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?

📜 Leaflet #calculation

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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?

📜 Leaflet #calculation

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

📜 Leaflet #divisibility

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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).

📜 Leaflet #Prime_number

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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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