[b]1)[/b] Prove that there are finite sequences, of any length, of nonegative integers having the property that the arithmetic mean of any choice of its elements is natural. [b]2)[/b] Study if there is an increasing infinite sequence of nonegative integers having the property that the arithmetic mean of any finite choice of its elements is natural.

Let $\mathcal{S}$ be a set consisting of $n \ge 3$ positive integers, none of which is a sum of two other distinct members of $\mathcal{S}$. Prove that the elements of $\mathcal{S}$ may be ordered as $a_1, a_2, \dots, a_n$ so that $a_i$ does not divide $a_{i - 1} + a_{i + 1}$ for all $i = 2, 3, \dots, n - 1$.

For any two positive real numbers $x_0 > 0$, $x_1 > 0$, a sequence of real numbers is defined recursively by $$x_{n+1} =\frac{4 \max\{x_n, 4\}}{x_{n-1}}$$ for $n \ge 1$. Find $x_{2010}$.

A biologist watches a chameleon. The chameleon catches flies and rests after each catch. The biologist notices that: [list=1][*]the first fly is caught after a resting period of one minute; [*]the resting period before catching the $2m^\text{th}$ fly is the same as the resting period before catching the $m^\text{th}$ fly and one minute shorter than the resting period before catching the $(2m+1)^\text{th}$ fly; [*]when the chameleon stops resting, he catches a fly instantly.[/list] [list=a][*]How many flies were caught by the chameleon before his first resting period of $9$ minutes in a row? [*]After how many minutes will the chameleon catch his $98^\text{th}$ fly? [*]How many flies were caught by the chameleon after 1999 minutes have passed?[/list]

Let $n$ be a positive integer and let $a_1, \ldots, a_{n-1} $ be arbitrary real numbers. Define the sequences $u_0, \ldots, u_n $ and $v_0, \ldots, v_n $ inductively by $u_0 = u_1 = v_0 = v_1 = 1$, and $u_{k+1} = u_k + a_k u_{k-1}$, $v_{k+1} = v_k + a_{n-k} v_{k-1}$ for $k=1, \ldots, n-1.$ Prove that $u_n = v_n.$

Determine all increasing sequences $\{a_n\}_{n=1}^\infty$ of natural numbers with the following property: for each two natural numbers $i$ and $j$ (not necessarily different), the numbers $i+j$ and $a_i+a_j$ have an equal number of distinct natural divisors.

Let ${(a_n)}_{n=0}^{\infty}$ and ${(b_n)}_{n=0}^{\infty}$ be real squences such that $a_0=40$, $b_0=41$ and for all $n\geq 0$ the given equalities hold. $$a_{n+1}=a_n+\frac{1}{b_n} \hspace{0.5 cm} \text{and} \hspace{0.5 cm} b_{n+1}=b_n+\frac{1}{a_n}$$ Find the least possible positive integer value of $k$ such that the value of $a_k$ is strictly bigger than $80$.

Let $f$ be a representation of the set $M = \{1, 2,..., 1988\}$ into $M$. For any natural $n$, let $x_1 = f(1)$, $x_{n+1} = f(x_n)$. Find out if there exists $m$ such that $x_{2m} = x_m$.

Denote by $a_n$ the greatest number that is not divisible by $3$ and that divides $n$. Consider the sequence $s_0 = 0, s_n = a_1 +a_2+\cdots+a_n, n \in \mathbb N$. Denote by $A(n)$ the number of all sums $s_k \ (0 \leq k \leq 3^n, k \in \mathbb N_0)$ that are divisible by $3$. Prove the formula \[A(n) = 3^{n-1} + 2 \cdot 3^{(n/2)-1} \cos \left(\frac{n\pi}{6}\right), \qquad n\in \mathbb N_0.\]

An infinite sequence of distinct real numbers is given. Prove that it contains a subsequence of $1999$ terms which is either monotonically increasing or monotonically decreasing.

Consider the sequence of natural numbers $a_n$ defined as $a_0=4$ and $a_{n+1}=\frac{a_n(a_n-1)}{2}$ for each $n\geq 0$. Define a new sequence $b_n$ as follows: $b_n=0$ if $a_n$ is even, and $b_n=1$ if $a_n$ is odd. Prove that for each natural $m$, the sequence \[b_m, b_{m+1}, b_{m+2},b_{m+3}, \dots\] is not periodic.

Let $a$ be a given positive integer. We consider the sequence $(x_n)_{n \ge 1}$ defined by $x_n=\frac{1}{1+na},$ for every positive integer $n.$ Prove that for any integer $k \ge 3,$ there exist positive integers $n_1<n_2<\ldots<n_k$ such that the numbers $x_{n_1},x_{n_2},\ldots,x_{n_k}$ are consecutive terms in an arithmetic progression.

