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?

📜 Leaflet #weighing

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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.)

📜 Leaflet #weighing

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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.

📜 Leaflet #weighing

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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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Can 4 sticks of length 1 cm, 4 sticks of length 2 cm, 7 sticks of length 3 cm, and 5 sticks of length 4 cm make a rectangle? (All sticks must be used, they must not be broken.)

📜 Leaflet #can

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David conceived a natural number, multiplied all its digits, and multiplied the result by the conceived number. Could he have gotten 1716?

📜 Leaflet #can

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🎉 Happy Pi Day!

Is it possible to arrange the chips in the cells of an 8×8 board so that in any two columns the number of chips is the same, and in any two rows different?

📜 Leaflet #can

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Find the sum of 123456789 + 234567891 + 345678912 + ... + 912345678.

📜 Leaflet #calculation

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In a 3×3 square all integers from 1 to 9 are arranged in some way. First, we counted the arithmetic mean of the numbers in four different 2×2 squares (they all turned out to be integers), and then we calculated the arithmetic mean of the four numbers obtained (it also turned out to be integers). What is the largest possible value of the last counted number?

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John wrote several different natural numbers on the board and divided the sum of these numbers by their product. Then he erased the smallest number and divided the sum of the remaining numbers by their product. The second result was 3 times as many as the first result. Which number did John erase?

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Can the sum of three different natural numbers be divisible by each summand?

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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_number

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

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Olivia drew a polygon and Liam cut it into a triangle and a quadrilateral. How many sides could Olivia's polygon have? Find all the variants and prove that there are no others.

📜 Leaflet #geometry

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