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-102839

Problems Fun Problems Word Problems

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-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-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)?

Problem #PRU-102879

Problems Methods Examples and counterexamples. Constructive proofs Pigeonhole principle Pigeonhole principle (other)

12–14 3.0

What is the maximum number of kings, that cannot capture each other, which can be placed on a chessboard of size \(8 \times 8\) cells?

Problem #PRU-103780

Problems Methods Algebraic methods Partitions into pairs and groups; bijections Pigeonhole principle Pigeonhole principle (other) Proof by exhaustion

11–13 3.0

In any group of 10 children, out of a total of 60 pupils, there will be three who are in the same class. Will it always be the case that amongst the 60 pupils there will be: 1) 15 classmates? 2) 16 classmates?

Problem #PRU-103787

Problems Methods Algebraic methods Counting in two ways Extremal principle Extremal principle (other) Pigeonhole principle Pigeonhole principle (other)

12–14 3.0

One term a school ran 20 sessions of an after-school Astronomy Club. Exactly five pupils attended each session and no two students encountered one another over all of the sessions more than once. Prove that no fewer than 20 pupils attended the Astronomy Club at some point during the term.

Problem #PRU-103802

Problems Algebra and arithmetic Number systems Decimal number system Number theory. Divisibility Divisibility of a number. General properties Set theory and logic Set theory Set cardinality. One-to-one correspondence

12–13 2.0

Which five-digit numbers are there more of: ones that are not divisible by 5 or those with neither the first nor the second digit on the left being a five?

Problem #PRU-103819

Problems Methods Pigeonhole principle Pigeonhole principle (other) Set theory and logic Theory of algotithms Theory of algorithms (other)

12–14 3.0

In Mexico, environmentalists have succeeded in enacting a law whereby every car should not be driven at least one day a week (the owner informs the police about their car registration number and the day of the week when this car will not be driven). In a certain family, all adults want to travel daily (each for their own business!). How many cars (at least) should the family have, if the family has a) 5 adults? b) 8 adults?

Problem #PRU-103824

Problems Geometry Solid geometry Spheres Circle on a sphere Visual geometry in space

11–13 2.0

A globe has 17 parallels and 24 meridians. How many parts is the globe’s surface divided into? The meridian is an arc connecting the North Pole with the South Pole. A parallel is a circle parallel to the equator (the equator is also a parallel).