We solve problems

Problem #PRU-32045

Problems Algebra and arithmetic Word problems Tables and tournaments Chessboards and chess pieces Set theory and logic Theory of algotithms Game theory Game theory (other)

10–12 2.0

Two grandmasters in turn put rooks on a chessboard (one turn – one rook) so that they cannot capture each other. The person who cannot put a rook on the chessboard loses. Who will win with the game – the first or second grandmaster?

Problem #PRU-32046

Problems Methods Algebraic methods Counting in two ways Partitions into pairs and groups; bijections Pigeonhole principle Pigeonhole principle (other)

11–13 3.0

You are given 25 numbers. The sum of any 4 of these numbers is positive. Prove that the sum of all 25 numbers is also positive.

Problem #PRU-32050

Problems Algebra and arithmetic Word problems Tables and tournaments Tournament tables

11–13 2.0

In a tournament by the Olympic system (the loser is eliminated), 50 boxers participate. What is the minimum number of matches needed to be conducted in order to identify the winner?

Problem #PRU-32068

Problems Combinatorics Painting problems Methods Pigeonhole principle Pigeonhole principle (area and volume)

11–13 3.0

A square area of size \(100\times 100\) is covered in tiles of size \(1\times 1\) in 4 different colours – white, red, black, and grey. No two tiles of the same colour touch one another, that is share a side or a corner. How many red tiles can there be?

Problem #PRU-32092

Problems Methods Algebraic methods Counting in two ways Algebra and arithmetic Word problems Tables and tournaments Number tables and its properties

10–12 2.0

In each square of a rectangular table of size \(M \times K\), a number is written. The sum of the numbers in each row and in each column, is 1. Prove that \(M = K\).

Problem #PRU-32095

Problems Geometry Plane geometry Visual geometry

10–13 2.0

Is it possible to draw this picture (see the figure), without taking your pencil off the paper and going along each line only once?

Problem #PRU-32104

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

10–13 2.0

One of five brothers baked a cake for their Mum. Alex said: “This was Vernon or Tom.” Vernon said: “It was not I and not Will who did it.” Tom said: “You’re both lying.” David said: “No, one of them told the truth, and the other was lying.” Will said: “No David, you’re wrong.” Mum knows that three of her sons always tell the truth. Who made the cake?

Problem #PRU-33134

Problems Algebra and arithmetic Number theory. Divisibility Odd and even numbers

11–13 2.0

Prove that the sum of

a) any number of even numbers is even;

b) an even number of odd numbers is even;

c) an odd number of odd numbers is odd.

Problem #PRU-33135

Problems Algebra and arithmetic Number theory. Divisibility Odd and even numbers

11–13 2.0

Prove that the product of

a) two odd numbers is odd;

b) an even number with any integer is even.

Problem #PRU-34859

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

2.0

On the selection to the government of the planet of liars and truth tellers \(12\) candidates gave a speech about themselves. After a while, one said: “before me only once did someone lie” Another said: “And now-twice.” “And now – thrice” – said the third, and so on until the \(12\)th, who said: “And now \(12\) times someone has lied.” Then the presenter interrupted the discussion. It turned out that at least one candidate correctly counted how many times someone had lied before him. So how many times have the candidates lied?