Problems

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 \(\sin 2x + \cos x > 1\).

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.

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.

Prove that for any natural number \(a_1> 1\) there exists an increasing sequence of natural numbers \(a_1, a_2, a_3, \dots\), for which \(a_1^2+ a_2^2 +\dots+ a_k^2\) is divisible by \(a_1+ a_2+\dots+ a_k\) for all \(k \geq 1\).

Is there a sequence of natural numbers in which every natural number occurs exactly once, and for any \(k = 1, 2, 3, \dots\) the sum of the first \(k\) terms of the sequence is divisible by \(k\)?

At all rational points of the real line, integers are arranged. Prove that there is a segment such that the sum of the numbers at its ends does not exceed twice the number on its middle.

Prove that for any positive integer \(n\) the inequality

is true.