Problems

Prove that for any natural number $a_1>1$ there exists an increasing sequence of natural numbers $a_1,a_2,a_3,\dots$, for which $a_1^2+a_2^2+\cdots +a_k^2$ is divisible by $a_1+a_2+\cdots +a_k$ for all $k\ge 1$.

The sequence of numbers $a_n$ is given by the conditions $a_1=1$, $a_{n+1}=a_n+1/a_n^2$ ($n\ge 1$).

Is it true that this sequence is limited?

The sequence of numbers $\{x_n\}$ is given by the following conditions: $x_1\ge -a$, $x_{n+1}=\sqrt{a+x_n}$. Prove that the sequence $x_n$ is monotonic and bounded. Find its limit.

We took several positive numbers and constructed the following sequence: $a_1$ is the sum of the initial numbers, $a_2$ is the sum of the squares of the original numbers, $a_3$ is the sum of the cubes of the original numbers, and so on.

a) Could it happen that up to $a_5$ the sequence decreases ($a_1>a_2>a_3>a_4>a_5$), and starting with $a_5$ – it increases ($a_5<a_6<a_7<\dots$)?

b) Could it be the other way around: before $a_5$ the sequence increases, and starting with $a_5$ – decreases?

At what value of $k$ is the quantity $A_k=({19}^k+{66}^k)/k!$ at its maximum?

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