We solve problems

Problem #PRU-103964

Problems Algebra and arithmetic Number theory. Divisibility Division with remainders. Arithmetic of remainders Arithmetic of remainders Methods Pigeonhole principle Pigeonhole principle (other)

12–14 3.0

There are $n$ integers. Prove that among them either there are several numbers whose sum is divisible by $n$ or there is one number divisible by $n$ itself.

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-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-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-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-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 $c$ .

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 $x^{2} + a x + b = 0$ , $x_{2}$ is the real root of the equation $x^{2} - a x - b = 0$ .

Prove that the equation $x^{2} + 2 a x + 2 b = 0$ has a real root, enclosed between $x_{1}$ and $x_{2}$ . ( $a$ and $b$ are real numbers).

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