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Prove that for any infinite continued fraction [$a_0$; $a_1$, …, $a_n$, …] there exists a limit of its suitable fractions – an irrational number α. Explain why if this number α is decomposed into an infinite continued fraction by the algorithm of task 60606, then an infinite continued fraction is obtained.

a$)$ There are 21 coins on a table with the tails side facing upwards. In one operation, you are allowed to turn over any 20 coins. Is it possible to achieve the arrangement were all coins are facing with the heads side upwards in a few operations?

b$)$ The same question, if there are 20 coins, but you are allowed to turn over 19.

In one urn there are two white balls, in another two black ones, in the third – one white and one black. On each urn there was a sign indicating its contents: WW, BB, WB. Someone rehung the signs so that now each sign indicating the contents of the urn is incorrect. It is possible to remove a ball from any urn without looking into it. What is the minimum number of removals required to determine the composition of all three urns?

To transmit messages by telegraph, each letter of the Russian alphabet $($а, б, в, г, д, е, ё, ж, з, и, й, к, л, м, н, о, п, р, с, т, у, ф, х, ц, ч, ш, щ, ъ, ы, ь, э, ю, я$)$ $($E and Ё are counted as identical$)$ is represented as a five-digit combination of zeros and ones corresponding to the binary number of the given letter in the alphabet $($letter numbering starts from zero$)$. For example, the letter A is represented in the form 00000, letter B-00001, letter Ч-10111, letter Я-11111. Transmission of the five-digit combination is made via a cable containing five wires. Each bit is transmitted on a separate wire. When you receive a message, Cryptos has confused the wires, so instead of the transmitted word, a set of letters ЭАВЩОЩИ is received. Find the word you sent.

A broken calculator carries out only one operation “asterisk”: $a*b = 1 – a/b$. Prove that using this calculator it is possible to carry out all four arithmetic operations (addition, subtraction, multiplication, division).

There are 8 glasses of water on the table. You are allowed to take any two of the glasses and make them have equal volumes of water (by pouring some water from one glass into the other). Prove that, by using such operations, you can eventually get all the glasses to contain equal volumes of water.

Three friends decide, by a coin toss, who goes to get the juice. They have one coin. How do they arrange coin tosses so that all of them have equal chances to not have to go and get the juice?

Hannah has 10 employees. Each month, Hannah raises the salary by 1 pound of exactly nine of her employees $($of her choice$)$. How can Hannah raise the salaries to make them equal? $($Salaries are an integer number of pounds.$)$

Your task is to find out a five-digit phone number, asking questions that can be answered with either “yes” or “no.” What is the smallest number of questions for which this can be guaranteed $($provided that the questions are answered correctly$)$?

Each of the three cutlets should be fried in a pan on both sides for five minutes each side. Only two cutlets can fit onto the frying pan. Is it possible to fry all three cutlets more quickly than in 20 minutes $($if the time to turn over and transfer the cutlets is neglected$)$?

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.$)$

How can one measure out 15 minutes, using an hourglass of 7 minutes and 11 minutes?

10 guests came to a party and each left a pair of shoes in the corridor $($all guests have the same shoes$)$. All pairs of shoes are of different sizes. The guests began to disperse one by one, putting on any pair of shoes that they could fit into $($that is, each guest could wear a pair of shoes no smaller than his own$)$. At some point, it was discovered that none of the remaining guests could find a pair of shoes so that they could leave. What was the maximum number of remaining guests?

A $99 \times 99$ chequered table is given, each cell of which is painted black or white. It is allowed $($at the same time$)$ to repaint all of the cells of a certain column or row in the colour of the majority of cells in that row or column. Is it always possible to have that all of the cells in the table are painted in the same colour?

Prove that multiplying the polynomial $(x + 1)^{n-1}$ by any polynomial different from zero, we obtain a polynomial having at least n nonzero coefficients.

It is known that a certain polynomial at rational points takes rational values. Prove that all its coefficients are rational.

Some person A thought of a number from 1 to 15. Some person B asks some questions to which you can answer ‘yes’ or ‘no’. Can B guess the number by asking a$)$ 4 questions; b$)$ 3 questions.

An $8 \times 8$ square is painted in two colours. You can repaint any $1 \times 3$ rectangle in its predominant colour. Prove that such operations can make the whole square monochrome.

Here’s a rather simple rebus:

$\\$

EX is four times larger than OJ.

AJ is four times larger than OX.

$\\$

Find the sum of all four numbers.

Thirty girls – 13 in red dresses and 17 in blue dresses – led a dance around the Christmas tree. Subsequently, each of them was asked if her neighbour on the right was in a blue dress. It turned out that those girls which answered correctly were only those who stood between two girls in dresses of the same color. How many girls could have said yes?