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#### Sequences

For which natural $n$ does the number $\frac{n^2}{1,001^n}$ reach its maximum value?

#### 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?

#### Boundedness, monotonicity , Sequnces

$a_1$, $a_2$, $a_3$, … is an increasing sequence of natural numbers. It is known that $a_{a_k} = 3k$ for any $k.$ Find a$)$ $a_{100}$; b$)$ $a_{2022}$.

#### 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.

#### Equations of higher order (other)

Determine all integer solutions of the equation $yk = x² + x$. Where $k$ is an integer  greater than $1.$

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