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Are there such irrational numbers $a$ and $b$ so that $a > 1,\, b > 1,$ and [$a^m$] is different from [$b^n$] for any natural numbers $m$ and $n$?

Some real numbers $ a_1, a_2, a_3,…,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.

Find the number of solutions in natural numbers of the equation $⌊x / 10⌋ = ⌊x / 11⌋ + 1.$

a$)$ Give an example of a positive number a such that ${a} + {1 / a} = 1.$

$\\$

b$)$ Can such an a be a rational number?

Prove that for every natural number $n > 1$ the equality: $[n^{1 / 2}] + [n^{1/ 3}] + … + [n^{1 / n}] = [log_{2}n] + [log_{3}n] + … + [log_{n}n]$ is satisfied.

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.

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?

Is there a line on the coordinate plane relative to which the graph of the function $y = 2^x$ is symmetric?

Does there exist a number $h$ such that for any natural number n the number [$h \times 2021^n$] is not divisible by [$h \times 2021^{n-1}$]?

How can you connect 50 cities with the least number of airlines so that from every city you can get to any other one by making no more than two transfers?

The numbers $[a],\, [2a],\, …,\, [Na]$ are all different, and the numbers $[1/a],\, [2/a],\,…,\, [M/a]$ are also all different. Find all such $a$.

The segment OA is given. From the end of the segment A there are 5 segments $AB_1, AB_2, AB_3, AB_4, AB_5$. From each point $B_i$ there can be five more new segments or not a single new segment, etc. Can the number of free ends of the constructed segments be 1001? By the free end of a segment we mean a point belonging to only one segment $($except point O$)$.

Author: A.K. Tolpygo

12 grasshoppers sit on a circle at various points. These points divide the circle into 12 arcs. Let’s mark the 12 mid-points of the arcs. At the signal the grasshoppers jump simultaneously, each to the nearest clockwise marked point. 12 arcs are formed again, and jumps to the middle of the arcs are repeated, etc. Can at least one grasshopper return to his starting point after he has made a) 12 jumps; b) 13 jumps?

Author: A. Khrabrov

Do there exist integers a and b such that

a) the equation $x^2 + ax + b = 0$ does not have roots, and the equation $[x^2] + ax + b = 0$ does have roots?

b) the equation $x^2 + 2ax + b = 0$ does not have roots, and the equation $[x^2] + 2ax + b = 0$ does have roots?

Note that here, square brackets represent integers and curly brackets represent non-integer values or 0.

A firm recorded its expenses in pounds for 100 items, creating a list of 100 numbers $($with each number having no more than two decimal places$)$. Each accountant took a copy of the list and found an approximate amount of expenses, acting as follows. At first, he arbitrarily chose two numbers from the list, added them, discarded the sum after the decimal point $($if there was anything$)$ and recorded the result instead of the selected two numbers. With the resulting list of 99 numbers, he did the same, and so on, until there was one whole number left in the list. It turned out that in the end all the accountants ended up with different results. What is the largest number of accountants that could work in the company?

It is known that $a > 1.$ Is it always true that $⌊\sqrt{⌊\sqrt{a}⌋}⌋ = ⌊\sqrt{4}{a}⌋$?

Author: E.V. Bakaev

Write in five circles natural numbers so that the following two conditions are fulfilled:

– if two circles are connected by a line, then the numbers in them must differ exactly by two or four times;

– if two circles are not connected by a line, then the ratio of the numbers written in them should not be equal to either 2 or 4.

Find the general formula for the coefficients of the series

$(1 – 4x)^{ ½} = 1 + 2x + 6x^2 + 20x^3 + … + a_nx^n + …$

A frog jumps over the vertices of the triangle ABC, moving each time to one of the neighbouring vertices.

How many ways can it get from A to A in n jumps?