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

Baron Munchausen claims that he can rearrange the numbers 1, 2, . . . N in a different order, and then write them all in a row without spaces, so that the result is a multi-digit palindrome number (it reads equally from left to right and from right to left).
Isn't the baron bragging?

📜 Leaflet #counterexample

There are three different numbers from 1 to 9 written on the board. With one move you can either add 1 to one of the numbers or subtract 1 from all the numbers.
Is it true that it is always possible to get only zeros left on the board by making no more than 23 moves?

📜 Leaflet #counterexample

Tom said: "If the cat hisses, the dog is around, and vice versa, if the dog isn't around, the cat doesn't hiss."
Did Tom say something unnecessary?

📜 Leaflet #logic2

Among the 5 students A, B, C, D, E, two always lie, and three always tell the truth.
Each of them knows who passed the test and who did not.

They made the following statements.
A: "B didn't pass the test."
B: "C didn't pass the test."
C: "A didn't pass the test."
D: "E didn't pass the test."
E: "D didn't pass the test."

How many of them passed?

📜 Leaflet #logic2

The school held a race with five athletes, and they all took different places. The next day each of them was asked what place they had taken, and each, of course, named one number from 1 to 5. The sum of their answers was 22.
What was the smallest number of liars?

📜 Leaflet #logic2

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A four-digit number is such that all its digits are different, and it is also known that numbers 5860, 1674, 9432, 3017 contain exactly two digits each belonging to this number, but none of them is in the same place as in this number. Find this number.

📜 Leaflet #logic2

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The 2023 inhabitants of the island of knights and liars stood in a circle. Each of them in turn uttered the phrase: Both my neighbors are liars. If the knight is lied to, he is offended and becomes a liar. If the liar is told the truth, he gets upset and becomes a knight. When were there more liars at the beginning, or at the end?

📜 Leaflet #logic2

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In the village, some houses are connected by wires. Neighbors are two people whose houses are connected by a wire.
Is it always possible to put one person - a liar or a knight - in each house, so that everyone would answer positively to the question, "Are there liars among your neighbors?" (Everyone knows about each of his neighbors whether he is a liar or a knight.)

📜 Leaflet #logic2

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1) Arrange 8 rooks on the chessboard so that they do not hit each other in three different ways.

2) How many ways are there in total?

📜 Leaflet #rook_placement

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On the island where knights and liars live, two locals, Harry and Larry, met.
"At least one of us is a knight," said Harry thoughtfully.
"But you're certainly a liar!" slyly replied Larry.

Determine what they are.

📜 Leaflet #Knights_and_Knaves

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Twelve jurors gathered in the room. After a long discussion of the case, one of them said, "There is not one honest man here. "There isn't more than one honest man here," said a second. The third said there were no more than two honest men, the fourth no more than three, and so on until the twelfth, who said there were no more than eleven honest men. How many honest men are there really among the jurors?

📜 Leaflet #Knights_and_Knaves

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