We solve problems

Problems Algebra Word Problems Percentages and ratios

A field that will be used to grow wheat has a rectangular shape. This year, the farmer responsible for this field decided to increase the length of one of the sides by \(20\%\) and decrease the length of another side by \(20\%\). The field remains rectangular. Will the harvest of wheat change this year and, if so, then by how much?

Problem #PRU-108737

Problems Set theory and logic Algorithm Theory Algorithm Theory

10–12 2.0

Solve problem number 108736 for the inscription \(A\), \(BC\), \(DEF\), \(CGH\), \(CBE\), \(EKG\).

Problem #PRU-109423

Problems Set theory and logic Mathematical logic Puzzles Algebra 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-109468

Problems Algebra Algebraic equations and systems of equations Algebraic equations and systems of equations

12–14 2.0

Replace \(a, b\) and \(c\) with integers not equal to \(1\) in the equality \((ay^b)^c = - 64y^6\), so it would become an identity.

Problem #PRU-109476

Problems Algebra 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-109754

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

Prove that for all \(x \in (0;\pi /2)\) for \(n > m\), where \(n, m\) are natural, we have the inequality \(2 | \sin^n x-\cos^n x | \leq 3 | \sin^m x-\cos^m x |\);

Problem #PRU-109909

Problems Set theory and logic Algorithm Theory Algorithm Theory

10–12 2.0

Members of the State parliament formed factions in such a way that for any two factions \(A\) and \(B\) (not necessarily different)

– also a faction (through

the set of all parliament members not included in \(C\) is denoted). Prove that for any two factions \(A\) and \(B\), \(A \cup % \includegraphics{https://problems-static.s3.eu-west-2.amazonaws.com/production/task_images/700/109909-3.png} B\) is also a faction.

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-111241

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

A New Year’s garland, hanging along the school corridor, consists of red and blue light bulbs. Next to each red light bulb there must necessarily be a blue one. What is the largest number of red light bulbs in this garland, if it consists of only 50 light bulbs?

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.