Airlines connect pairs of cities. How can you connect 50 cities with the fewest number of airlines so that from every city you can get to any other city by taking at most two flights?
Does there exist a number \(h\) such that for any natural number \(n\) the number \(\lfloor h \times 2021^n\rfloor\) is not divisible by \(\lfloor h \times 2021^{n-1}\rfloor\)?
Is there a line on the coordinate plane relative to which the graph of the function \(y = 2^x\) is symmetric?
In a dark room on a shelf there are 4 pairs of socks of two different sizes and two different colours that are not arranged in pairs. What is the minimum number of socks necessary to move from the drawer to the suitcase, without leaving the room, so that there are two pairs of socks of different sizes and colours in the suitcase?
Aladdin visited all of the points on the equator, moving to the east, then to the west, and sometimes instantly moving to the diametrically opposite point on Earth. Prove that there was a period of time during which the difference in distances traversed by Aladdin to the east and to the west was not less than half the length of the equator.
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?
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.
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 \(n\)?
The board has the form of a cross, which is obtained if corner boxes of a square board of \(4 \times 4\) are erased. Is it possible to go around it with the help of the knight chess piece and return to the original cell, having visited all the cells exactly once?