We solve problems

Problems Fun Problems 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 Discrete Mathematics Algorithm Theory Theory of algorithms (other)

10–12 2.0

Solve problem number 108736 for the inscription $A$ , $B C$ , $D E F$ , $C G H$ , $C B E$ , $E K G$ .

Problem #PRU-109018

Problems Discrete Mathematics Combinatorics Dissections, partitions, covers and tilings Covers Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

12–14 3.0

A circle is covered with several arcs. These arcs can overlap one another, but none of them cover the entire circumference. Prove that it is always possible to select several of these arcs so that together they cover the entire circumference and add up to no more than $720^{\circ}$ .

Problem #PRU-109423

Problems Discrete Mathematics Set theory and logic Mathematical logic Puzzles 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 (other)

12–14 2.0

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

Problem #PRU-109476

Problems Fun Problems Word Problems Solving problems by inequalities. Going through the cases. Discrete Mathematics 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-109482

Problems Discrete Mathematics Algorithm Theory Balance puzzles

10–13 3.0

A row of 4 coins lies on the table. Some of the coins are real and some of them are fake (the ones which weigh less than the real ones). It is known that any real coin lies to the left of any false coin. How can you determine whether each of the coins on the table is real or fake, by weighing once using a balance scale?

Problem #PRU-109527

Problems Methods Pigeonhole principle Pigeonhole principle (other) Proof by contradiction Fun Problems 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-109754

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

Prove that for all $x \in (0; π / 2)$ 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-109766

Problems Fun Problems Word Problems Tables and tournaments Number tables and its properties Methods Pigeonhole principle Pigeonhole principle (other)

12–14 4.0

Is it possible to arrange natural numbers from 1 to $2002^{2}$ in the cells of a $2002 \times 2002$ table so that for each cell of this table one could choose a triplet of numbers, from a row or column, where one of the numbers is equal to the product of the other two?