We solve problems

Problems Algebra and arithmetic Algebraic equations and systems of equations

Solve the equation \(xy = x + y\) in integers.

There are three sets of dominoes of different colours. How can you put the dominoes from all three sets into a chain (according to the rules of the game) so that every two neighbouring dominoes are of a different colour?

Problem #PRU-35295

Problems Methods Colouring Colouring (other) Pigeonhole principle Pigeonhole principle (other) Algebra and arithmetic Word problems Tables and tournaments Tables and tournaments (other)

13–15 3.0

On every cell of a \(9 \times 9\) board there is a beetle. At the sound of a whistle, every beetle crawls onto one of the diagonally neighbouring cells. Note that, in some cells, there may be more than one beetle, and some cells will be unoccupied.

Prove that there will be at least 9 unoccupied cells.

Problem #PRU-35296

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

In the TV series “The Secret of Santa Barbara” there are 20 characters. Each episode contains one of the events: some character discovers the Mystery, some character discovers that someone knows the Mystery, some character discovers that someone does not know the Mystery. What is the maximum number of episodes that this tv series can last?

Problem #PRU-35298

Problems Algebra and arithmetic Algebraic equations and systems of equations Higher order equations. Palindromic polynomial equations Equations of higher order (other)

13–15 3.0

Determine all solutions of the equation \((n + 2)! - (n + 1)! - n! = n^2 + n^4\) in natural numbers.

Problem #PRU-35335

Problems Set theory and logic Algorithm Theory Game Theory Symmetric strategies

12–14 2.0

Two boys play the following game: they take turns placing rooks on a chessboard. The one who wins is the one whose last move leaves all the board cells filled. Who wins if both try to play with the best possible strategy?

Problem #PRU-35339

Point symmetry Problems Geometry Solid geometry Space transformations Isometry Isometry helps solve the problem Methods Pigeonhole principle Pigeonhole principle (area and volume)

13–15 2.0

On the planet Tau Ceti, the landmass takes up more than half the surface area. Prove that the Tau Cetians can drill a hole through the centre of their planet that connects land to land.

Problem #PRU-35342

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

11–13 2.0

A traveller who came to the planet hired a local as a guide. They went for a walk and saw another alien. The traveller sent the guide to find out to whether this native is a liar or truth teller. The guide returned and said: “The native says that they are a truth teller.” Who was the guide: a liar or a truth teller?

Problem #PRU-35344

Problems Algebra and arithmetic Number theory. Divisibility Division with remainders. Arithmetic of remainders Division with remainder Methods Pigeonhole principle Pigeonhole principle (other)

13–14 3.0

Prove that if \(a, b, c\) are odd numbers, then at least one of the numbers \(ab-1\), \(bc-1\), \(ca-1\) is divisible by 4.

Problem #PRU-35347

Problems Set theory and logic Algorithm Theory Balance puzzles

12–13 2.0

In a basket there are 13 apples. There are scales, with which you can find out the total weight of any two apples. Think of a way to find out from 8 weighings the total weight of all the apples.