We solve problems

Problem #PRU-109438

Problems Calculus Functions of one variable. Continuity Certain properties of a function and recurrence relations. Algebra Exponential functions and logarithms Exponential functions and logarithms (other)

13–16 3.0

The function $f$ is such that for any positive $x$ and $y$ the equality $f (x y) = f (x) + f (y)$ holds. Find $f (2007)$ if $f (1 / 2007) = 1$ .

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

Problem #PRU-109441

Problems Methods Pigeonhole principle Pigeonhole principle (other)

13–16 4.0

We are given a $100 \times 100$ square grid and $N$ counters. All of the possible arrangements of the counters on the grid which follow the following rule are considered: no two counters lie in adjacent squares.

What is the largest value of $N$ for which, in every single possible arrangement of counters following this rule, it is possible to find at least one counter such that moving it to an adjacent square does not break the rule. Squares are considered adjacent if they share a side.

Problem #PRU-109468

Problems Algebra Algebraic equations and systems of equations Algebraic equations and systems of equations (other)

12–14 2.0

Replace $a, b$ and $c$ with integers not equal to $1$ in the equality $(a y^{b})^{c} = - 64 y^{6}$ , so it would become an identity.

Problem #PRU-109476

Problems Fun Problems Word Problems Solving problems by inequalities. Going through the cases. Discrete Mathematics Set theory and logic Mathematical logic Mathematical logic (other)

11–14 2.0

Sarah believes that two watermelons are heavier than three melons, Anna believes that three watermelons are heavier than four melons. It is known that one of the girls is right, and the other is mistaken. Is it true that 12 watermelons are heavier than 18 melons? (It is believed that all watermelons weigh the same and all melons weigh the same.)

Problem #PRU-109493

Problems Methods Algebraic methods Iterations Mathematical induction Mathematical induction (other) Processes and operations Calculus Real numbers Rational and irrational numbers Discrete Mathematics Algorithm Theory Theory of algorithms (other)

14–16 4.0

With a non-zero number, the following operations are allowed: $x \to \frac{1 + x}{x}$ , $x \to \frac{1 - x}{x}$ . Is it true that from every non-zero rational number one can obtain each rational number with the help of a finite number of such operations?

Problem #PRU-109534

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

13–15 3.0

Seven triangular pyramids stand on the table. For any three of them, there is a horizontal plane that intersects them along triangles of equal area. Prove that there is a plane intersecting all seven pyramids along triangles of equal area.

Problem #PRU-109573

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

13–15 3.0

Prove that for all $x$ , $0 < x < π / 3$ , we have the inequality $\sin 2 x + \cos x > 1$ .

Problem #PRU-109574

Problems Methods Mathematical induction Mathematical induction (other) Pigeonhole principle Pigeonhole principle (other)

13–16 4.0

On a particular day it turned out that every person living in a particular city made no more than one phone call. Prove that it is possible to divide the population of this city into no more than three groups, so that within each group no person spoke to any other by telephone.

Problem #PRU-109577

Problems Calculus Functions of one variable. Continuity Certain properties of a function and recurrence relations.

14–16 4.0

The function $f (x)$ is defined and satisfies the relationship $(x - 1) f ((x = 1) / (x - 1)) - f (x) = x$ for all $x \neq 1$ . Find all such functions.