We solve problems

Problems Set theory and logic Mathematical logic Mathematical logic (other)

Several natives of an island met up (each either a liar or a knight), and everyone said to everyone else: “You are all liars.” How many knights were there among them?

Problem #PRU-98670

Problems Methods Examples and counterexamples. Constructive proofs

11–12 2.0

Among 4 people there are no three with the same name, the same middle name and the same surname, but any two people have either the same first name, middle name or surname. Can this be so?

Problem #PRU-98696

Problems Set theory and logic Joke problem

10–11 1.0

In a 10-storey house, 1 person lives on the first floor, 2 on the second floor, 3 on the third, 4 on the fourth, ..., 10 on the tenth. On which floor does the elevator stop most often?

Problem #PRU-102815

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

Propose a method for measuring the diagonal of a conventional brick, which is easily realied in practice (without the Pythagorean theorem).

Problem #PRU-102822

Problems Set theory and logic Mathematical logic Mathematical logic (other)

11–12 2.0

The seller with weights. With four weights the seller can weigh any integer number of kilograms, from 1 to 40 inclusive. The total mass of the weights is 40 kg. What are the weights available to the seller?

Problem #PRU-102845

Problems Combinatorics Dissections, partitions, covers and tilings Dissections with certain properties

11–12 2.0

Cutting into four parts. Cut each of the figures below into four equal parts (you can cut along the sides and diagonals of cells).

Problem #PRU-102852

Problems Set theory and logic Mathematical logic Mathematical logic (other)

11–12 2.0

We are looking for the correct statement. In a notebook one hundred statements are written:

1) There is exactly one false statement in this notebook.

2) There are exactly two false statements in this notebook.

...

100) There are exactly one hundred false statements in this notebook.

Which of these statements is true, if it is known that only one is true?

Problem #PRU-102854

Problems Set theory and logic Mathematical logic Puzzles

11–12 2.0

Solve the rebus \(AC \times CC \times K = 2002\) (different letters correspond to different integers and vice versa).

Problem #PRU-102861

Problems Set theory and logic Mathematical logic Puzzles

11–12 2.0

Rebus. Solve the numerical rebus \(AAAA-BBB + SS-K = 1234\) (different letters correspond to different numbers, but the same letters each time correspond to the same numbers)

Problem #PRU-102964

Problems Set theory and logic Theory of algotithms Game theory Game theory (other) Algebra and arithmetic Number theory. Divisibility Odd and even numbers

10–12 2.0

Petya and Misha play such a game. Petya takes in each hand a coin: one – 10 pence, and the other – 15. After that, the contents of the left hand are multiplied by 4, 10, 12 or 26, and the contents of the right hand – by 7, 13, 21 or 35. Then Petya adds the two results and tells Misha the result. Can Misha, knowing this result, determine which hand – the right or left – contains the 10 pence coin?