Filter Problems

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$a_1$, $a_2$, $a_3$, … is an increasing sequence of natural numbers. It is known that $a_{a_k}$ = 3k for any k. Find a$)$ $a_{100}$; b$)$ $a_{1983}$.

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 $(a_5 < a_6 < a_7 <...)$?

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

On the occasion of the beginning of the winter holidays all of the boys from class 8B went to the shooting range. It is known that there are n boys in 8B. There are n targets at the shooting range which the class attended. Each of the boys randomly chooses a target, while some of the boys could choose the same target. After this, all of the boys simultaneously attempt to shoot their target. It is known that each of the boys hits their target. The target is considered to be affected if at least one boy has hit it.

a) Find the average number of affected targets.

b) Can the average number of affected targets be less than n/2?

The sequence of numbers {$x_n$} is given by the following conditions:

$x_1$ $\geq$ – a, $x_{n + 1}$ = $\sqrt{a + x_n}$.

Prove that the sequence {$x_n$} is monotonic and bounded. Find its limit.

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?

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.