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

#### Central angle. Arc length and circumference , Integer and fractional parts. Archimedean property , Pigeonhole principle (angles and lengths) , Sequnces , Symmetry and involutions

For each pair of real numbers a and b, consider the sequence of numbers $p_n$ = [2 {an + b}]. Any k consecutive terms of this sequence will be called a word. Is it true that any ordered set of zeros and ones of length k is a word of the sequence given by some a and b for k = 4; when k = 5?
Note: [c] is the integer part, {c} is the fractional part of the number c.

#### 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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