We solve problems

Problem #PRU-116573

Problems Geometry Plane geometry Circles Diameter, chords, and secants Chords and secants (other) Polygons Regular polygons

15–17 3.0

What is the maximum number of pairwise non-parallel segments with endpoints at the vertices of a regular $n$ -gon?

Problem #PRU-60514

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

12–15 3.0

Numbers $a, b, c$ are integers with $a$ and $b$ being coprime. Let us assume that integers $x_{0}$ and $y_{0}$ are a solution for the equation $a x + b y = c$ .

Prove that every solution for this equation has the same form $x = x_{0} + k b$ , $y = y_{0} - k a$ , with $k$ being a random integer.

Problem #PRU-61350

Problems Algebra Algebraic equations and systems of equations Discrete Mathematics Set theory and logic Mathematical logic Weightings

15–17 5.0

There are 13 weights. It is known that any 12 of them could be placed in 2 scale cups with 6 weights in each cup in such a way that balance will be held.

Prove the mass of all the weights is the same, if it is known that:

a) the mass of each weight in grams is an integer;

b) the mass of each weight in grams is a rational number;

c) the mass of each weight could be any real (not negative) number.

Problem #PRU-102995

Problems Algebra Functional equations

14–16 4.0

Can there exist two functions $f$ and $g$ that take only integer values such that for any integer $x$ the following relations hold:

a) $f (f (x)) = x$ , $g (g (x)) = x$ , $f (g (x)) > x$ , $g (f (x)) > x$ ?

b) $f (f (x)) < x$ , $g (g (x)) < x$ , $f (g (x)) > x$ , $g (f (x)) > x$ ?

Problem #PRU-107749

Problems Algebra and arithmetic Number systems Decimal number system 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 Algebra and arithmetic Number systems Decimal number system 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-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 $1 / x$ .