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#### Boundedness, monotonicity , Quadratic inequaities and systems of inequalities

For which natural K does the number reach its maximum value?

#### 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$)$ $a_{1983}$.

#### Boundedness, monotonicity

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

#### Boundedness, monotonicity , Recurrent relations (other)

The sequence of numbers $a_n$ is given by the conditions $a_1$ = 1, $a_{n + 1}$ = $a_n$ + 1/$a^2_n$ $($n $\geq$ 1$)$
Is it true that this sequence is limited?

#### Boundedness, monotonicity , Divisibility of a number. General properties , Examples and counterexamples. Constructive proofs , Identical transformations , Sequnces

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

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