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

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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.

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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.

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