We solve problems

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-109482

Problems Set theory and logic 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 Algebra 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 \(x \in (0;\pi /2)\) 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-109766

Problems Algebra 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 \(2002\times2002\) 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?

Problem #PRU-109808

Problems Combinatorics Geometry on grid paper Integer lattices Integer lattices (other) Painting problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

12–14 3.0

All of the points with whole number co-ordinates in a plane are plotted in one of three colours; all three colours are present. Prove that there will always be possible to form a right-angle triangle from these points so that its vertices are of three different colours.

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-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-110072

Problems Combinatorics Partitions Dissections with certain properties Geometry on grid paper Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

12–14 4.0

A target consists of a triangle divided by three families of parallel lines into 100 equilateral unit triangles. A sniper shoots at the target. He aims at a particular equilateral triangle and either hits it or hits one of the adjacent triangles that share a side with the one he was aiming for. He can see the results of his shots and can choose when to stop shooting. What is the largest number of triangles that the sniper can guarantee he can hit exactly 5 times?

Problem #PRU-110083

Problems Combinatorics Painting problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Algebra Word Problems Tables and tournaments Tables and tournaments (other)

12–14 3.0

Can the cells of a \(5 \times 5\) board be painted in 4 colours so that the cells located at the intersection of any two rows and any two columns are painted in at least three colours?