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Problems Geometry Solid geometry Visual geometry in space

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?

Problem #PRU-35430

Problems Set theory and logic Theory of algotithms Game theory Game theory (other) Algebra and arithmetic Number theory. Divisibility Odd and even numbers

12–13 3.0

Given a board (divided into squares) of the size: a) $10 \times 12$ , b) $9 \times 10$ , c) $9 \times 11$ , consider the game with two players where: in one turn a player is allowed to cross out any row or any column if there is at least one square not crossed out. The loser is the one who cannot make a move. Is there a winning strategy for one of the players?

Problem #PRU-35438

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

2.0

In a group of friends, each two people have exactly five common acquaintances. Prove that the number of pairs of friends is divisible by 3.

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-35479

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

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-35541

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 $2 x^{x} = \sqrt{2}$ for positive numbers.