We solve problems

Problem #PRU-32038

Problems Methods Examples and counterexamples. Constructive proofs Geometry Plane geometry Lines, segments, rays, and angles Measurement of segments and angles. Adjacent angles.

11–13 2.0

Is it possible to draw five lines from one point on a plane so that there are exactly four acute angles among the angles formed by them? Angles between not only neighboring rays, but between any two rays, can be considered.

Problem #PRU-32041

Problems Methods Pigeonhole principle Pigeonhole principle (other)

10–13 2.0

A group of 20 tourists go on a trip. The oldest member of the group is 35, the youngest is 20. Is it true that there are members of the group that are the same age?

Problem #PRU-32045

Problems Algebra Word Problems Tables and tournaments Chessboards and chess pieces Set theory and logic Algorithm Theory Game Theory Game Theory

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 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 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-32793

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

11–12 2.0

In a certain realm there are magicians, sorcerers and wizards. The following is known about them: firstly, not all magicians are sorcerers, and secondly, if the wizard is not a sorcerer, then he is not a magician. Is it true that not all magicians are wizards?