We solve problems

Problems Algebra Algebraic equations and systems of equations

Show that $\frac{x}{y} + \frac{y}{z} + \frac{z}{x} = 1$ is not solvable in natural numbers.

Problems Algebra 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

Solve the equation $⌊ x^{3} ⌋ + ⌊ x^{2} ⌋ + ⌊ x ⌋ = {x} - 1$ .

Problem #PRU-103757

Problems Algebra 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

A resident of one foreign intelligence agency informed the centre about the forthcoming signing of a number of bilateral agreements between the fifteen former republics of the USSR. According to his report, each of them will conclude an agreement exactly with three others. Should this resident be trusted?

Problem #PRU-103966

Problems Algebra 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

In Mongolia there are in circulation coins of 3 and 5 tugriks. An entrance ticket to the central park costs 4 tugriks. One day before the opening of the park, a line of 200 visitors queued up in front of the ticket booth. Each of them, as well as the cashier, has exactly 22 tugriks. Prove that all of the visitors will be able to buy a ticket in the order of the queue.

Problem #PRU-104069

Problems Algebra Algebraic equations and systems of equations Algebra and arithmetic Arithmetics. Mental maths

10–12 2.0

There are two numbers $x$ and $y$ being added together. The number $x$ is less than the sum $x + y$ by 2000. The sum $x + y$ is bigger than $y$ by 6. What are the values of $x$ and $y$ ?

Problem #PRU-107749

Problems Number Theory Numeral systems Decimal number system Algebra Algebraic equations and systems of equations Diophantine equations

13–15 3.0

The student did not notice the multiplication sign between two three-digit numbers and wrote one six-digit number, which turned out to be seven times bigger than their product. Determine these numbers.

Problem #PRU-107760

Problems Number Theory Numeral systems Decimal number system Algebra Algebraic equations and systems of equations Diophantine equations

13–15 3.0

The student did not notice the multiplication sign between two seven-digit numbers and wrote one fourteen-digit number, which turned out to be three times bigger than their product. Determine these numbers.

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-109946

Problems Algebra Algebraic equations and systems of equations Algebraic equations and systems of equations (other) Calculus Real numbers Integer and fractional parts. Archimedean property Polynomials Quadratic polynomials Investigating the quadratic polynomial

14–15 3.0

Solve the equation $(x + 1)^{3} = x^{3}$ .

Problem #PRU-30416

Problems Algebra 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

The board has the form of a cross, which is obtained if corner boxes of a square board of $4 \times 4$ are erased. Is it possible to go around it with the help of the knight chess piece and return to the original cell, having visited all the cells exactly once?