We solve problems

Problems Set theory and logic Theory of algotithms Balance puzzles

In a physics club, the teacher created the following experiment. He spread out 16 weights of weight 1, 2, 3, ..., 16 grams onto weighing scales, so that one of the bowls outweighed the other. Fifteen students in turn left the classroom and took with them one weight each, and after each student’s departure, the scales changed their position and outweighed the opposite bowl of the scales. What weight could remain on the scales?

Problem #PRU-105052

Problems Methods Pigeonhole principle Pigeonhole principle (other) Proof by contradiction Algebra and arithmetic Word problems Tables and tournaments Tables and tournaments (other)

11–13 3.0

In a chess tournament, each participant played two games with each of the other participants: one with white pieces, the other with black. At the end of the tournament, it turned out that all of the participants scored the same number of points (1 point for a victory, $\frac{1}{2}$ a point for a draw and 0 points for a loss). Prove that there are two participants who have won the same number of games using white pieces.

Problem #PRU-107703

Problems Combinatorics Packing problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

11–13 2.0

Is it possible to place 12 identical coins along the edges of a square box so that touching each edge there were exactly: a) 2 coins, b) 3 coins, c) 4 coins, d) 5 coins, e) 6 coins, f) 7 coins.

You are allowed to place coins on top of one another. In the cases where it is possible, draw how this could be done. In the other cases, prove that doing so is impossible.

Problem #PRU-109423

Problems Set theory and logic Mathematical logic Puzzles Algebra and arithmetic Number theory. Divisibility The fundamental theorm of arithmetic. Prime factorisation.

11–12 2.0

At the end of the term, Billy wrote out his current singing marks in a row and put a multiplication sign between some of them. The product of the resulting numbers turned out to be equal to 2007. What is Billy’s term mark for singing? (The marks that he can get are between 2 and 5, where 5 is the highest mark).

Problem #PRU-109476

Problems Algebra and arithmetic Word problems Solving problems by inequalities. Going through the cases. Set theory and logic Mathematical logic Mathematical logic (other)

11–14 2.0

Sarah believes that two watermelons are heavier than three melons, Anna believes that three watermelons are heavier than four melons. It is known that one of the girls is right, and the other is mistaken. Is it true that 12 watermelons are heavier than 18 melons? (It is believed that all watermelons weigh the same and all melons weigh the same.)

Problem #PRU-109527

Problems Methods Pigeonhole principle Pigeonhole principle (other) Proof by contradiction Algebra and arithmetic Word problems Tables and tournaments Tables and tournaments (other)

11–13 3.0

What is the largest number of counters that can be put on the cells of a chessboard so that on each horizontal, vertical and diagonal (not only on the main ones) there is an even number of counters?

Problem #PRU-109877

Problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

11–13 3.0

A regular hexagon with sides of length $5$ is divided by straight lines, that are parallel to its sides, to form regular triangles with sides of length 1. We call the vertices of all such triangles nodes. It is known that more than half of the nodes are marked. Prove that there are five marked nodes lying on one circle.

Problem #PRU-110920

Problems Methods Pigeonhole principle Pigeonhole principle (other) Algebraic methods Proof by exhaustion

11–13 2.0

Peter has some coins in his pocket. If Peter pulls $3$ coins from his pocket, without looking, there will always be a £1 coin among them. If Peter pulls $4$ coins from his pocket, without looking, there will always be a £2 coin among them. Peter pulls $5$ coins from his pocket. Identify these coins.

Problem #PRU-115374

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

10–12 2.0

In Neverland, only elves and gnomes live. Gnomes lie about their gold, but in any other instances they tell the truth. Elves lie when talking about gnomes, but in other instances they tell the truth. One day two neverlandians said:

$A$ : All my gold I stole from the Dragon.

$B$ : You’re lying.

Determine whether each of them is an elf or a gnome.

Problem #PRU-115379

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

11–12 2.0

Hannah has a calculator that allows you to multiply a number by 3, add 3 to the number or (4 if the number is divisible by 3 to make a whole number) divide by 3. How can the number 11 be made on this calculator from the number 1?