We solve problems

Problem #PRU-35457

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

13–15 3.0

Prove that the equation $a_{1} \sin x + b_{1} \cos x + a_{2} \sin 2 x + b_{2} \cos 2 x + \dots + a_{n} \sin n x + b_{n} \cos n x = 0$ has at least one root for any values of $a_{1}, b_{1}, a_{2}, b_{2}, \dots, a_{n}, b_{n}$ .

Problem #PRU-35474

Problems Discrete Mathematics Algorithm Theory Theory of algorithms (other)

12–13 2.0

A journalist came to a company which had $N$ people. He knows that this company has a person $Z$ , who knows all the other members of the company, but nobody knows him. A journalist can address each member of the company with the question: “Do you know such and such?” Find the smallest number of questions sufficient to surely find $Z$ . (Everyone answers the questions truthfully. One person can be asked more than one question.)

Problem #PRU-35476

Problems Calculus Functions of one variable. Continuity Methods Algebraic methods Iterations

15–16 3.0

Let $f (x)$ be a polynomial about which it is known that the equation $f (x) = x$ has no roots. Prove that then the equation $f (f (x)) = x$ does not have any roots.

Problem #PRU-35479

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

At a round table, 10 boys and 15 girls were seated. It turned out that there are exactly 5 pairs of boys sitting next to each other.

How many pairs of girls are sitting next to each other?

Problem #PRU-35483

Problems Methods Pigeonhole principle Pigeonhole principle (other)

14–16 4.0

Prove that there exist numbers, that can be presented in no fewer than 100 ways in the form of a summation of 20001 terms, each of which is the 2000th power of a whole number.

Problem #PRU-35501

Problems Geometry Plane geometry Polygons Polygons (other)

12–14 2.0

Arrows are placed on the sides of a polygon. Prove that the number of vertices in which two arrows converge is equal to the number of vertices from which two arrows emerge.

Problem #PRU-35503

Problems Discrete Mathematics Set theory and logic Mathematical logic Mathematical logic (other)

12–14 2.0

In the government of the planet of liars and truth tellers there are $101$ ministers. In order to reduce the budget, it was decided to reduce the number of ministers by $1.$ But each of the ministers said that if they were to be removed from the government, then the majority of the remaining ministers would be liars. How many truth tellers and how many liars are there in the government?

Problem #PRU-35506

Problems Fun Problems Word Problems Tables and tournaments Number tables and its properties Methods Pigeonhole principle Pigeonhole principle (other)

12–13 2.0

Is it possible to fill an $n \times n$ table with the numbers $- 1$ , $0$ , $1$ , such that the sums of all the rows, columns, and diagonals are unique?

Problem #PRU-35512

Problems Methods Pigeonhole principle Pigeonhole principle (other)

12–14 3.0

Prove that in any group of friends there will be two people who have the same number of friends.

Problem #PRU-35534

Problems Methods Pigeonhole principle Pigeonhole principle (other)

12–14 2.0

In chess, ‘check’ is when the king is under threat of capture from another piece. What is the largest number of kings that it is possible to place on a standard $8 \times 8$ chess board so that no two check one another.