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

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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?

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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.

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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.

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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.

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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).

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

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

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Find all pairs of prime numbers p and q such that (p + q)² - pq is a complete square.

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From which of the largest number of coins will it be possible to find a fake coin, which is lighter than the rest, in 4 weighings?

📜 Leaflet #weighing

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There are 30 real coins and 31 counterfeit coins. A real coin is 3 grams heavier than a counterfeit coin. The magician chooses a random coin and then determines in one weighing on a scale with an arrow whether it is real or not. How does he do this?

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Four coins and a cup scale are given. It is known that among them there is 1 counterfeit coin, which somehow differs in weight from the real one, but looks the same. What is the least amount of weighing that can be done to determine it?

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There are 60 identical coins, one of which is a counterfeit of a different weight. How can you tell if a counterfeit coin is lighter or heavier in two weighings on a cup scale?

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The cabin boy dug out the treasure chest and found three bags of coins in it. The bags contained 18, 22, and 28 coins. There was a note in the chest that one of the bags contained a fake coin that, which differs in weight from the real ones.
If the cabin boy brings a bag with a fake coin to the captain of the ship, he will be executed.
How can you find a sack with all the real coins in it in 1 weighing on a cup scale without weights?

📜 Leaflet #weighing

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There are 25 gold and 26 silver coins, of which exactly one is false. It is known that if the counterfeit coin is gold, it is lighter than the real coin because it is made of less gold, and if the counterfeit coin is silver, it is heavier than the real coin because it is made of a cheaper and heavier metal. What is the minimum number of weighings needed on a cup scale to find a counterfeit coin? (Real gold coins weigh the same and real silver coins weigh the same.)

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Dale accidentally broke an electronic scale, but he has a cup scale. He wants to learn how to measure every number of grams from 1 to 10. Chip says he can give him three screws of any weight. Can you help Dale decide on a scale?

📜 Leaflet #weighing

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The cabin boy found a new treasure chest, this time it contained 80 bags of gold of various weights. His real weight was written on each bag. Just in case, the cabin boy wrote down all the weights, and for good reason, since the innoscriptions on the bags were erased during the storm. He needs to prove to the captain how much each bag weighs, using only two-cup scales that show the difference between the bowls (in grams).

a) Prove that 4 weighings are enough for him for this purpose.
b) Prove that three weighings are not enough.

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The natural numbers x, y, z are such that x² + y² + z² is divisible by 9. Prove that one can choose two of these three numbers such that the difference of their squares is also divisible by 9.

📜 Leaflet #modular_arithmetic

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