We solve problems

Problem #PRU-35154

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

Prove that multiplying the polynomial $(x + 1)^{n - 1}$ by any polynomial different from zero, we obtain a polynomial having at least $n$ nonzero coefficients.

Problem #PRU-35176

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

Find the number of zeros in which the number $11^{100} - 1$ ends.

Problem #PRU-35186

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

You are given a table of size $m \times n$ ( $m, n > 1$ ). In it, the centers of all cells are marked. What is the largest number of marked centers that can be chosen so that no three of them are the vertices of a right triangle?

Problem #PRU-35199

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

There are several cities (more than one) in a country; some pairs of cities are connected by roads. It is known that you can get from every city to any other city by driving along several roads. In addition, the roads do not form cycles, that is, if you leave a certain city on some road and then move so as not to pass along one road twice, it is impossible to return to the initial city. Prove that in this country there are at least two cities, each of which is connected by a road with exactly one city.

Problem #PRU-35282

Problems Algebra and arithmetic Algebraic equations and systems of equations

12–14

Solve the equation $x y = x + y$ in integers.

Problem #PRU-35296

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

In the TV series “The Secret of Santa Barbara” there are 20 characters. Each episode contains one of the events: some character discovers the Mystery, some character discovers that someone knows the Mystery, some character discovers that someone does not know the Mystery. What is the maximum number of episodes that this tv series can last?

Problem #PRU-35298

Problems Algebra and arithmetic Algebraic equations and systems of equations Higher order equations. Palindromic polynomial equations Equations of higher order (other)

13–15 3.0

Determine all solutions of the equation $(n + 2)! - (n + 1)! - n! = n^{2} + n^{4}$ in natural numbers.

Problem #PRU-35362

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

A raisin bag contains 2001 raisins with a total weight of 1001 g, and no raisin weighs more than 1.002 g.

Prove that all the raisins can be divided onto two scales so that they show a difference in weight not exceeding 1 g.

Problem #PRU-35410

Problems Algebra and arithmetic 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

10 numbers are written around the circle, the sum of which is equal to 100. It is known that the sum of every three numbers standing side by side is not less than 29.

Specify the smallest number $A$ such that in any such set of numbers each of the numbers does not exceed $A$ .

Problem #PRU-35427

Problems Set theory and logic Theory of algotithms Game theory Symmetric strategies

12–14 2.0

A daisy has a) 12 petals; b) 11 petals. Consider the game with two players where: in one turn a player is allowed to remove either exactly one petal or two petals which are next to each other. The loser is the one who cannot make a turn. How should the second player act, in cases a) and b), in order to win the game regardless of the moves of the first player?