1) Arrange 8 rooks on the chessboard so that they do not hit each other in three different ways.

2) How many ways are there in total?

📜 Leaflet #rook_placement

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On the island where knights and liars live, two locals, Harry and Larry, met.
"At least one of us is a knight," said Harry thoughtfully.
"But you're certainly a liar!" slyly replied Larry.

Determine what they are.

📜 Leaflet #Knights_and_Knaves

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Twelve jurors gathered in the room. After a long discussion of the case, one of them said, "There is not one honest man here. "There isn't more than one honest man here," said a second. The third said there were no more than two honest men, the fourth no more than three, and so on until the twelfth, who said there were no more than eleven honest men. How many honest men are there really among the jurors?

📜 Leaflet #Knights_and_Knaves

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There are three types of citizens living in a certain state:
a) a fool thinks everyone is a fool, and he considers himself a smart guy;
b) a modest smart guy knows about everyone correctly, and considers himself a fool;
c) a confident smart guy knows about everyone correctly, and considers himself a smart guy.

There are 200 senators in Congress. The vice president conducted an anonymous poll of the senators: how many smart people are in the room right now? He could not find out the number of smart ones from the questionnaires. But then the only senator who did not take part in the survey returned from his trip. He had filled out a questionnaire about the whole Congress, including himself, and after reading it, the vice president understood everything.
How many smart people could there be in Congress (including the traveler)?

📜 Leaflet #Knights_and_Knaves

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A tourist decided to conduct a complete census of the population. To do this, he talked to every inhabitant of the island. Some of the natives said that the island has an even number of Knights, the rest said that the island has an odd number of Knaves. So is the number of the islanders even or odd?

📜 Leaflet #Knights_and_Knaves

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The Niagara Falls excursion goes 30 schoolchildren along with their parents, some of whom are driving cars. Five people, including the driver, fit in each of the cars. What is the smallest number of parents that should be invited on the field trip?

📜 Leaflet #estimation_plus_example

📚 Estimation plus example is a special mathematical reasoning that applies in some problems on finding the greatest or least values. In all problems of this kind, two things must be done:
1) give an example to prove that the answer you get is correct;
2) come up with an estimate that proves that the smaller/larger number somehow contradicts the condition.

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What is the maximum number of consecutive five-digit numbers that have the following property: each number cannot be represented as the product of 2 three-digit numbers?

📜 Leaflet #estimation_plus_example

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An electronic clock shows the time in a standard format (e.g., 20:27). Find the largest possible value of the product of digits on such a clock.

📜 Leaflet #estimation_plus_example

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Replace the asterisks in the number 13**4* with digits so that the number is divisible by 396. Specify all possible variations.

📜 Leaflet #divisibility

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Find the smallest number whose record consists only of zeros and twos that is divisible by 1125.

📜 Leaflet #divisibility

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What is the greatest natural degree of the number 2023 that is a divisor of 2023! (factorial)?

📜 Leaflet #divisibility

How many ways can you put 7 coins of (a) different denominations into 3 piggy banks? What if the coins are (b) the same? The piggy banks may remain empty.

📜 Leaflet #combinatorics

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Twelve athletes participated in the sprint race. How many ways can medals be distributed if two athletes cannot take the same place?

📜 Leaflet #combinatorics

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There are 20 computers on the same network. From each computer, files were sent to 10 others. Prove that at least 10 pairs of computers exchanged files with each other.

📜 Leaflet #combinatorics

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Find the number of ways to represent the number 5³·7⁴·11⁶ as a product of three positive integers a·b·c.

📜 Leaflet #combinatorics

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The 15 members of the choir must perform at the concert. To do this, they were given concert costumes of three colors: blue, yellow, and green. There are five costumes of each color. To perform, the singers had to line up in a row. The choir director does not like it when 2 people in yellow stand next to each other. How many ways are there to line up the singers so that the choir director likes it?

📜 Leaflet #combinatorics

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A tennis player plays at least 1 game per day and no more than 12 games per week. A week is any 7 consecutive days.

Prove that there is a period of time (counted in full days) in which he will play exactly 20 games.

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Find all prime numbers for which the square of this number increased by 4 and the square of this number increased by 6 are also prime numbers.

📜 Leaflet #prime_numbers

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Find all prime numbers p for which the following holds: p = a + b = c - d, where a, b, c, d are prime numbers (not necessarily distinct).

📜 Leaflet #prime_numbers

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Find all prime p such that p = a + b, where a, b are composite.

📜 Leaflet #prime_numbers

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Is there such a number p that all three numbers p-2023, p, p+2023 are prime?

📜 Leaflet #prime_numbers

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