps $($jumping over five to land on the sixth$)$, seven steps or eight steps. Note that Vivian does not jump onto the same step twice. Will she be able to go down the stairs?

Jack and Ben had a bicycle on which they went to a neighbouring village. They rode it in turns, but whenever one rode, the other walked and did not run. They managed to arrive in the village at the same time and almost twice as fast than if they had both walked. How did they do it?

Fred and George together with their mother were decorating the Christmas tree. So that they would not fight, their mother gave each brother the same number of decorations and branches. Fred tried to hang one decoration on each branch, but he needed one more branch for his last decoration. George tried to hang two toys on each branch, but one branch was empty. What do you think, how many branches and how many decorations did the mother give to her sons?

A group of $2n$ people were gathered together, of whom each person knew no less than $n$ of the other people present. Prove that it is possible to select 4 people and seat them around a table so that each person sits next to people they know. ( $n \geq 2$)

The number $A$ is divisible by $1, 2, 3, …, 9$. Prove that if $2A$ is presented in the form of a sum of some natural numbers smaller than 10, $2A= a_1 +a_2 +…+a_k$, then we can always choose some of the numbers $a_1, a_2, …, a_k$ so that the sum of the chosen numbers is equal to $A$.

100 children were each given a bowl with 100 pieces of pasta. However, these children did not want to eat and instead started to play. One of the children started to place one piece of her pasta into other children’s bowls $($to whomever she wants$)$. What is the least amount of transfers needed so that everyone has a different number of pieces of pasta in their bowl?

10 children were each given a bowl with 100 pieces of pasta. However, these children did not want to eat and instead started to play. One of the children started to place one piece of pasta into every other child’s bowl. What is the least amount of transfers needed so that everyone has a different number of pieces of pasta in their bowl?

a$)$ There is an unlimited set of cards with the words “abc”, “bca”, “cab” written. Of these, the word written is determined according to this rule. For the initial word, any card can be selected, and then on each turn to the existing word, you can either add on a card to the left or to the right, or cut the word anywhere $($between the letters$)$ and put a card there. Is it possible to make a palindrome with this method?

b$)$ There is an unlimited set of red cards with the words “abc”, “bca”, “cab” and blue cards with the words “cba”, “acb”, “bac”. Using them, according to the same rules, a palindrome was made. Is it true that the same number of red and blue cards were used?

A castle is surrounded by a circular wall with nine towers, at which there are knights on duty. At the end of each hour, they all move to the neighbouring towers, each knight moving either clockwise or counter-clockwise. During the night, each knight stands for some time at each tower. It is known that there was an hour when at least two knights were on duty at each tower, and there was an hour when there was precisely one knight on duty on each of exactly five towers. Prove that there was an hour when there were no knights on duty on one of the towers.

Sam and Lena have several chocolates, each weighing not more than 100 grams. No matter how they share these chocolates, one of them will have a total weight of chocolate that does not exceed 100 grams. What is the maximum total weight of all of the chocolates?

There were seven boxes. In some of them, seven more boxes were placed inside $($not nested in each other$)$, etc. As a result, there are 10 non-empty boxes.

How many boxes are there now in total?

In order to glaze 15 windows of different shapes and sizes, 15 pieces of glass are prepared exactly for the size of the windows $($windows are such that each window should have one glass$)$. The glazier, not knowing that the glass is specifically selected for the size of each window, works like this: he approaches a certain window and sorts out the unused glass until he finds one that is large enough $($that is, either an exactly suitable piece or one from which the right size can be cut$)$, if there is no such glass, he goes to the next window, and so on, until he has assessed each window. It is impossible to make glass from several parts. What is the maximum number of windows which can be left unglazed?

We are given a table of size $n \times n$. $n-1$ of the cells in the table contain the number $1$. The remainder contain the number $0$. We are allowed to carry out the following operation on the table:

1. Pick a cell.

2. Subtract 1 from the number in that cell.

3. Add 1 to every other cell in the same row or column as the chosen cell.

Is it possible, using only this operation, to create a table in which all the cells contain the same number?