We solve problems

Problem #PRU-30429

Problems Algebra and arithmetic Algebraic equations and systems of equations Higher order equations. Palindromic polynomial equations Equations of higher order (other) Calculus Real numbers Integer and fractional parts. Archimedean property

12–14 2.0

In the Far East, the only type of transport is a carpet-plane. From the capital there are 21 carpet-planes, from the city of Dalny there is one carpet-plane, and from all of the other cities there are 20. Prove that you can fly from the capital to Dalny (possibly with interchanges).

Problem #PRU-30430

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

In a country coming out of each city there are 100 roads and from each city it is possible to reach any other. One road was closed for repairs. Prove that even now you can get from every city to any other.

Problem #PRU-30438

Problems Set theory and logic Theory of algotithms Game theory Fun games

12–14 3.0

The numbers 25 and 36 are written on a blackboard. Consider the game with two players where: in one turn, a player is allowed to write another natural number on the board. This number must be the difference between any two of the numbers already written, such that this number does not already appear on the blackboard. The loser is the player who cannot make a move.

Problem #PRU-30439

Problems Set theory and logic Theory of algotithms Game theory Fun games

12–14 3.0

Consider a chessboard of size (number of rows $\times$ number of columns): a) $9 \times 10$ ; b) $10 \times 12$ ; c) $9 \times 11$ . Two people are playing a game where: in one turn a player is allowed to cross out any row or column as long as there it contains at least one square that is not crossed out. The loser is the player who cannot make a move. Which player will win?

Problem #PRU-30441

Problems Set theory and logic Theory of algotithms Game theory Symmetric strategies

12–14 3.0

Two people take turns placing bishops on a chessboard such that the bishops cannot attack each other. Here, the colour of the bishops does not matter. (Note: bishops move and attack diagonally.) Which player wins the game, if the right strategy is used?

Problem #PRU-30442

Problems Set theory and logic Theory of algotithms Game theory Symmetric strategies

12–13 3.0

There are two piles of rocks, each with 7 rocks. Consider the game with two players where: in one turn you can take any amount of rocks, but only from one pile. The loser is the one who has no rocks left to take.

Problem #PRU-30443

Problems Set theory and logic Theory of algotithms Game theory Symmetric strategies

12–13 3.0

Two people take turns placing knights on a chessboard such that the knights cannot attack each other. The loser is the player who cannot make a move. Which player wins the game, if the right strategy is used?

Problem #PRU-30444

Problems Set theory and logic Theory of algotithms Game theory Symmetric strategies

12–13 3.0

Two people take turns placing kings on squares of a $9 \times 9$ chessboard such that the kings cannot attack each other. The loser is the player who cannot make a move. Which player wins the game, if the right strategy is used?

Problem #PRU-30445

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

11–12 2.0

a) Two in turn put bishops in the cells of a chessboard. The next move must beat at least one empty cell. The bishop also beats the cell in which it is located. The player who loses is the one who cannot make a move.

b) Repeat the same, but with rooks.

Problem #PRU-30446

Problems Set theory and logic Theory of algotithms Game theory Symmetric strategies

12–13 3.0

There is a board of $10 \times 10$ squares. Consider the game with two players where: in one turn a player is allowed to cover any two adjacent squares with a domino (a $1 \times 2$ rectangle) so that the domino doesnt cover another domino. The loser is the one who cannot make a move. Which player would win, if the right strategy was used?