Prove that for every natural number \(n > 1\) the equality: \[\lfloor n^{1 / 2}\rfloor + \lfloor n^{1/ 3}\rfloor + \dots + \lfloor n^{1 / n}\rfloor = \lfloor \log_{2}n\rfloor + \lfloor \log_{3}n\rfloor + \dots + \lfloor \log_{n}n\rfloor\] is satisfied.
a) Give an example of a positive number \(a\) such that \(\{a\} + \{1 / a\} = 1\).
b) Can such an \(a\) be a rational number?
Find the number of solutions in natural numbers of the equation \(\lfloor x / 10\rfloor = \lfloor x / 11\rfloor + 1\).
Let \(n\) numbers are given together with their product \(p\). The difference between \(p\) and each of these numbers is an odd number.
Prove that all \(n\) numbers are irrational.
Some real numbers \(a_1, a_2, a_3,\dots ,a _{2022}\) are written in a row. Prove that it is possible to pick one or several adjacent numbers, so that their sum is less than 0.001 away from a whole number.
Are there such irrational numbers \(a\) and \(b\) so that \(a > 1\), \(b > 1\), and \(\lfloor a^m\rfloor\) is different from \(\lfloor b^n\rfloor\) for any natural numbers \(m\) and \(n\)?
We have a very large chessboard, consisting of white and black squares. We would like to place a stain of a specific shape on this chessboard and we know that the area of this stain is less than the area of one square of the chessboard. Show that it is always possible to place the stain in such a way that it does not cover a vertex of any square.