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#### Integer and fractional parts. Archimedean property

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 $n$?

#### Integer and fractional parts. Archimedean property , Pigeonhole principle

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.

#### Equations in integer numbers , Integer and fractional parts. Archimedean property

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

#### Integer and fractional parts. Archimedean property

a$)$ Give an example of a positive number a such that ${a} + {1 / a} = 1.$
$\\$
b$)$ Can such an a be a rational number?

#### Counting in two ways , Exponential functions and logarithms (other) , Integer and fractional parts. Archimedean property

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.

#### Integer and fractional parts. Archimedean property , Pigeonhole principle (angles and lengths)

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

#### Divisibility of a number. General properties , Examples and counterexamples. Constructive proofs , Integer and fractional parts. Archimedean property

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

#### Graph theory , Integer and fractional parts. Archimedean property

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?

#### Integer and fractional parts. Archimedean property

Solve the equation $x^3 – [x] = 3.$

#### Integer and fractional parts. Archimedean property , Pigeonhole principle (other)

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

#### Integer and fractional parts. Archimedean property , Irrational inequalities.

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

#### Integer and fractional parts. Archimedean property , Pigeonhole principle (angles and lengths)

Find the general formula for the coefficients of the series
$(1 – 4x)^{ ½} = 1 + 2x + 6x^2 + 20x^3 + … + a_nx^n + …$

#### Equations of higher order (other) , Integer and fractional parts. Archimedean property

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?

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