We solve problems

Problems Algebra and arithmetic Algebraic equations and systems of equations Diophantine equations

Prove there are no integer solutions for the equation \(4^k - 4^l = 10^n\).

Problems Geometry Solid geometry Visual geometry in space

12 teams played a volleyball tournament in one round. Two teams scored exactly 7 wins.

Prove that there are teams \(A\), \(B\), \(C\) where \(A\) won against \(B\), \(B\) won against \(C\), and \(C\) won against \(A\).

Problem #PRU-32793

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

11–12 2.0

In a certain realm there are magicians, sorcerers and wizards. The following is known about them: firstly, not all magicians are sorcerers, and secondly, if the wizard is not a sorcerer, then he is not a magician. Is it true that not all magicians are wizards?

Problem #PRU-32794

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

11–12 2.0

A traveller on the planet of liars and truth tellers met four people and asked them: “Who are you?”. They received the following answers:

1st: “We are all liars.”
2nd: “Among us is exactly one liar.”
3rd: “Among us there are two liars.”
4th: “I have never lied and I’m not lying”.

The traveller quickly realised who the fourth resident was. How did they do it?

Problem #PRU-32807

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

11–12 2.0

In the lower left corner of an 8 by 8 chessboard is a chip. Two in turn move it one cell up, right or right-up diagonally. The one who puts the chip in the upper right corner wins. Who will win in a regular game?

Problem #PRU-32817

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

11–12 2.0

a) There are 10 coins. It is known that one of them is fake (by weight, it is heavier than the real ones). How can you determine the counterfeit coin with three weighings on scales without weights?

b) How can you determine the counterfeit coin with three weighings, if there are 27 coins?

Problem #PRU-32901

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

Find a natural number greater than one that occurs in the Pascal triangle a) more than three times; b) more than four times.

Problem #PRU-32904

Problems Algebra and arithmetic Algebraic equations and systems of equations Higher order equations. Palindromic polynomial equations Equations of higher order (other) Calculus Real numbers Integer and fractional parts. Archimedean property

12–14 2.0

Can you find

a) in the 100th line of Pascal’s triangle, the number \(1 + 2 + 3 + \dots + 98 + 99\)?

b) in the 200th line the sum of the squares of the numbers in the 100th line?

Problem #PRU-32987

Problems Algebra and arithmetic Algebraic equations and systems of equations

11–13 2.0

Prove there are no integer solutions for the equation \(3x^2 + 2 = y^2\).

Problem #PRU-34931

Problems Algebra and arithmetic Algebraic equations and systems of equations Higher order equations. Palindromic polynomial equations Equations of higher order (other) Calculus Real numbers Integer and fractional parts. Archimedean property

12–14 2.0

On the dining room table, there is a choice of six dishes. Every day Valentina takes a certain set of dishes (perhaps, she does not take a single dish), and this set of dishes should be different from all of the sets that she took in the previous days. What is the maximum number of days that Valentina will be able to eat according to such rules and how many meals will she eat on average during the day?