There are several natural numbers written in a circle, each of which is not greater than one of its neighbors. Prove that among these numbers there are at least two equal ones.

📜 Leaflet #extreme

There are several numbers written in a circle, each number is equal to the arithmetic mean of two neighboring numbers. Prove that all these numbers are equal.

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Eight mushroom pickers collected 37 mushrooms. It is known that no two of them collected mushrooms equally and each found at least one mushroom. Prove that some two of them collected more than some five.

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There are several cities in the country. A crazy traveler goes from city A to the farthest city B. Then he goes to the farthest from B city C, and so on. Prove that if city C is not the same as city A, then the traveler will never go back to city A.

📜 Leaflet #extreme

There are 2023 asteroids flying in outer space, each of which has an astronomer sitting on it. All distances between asteroids are different. Every astronomer watches the nearest asteroid. Prove that no one is watching one of the asteroids.

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James conceived four non-negative numbers and counted all their possible pairwise sums (6 total). What numbers did he conceive if these sums are 1, 2, 3, 4, 5, 6?

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In a 7×7 square, paint some cells so that there are exactly 3 painted cells in each row and each column.

📜 Leaflet #coloring

Merry Christmas!

There are natural numbers in the cells of the chessboard such that in each row and in each column the sum of numbers is even.
Prove that the sum of numbers in black cells is even.

📜 Leaflet #coloring

Can a hexagonal cake be cut into 23 equal pieces along these lines?

📜 Leaflet #coloring

Each face of a cube with an edge of 4 cm is divided into cells with a side of 1 cm.
Is it possible to paste all three sides of the cube that have a common vertex, with sixteen 1×3 rectangular paper strips so that the borders of the strips coincide with the borders of the cells?

📜 Leaflet #coloring

The paper is crossed out into squares with a side of 1.
Theodore cut a rectangle out of it and found its area and perimeter.
Sophia took the scissors from him and with the words "Look, a trick!" she cut a square from the edge of the rectangle by the cells, threw the square away, and announced: "Now the remaining figure has the same perimeter as the area of the rectangle, and the area is what the perimeter was!"
Theodore was convinced that Sophia was right.

1) What size square did Sophia cut out and throw away?
2) Give an example of such a rectangle and such a square.
3) What size rectangle did Theodore cut out?

📜 Leaflet #coloring

Give a counterexample to each of the following statements.

1) All numbers divisible by 4 and by 6 are divisible by 24.
2) All rectangles are squares.
3) All quadrilaterals that have all sides equal are squares.

📜 Leaflet #counterexample

If the statement is always true, prove it, and if it is wrong in at least one case, show what that case is (give a counterexample).

Oliver thinks that if the area of the first rectangle is larger than the area of the second rectangle, and the perimeter of the first rectangle is larger than the perimeter of the second rectangle, then the second rectangle can be cut out of the first rectangle.
Is he right?

📜 Leaflet #counterexample

Choose 24 cells in a 5×8 rectangle and draw one of the diagonals in each chosen cell such that no two diagonals have common ends.

📜 Leaflet #counterexample