In the number \(a = 0.12457 \dots\) the \(n\)th digit after the decimal point is equal to the digit to the left of the decimal point in the number. Prove that \(\alpha\) is an irrational number.
The function \(f\) is such that for any positive \(x\) and \(y\) the equality \(f (xy) = f (x) + f (y)\) holds. Find \(f (2007)\) if \(f (1/2007) = 1\).
Solve the equation: \((x^3 - 2) (2^{\sin x} - 1) + (2^{x^3} - 4) \sin x = 0\).
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.
Replace \(a, b\) and \(c\) with integers not equal to \(1\) in the equality \((ay^b)^c = - 64y^6\), so it would become an identity.
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.)
With a non-zero number, the following operations are allowed: \(x \rightarrow \frac{1+x}{x}\), \(x \rightarrow \frac{1-x}{x}\). 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?
Find all functions \(f (x)\) defined for all positive \(x\), taking positive values and satisfying the equality \(f (x^y) = f (x)^f (y)\) for any positive \(x\) and \(y\).
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.
Prove that for all \(x\), \(0 < x < \pi /3\), we have the inequality \(\sin 2x + \cos x > 1\).