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.
At the end of the term, Billy wrote out his current singing marks in a row and put a multiplication sign between some of them. The product of the resulting numbers turned out to be equal to 2007. What is Billy’s term mark for singing? (The marks that he can get are between 2 and 5, where 5 is the highest mark).
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.)
A row of 4 coins lies on the table. Some of the coins are real and some of them are fake (the ones which weigh less than the real ones). It is known that any real coin lies to the left of any false coin. How can you determine whether each of the coins on the table is real or fake, by weighing once using a balance scale?
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 \(y\).