We solve problems

Problem #PRU-102797

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-104104

Problems Calculus Functions of one variable. Continuity Certain properties of a function and recurrence relations. Algebra Polynomials Quadratic polynomials

12–14 2.0

Find all functions $f (x)$ such that $f (2 x + 1) = 4 x^{2} + 14 x + 7$ .

Problem #PRU-107768

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

13–15 3.0

A cherry which is a ball of radius r is dropped into a round glass whose axial section is the graph of the function $y = x^{4}$ . At what maximum r will the ball touch the most bottom point of the bottom of the glass? (In other words, what is the maximum radius r of a circle lying in the region $y \geq x^{4}$ and containing the origin?).

Problem #PRU-107838

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

13–15 3.0

Consider the powers of the number five: 1, 5, 25, 125, 625, ... We form the sequence of their first digits: 1, 5, 2, 1, 6, ...

Prove that any part of this sequence, written in reverse order, will occur in the sequence of the first digits of the powers of the number two (1, 2, 4, 8, 1, 3, 6, 1, ...).

Problem #PRU-107841

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

13–15 3.0

Three functions are written on the board: $f_{1} (x) = x + 1 / x$ , $f_{2} (x) = x^{2}, f_{3} (x) = (x - 1)^{2}$ . You can add, subtract and multiply these functions (and you can square, cube, etc. them). You can also multiply them by an arbitrary number, add an arbitrary number to them, and also do these operations with the resulting expressions. Therefore, try to get the function $1 / x$ .

Prove that if you erase any of the functions $f_{1}, f_{2}, f_{3}$ from the board, it is impossible to get $1 / x$ .

Problem #PRU-107864

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-107987

Problems Calculus Functions of several variables Algebra 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 $p_{n} = ⌊ 2 {a n + b} ⌋$ . 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: $⌊ c ⌋$ is the integer part, ${c}$ is the fractional part of the number $c$ .