We solve problems

Problems Geometry Solid geometry Visual geometry in space

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

Problem #PRU-102818

Problems Set theory and logic Algorithm Theory Game Theory Symmetric strategies

12–13 3.0

A two-player game with matches. There are 37 matches on the table. In each turn, a player is allowed to take no more than 5 matches. The winner of the game is the player who takes the final match. Which player wins, if the right strategy is used?

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-102839

12–14 2.0

In the US, it is customary to record the date as follows: the number of the month, then the number of the day and then the year. In Europe, the number comes first, then the month and then the year. How many days are there in the year, the date of which can be read definitively, without knowing how it was written?

Problem #PRU-102841

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

11–12 2.0

Two weighings. There are 7 coins which are identical on the surface, including 5 real ones (all of the same weight) and 2 counterfeit coins (both of the same weight, but lighter than the real ones). How can you find the 3 real coins with the help of two weighings on scales without weights?

Problem #PRU-102844

Problems Combinatorics Painting problems Methods Pigeonhole principle Pigeonhole principle (other) Proof by contradiction Algebra and arithmetic Word problems Tables and tournaments Tables and tournaments (other)

12–14 3.0

The cells of a \(15 \times 15\) square table are painted red, blue and green. Prove that there are two lines which at least have the same number of cells of one colour.

Problem #PRU-102845

Problems Combinatorics Partitions 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-102859

Problems Set theory and logic Mathematical logic Puzzles

12–13 2.0

Can the equality \(K \times O \times T\) = \(U \times W \times E \times H \times S \times L\) be true if instead of the letters in it we substitute integers from 1 to 9 (different letters correspond to different numbers)?