We solve problems

Problem #PRU-107793

Problems Methods Examples and counterexamples. Constructive proofs Pigeonhole principle Pigeonhole principle (other) Set theory and logic Theory of algotithms Theory of algorithms (other)

13–16 4.0

Cut the interval \([-1, 1]\) into black and white segments so that the integrals of any a) linear function; b) a square trinomial in white and black segments are equal.

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-109041

Problems Methods Examples and counterexamples. Constructive proofs Pigeonhole principle Pigeonhole principle (other) Set theory and logic Theory of algotithms Theory of algorithms (other)

13–16 4.0

\(x_1\) is the real root of the equation \(x^2 + ax + b = 0\), \(x_2\) is the real root of the equation \(x^2 - ax - b = 0\).

Prove that the equation \(x^2 + 2ax + 2b = 0\) has a real root, enclosed between \(x_1\) and \(x_2\). (\(a\) and \(b\) are real numbers).

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-109493

Problems Methods Algebraic methods Iterations Mathematical induction Mathematical induction (other) Processes and operations Calculus Real numbers Rational and irrational numbers Set theory and logic Theory of algotithms Theory of algorithms (other)

14–16 4.0

With a non-zero number, the following operations are allowed: \(x \rightarrow \frac{1+x}{x}\), \(x \rightarrow \frac{1-x}{x}\). Is it true that from every non-zero rational number one can obtain each rational number with the help of a finite number of such operations?

Problem #PRU-109780

Problems Calculus Real numbers Set theory and logic Set theory

14–16 4.0

A number set \(M\) contains \(2003\) distinct positive numbers, such that for any three distinct elements \(a, b, c\) in \(M\), the number \(a^2 + bc\) is rational. Prove that we can choose a natural number \(n\) such that for any \(a\) in \(M\) the number \(a\sqrt{n}\) is rational.

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.

Problem #PRU-111236

Problems Set theory and logic Mathematical logic Mathematical logic (other)

11–13 3.0

One day all the truth tellers on the planet decided to carry a clearly visible mark of truth in order to be distinguished from liars. Two truth tellers and two liars met and looked at each other. Which of them could say the phrase:

“All of us are truth tellers.”
“Only one of you is a truth teller.”
“Exactly two of you are truth tellers.”

Problem #PRU-111360

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

10–12 2.0

A magician with a blindfold gives a spectator five cards with the numbers from 1 to 5 written on them. The spectator hides two cards, and gives the other three to the assistant magician. The assistant indicates to the spectator two of them, and the spectator then calls out the numbers of these cards to the magician (in the order in which he wants). After that, the magician guesses the numbers of the cards hidden by the spectator. How can the magician and the assistant make sure that the trick always works?