We solve problems

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

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

Problems Calculus Real numbers Integer and fractional parts. Archimedean property Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

13–15 3.0

A continuous function \(f\) has the following properties:

1. \(f\) is defined on the entire number line;

2. \(f\) at each point has a derivative (and thus the graph of f at each point has a unique tangent);

3. the graph of the function \(f\) does not contain points for which one of the coordinates is rational and the other is irrational.

Does it follow that the graph of \(f\) is a straight line?

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-107987

Problems Calculus Functions of several variables Algebra and arithmetic Polynomials Algebraic identities for polynomials Identical transformations

13–15 4.0

To each pair of numbers \(x\) and \(y\) some number \(x * y\) is placed in correspondence. Find \(1993 * 1935\) if it is known that for any three numbers \(x, y, z\), the following identities hold: \(x * x = 0\) and \(x * (y * z) = (x * y) + z\).

Problem #PRU-107992

Problems Calculus Real numbers Integer and fractional parts. Archimedean property Methods Pigeonhole principle Algebra Sequences Algebraic methods Symmetry and involutions

14–16 5.0

For each pair of real numbers \(a\) and \(b\), consider the sequence of numbers \(p_n = \lfloor 2 \{an + b\}\rfloor\). Any \(k\) consecutive terms of this sequence will be called a word. Is it true that any ordered set of zeros and ones of length \(k\) is a word of the sequence given by some \(a\) and \(b\) for \(k = 4\); when \(k = 5\)?

Note: \(\lfloor c\rfloor\) is the integer part, \(\{c\}\) is the fractional part of the number \(c\).

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-109196

Problems Calculus Real numbers Rational and irrational numbers

14–16 4.0

In the number \(a = 0.12457 \dots\) the \(n\)th digit after the decimal point is equal to the digit to the left of the decimal point in the number. Prove that \(\alpha\) is an irrational number.

Problem #PRU-109440

Problems Calculus Real numbers Integer and fractional parts. Archimedean property Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

13–15 3.0

Solve the equation: \((x^3 - 2) (2^{\sin x} - 1) + (2^{x^3} - 4) \sin x = 0\).

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-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?