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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There are three types of citizens living in a certain state:
a) a fool thinks everyone is a fool, and he considers himself a smart guy;
b) a modest smart guy knows about everyone correctly, and considers himself a fool;
c) a confident smart guy knows about everyone correctly, and considers himself a smart guy.

There are 200 senators in Congress. The vice president conducted an anonymous poll of the senators: how many smart people are in the room right now? He could not find out the number of smart ones from the questionnaires. But then the only senator who did not take part in the survey returned from his trip. He had filled out a questionnaire about the whole Congress, including himself, and after reading it, the vice president understood everything.
How many smart people could there be in Congress (including the traveler)?

📜 Leaflet #Knights_and_Knaves

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A tourist decided to conduct a complete census of the population. To do this, he talked to every inhabitant of the island. Some of the natives said that the island has an even number of Knights, the rest said that the island has an odd number of Knaves. So is the number of the islanders even or odd?

📜 Leaflet #Knights_and_Knaves

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The Niagara Falls excursion goes 30 schoolchildren along with their parents, some of whom are driving cars. Five people, including the driver, fit in each of the cars. What is the smallest number of parents that should be invited on the field trip?

📜 Leaflet #estimation_plus_example

📚 Estimation plus example is a special mathematical reasoning that applies in some problems on finding the greatest or least values. In all problems of this kind, two things must be done:
1) give an example to prove that the answer you get is correct;
2) come up with an estimate that proves that the smaller/larger number somehow contradicts the condition.

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What is the maximum number of consecutive five-digit numbers that have the following property: each number cannot be represented as the product of 2 three-digit numbers?

📜 Leaflet #estimation_plus_example

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