We solve problems

Problems Algebra and arithmetic Rational functions Rational equations

Solve the equation: \[x + \frac{x}{x} + \frac{x}{x+\frac{x}{x}} = 1\]

Problems Geometry Plane geometry Visual geometry

In a board, 20 pins are placed (see the picture). The distance between any adjacent pins is 1 inch. Pull a string of length 19 inches from the first pin to the second one, so that it goes through all the pins.

Problem #PRU-104122

Problems Algebra and arithmetic Arithmetics. Mental maths Set theory and logic Joke problem

12–13 2.0

Henry did not manage to get into the elevator on the first floor of the building and decided to go up the stairs. It takes 2 minutes to rise to the third floor. How long does it take to rise to the ninth floor?

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-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-111245

Problems Set theory and logic

16–16 2.0

Let’s denote any two digits with the letters \(A\) and \(X\). Prove that the six-digit number \(XAXAXA\) is divisible by 7 without a remainder.

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?