We solve problems

Problems Geometry Solid geometry Visual geometry in space

All of the sweets of different sorts in stock are arranged in \(n\) boxes, for which prices are set at \(1, 2, \dots , n\), respectively. It is required to buy such \(k\) of these boxes of the least total value, which contain at least \(k/n\) of the mass of all of the sweets. It is known that the mass of sweets in each box does not exceed the mass of sweets in any more expensive box.

a) What boxes should I buy when \(n = 10\) and \(k = 3\)?

b) The same question for arbitrary natural numbers \(n \geq k\).

Problem #PRU-107628

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

10–12 2.0

The bank of the Nile was approached by a group of six people: three Bedouins, each with his wife. At the shore is a boat with oars, which can withstand only two people at a time. A Bedouin can not allow his wife to be without him whilst in the company of another man. Can the whole group cross to the other side?

Problem #PRU-107704

Problems Algebra and arithmetic Arithmetics. Mental maths Set theory and logic Set theory Inclusion-exclusion principle

11–13 3.0

In a class there are 50 children. Some of the children know all the letters except “h” and they miss this letter out when writing. The rest know all the letters except “c” which they also miss out. One day the teacher asked 10 of the pupils to write the word “cat”, 18 other pupils to write “hat” and the rest to write the word “chat”. The words “cat” and “hat” each ended up being written 15 times. How many of the pupils wrote their word correctly?

Problem #PRU-107845

Problems Algebra and arithmetic Number theory. Divisibility Equations in integer numbers

11–14 3.0

Is it possible to find natural numbers \(x\), \(y\) and \(z\) which satisfy the equation \(28x+30y+31z=365\)?

Problem #PRU-107986

Graph theory Problems Methods Pigeonhole principle Set theory and logic

12–15 4.0

Peter has 28 classmates. Each 2 out of these 28 have a different number of friends in the class. How many friends does Peter have?

Problem #PRU-108737

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

10–12 2.0

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

Problem #PRU-109468

Problems Algebra and arithmetic 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 \((ay^b)^c = - 64y^6\), so it would become an identity.

Problem #PRU-109482

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

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 \(B\), \(A \cup % \includegraphics{https://problems-static.s3.eu-west-2.amazonaws.com/static/test/task_images/82/109909-3.png} B\) is also a faction.