The sequence of reals $a_1, a_2, a_3, \ldots$ is defined recursively by the recurrence: $$\dfrac{a_{n+1}}{a_n} - 3 = a_n(a_n - 3)$$ Given that $a_{2021} = 2021$, find $a_1$.

Let $a_1=2020$ and let $a_{n+1}=\sqrt{2020+a_n}$ for $n\ge 1$. How much is $\left\lfloor a_{2020}\right\rfloor$? Note: $\lfloor x\rfloor$ denotes the integer part of a number, that is that is, the immediate integer less than $x$. For example, $\lfloor 2.71\rfloor=2$ and $\lfloor \pi\rfloor=3$.

Let $N$ be a positive integer. A non-decreasing sequence $a_1 \le a_2 \le \dots$ of positive integers is said to be $N$-rioplatense if there exists an index $i$ such that $N = \frac{i}{a_i}$. Show that every sequence $2024$-rioplatense is $k$-rioplatense for $k=1, 2, 3, \dots, 2023$.

Define the sequence $a_1, a_2, \ldots$ and $b_1, b_2, \ldots$ as $a_0=1,a_1=4,a_2=49$ and for $n \geq 0$ $$ \begin{cases} a_{n+1}=7a_n+6b_n-3, \\ b_{n+1}=8a_n+7b_n-4. \end{cases} $$ Prove that for any non-negative integer $n,$ $a_n$ is a perfect square.

Let be the sequence $ \left( x_n \right)_{n\ge 1} $ defined as $$ x_n= \frac{4009}{4018020} x_{n-1} -\frac{1}{4018020} x_{n-2} + \left( 1+\frac{1}{n} \right)^n. $$ Prove that $ \left( x_n \right)_{n\ge 1} $ is convergent and determine its limit.

The sequence $a_0$, $a_1$, $a_2,$ $\ldots$ is defined as follows: \[a_0=2, \qquad a_{k+1}=2a_k^2-1 \quad\text{for }k \geq 0.\] Prove that if an odd prime $p$ divides $a_n$, then $2^{n+3}$ divides $p^2-1$. [hide="comment"] Hi guys , Here is a nice problem: Let be given a sequence $a_n$ such that $a_0=2$ and $a_{n+1}=2a_n^2-1$ . Show that if $p$ is an odd prime such that $p|a_n$ then we have $p^2\equiv 1\pmod{2^{n+3}}$ Here are some futher question proposed by me :Prove or disprove that : 1) $gcd(n,a_n)=1$ 2) for every odd prime number $p$ we have $a_m\equiv \pm 1\pmod{p}$ where $m=\frac{p^2-1}{2^k}$ where $k=1$ or $2$ Thanks kiu si u [i]Edited by Orl.[/i] [/hide]

Find the number of sequences with $2022$ natural numbers $n_1, n_2, n_3, \ldots, n_{2022}$, such that in every sequence: $\bullet$ $n_{i+1}\geq n_i$ $\bullet$ There is at least one number $i$, such that $n_i=2022$ $\bullet$ For every $(i, j)$ $n_1+n_2+\ldots+n_{2022}-n_i-n_j$ is divisible to both $n_i$ and $n_j$

A student firstly wrote $x=3$ on the board. For each procces, the stutent deletes the number x and replaces it with either $(2x+4)$ or $(3x+8)$ or $(x^2+5x)$. Is this possible to make the number $(20^{17}+2016)$ on the board? \\ (Explain your answer) \\ [hide=Note]This type of the question is well known but I am going to make a collection so, :blush: [/hide]

Let \(n\) be a non-negative integer and define \(a_n = 2^n - n\). Determine all non-negative integers \(m\) such that \(s_m = a_0 + a_1 + \dots + a_m\) is a power of 2.

Let $F_n$ be the nth Fibonacci number, defined by $F_1 = F_2 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for $n > 2$. Let $A_0, A_1, A_2,\cdots$ be a sequence of points on a circle of radius $1$ such that the minor arc from $A_{k-1}$ to $A_k$ runs clockwise and such that \[\mu(A_{k-1}A_k)=\frac{4F_{2k+1}}{F_{2k+1}^2+1}\] for $k \geq 1$, where $\mu(XY )$ denotes the radian measure of the arc $XY$ in the clockwise direction. What is the limit of the radian measure of arc $A_0A_n$ as $n$ approaches infinity?

Let $ a_1 \geq a_2 \geq \ldots \geq a_n$ be real numbers such that for all integers $ k > 0,$ \[ a^k_1 \plus{} a^k_2 \plus{} \ldots \plus{} a^k_n \geq 0.\] Let $ p \equal{}\max\{|a_1|, \ldots, |a_n|\}.$ Prove that $ p \equal{} a_1$ and that \[ (x \minus{} a_1) \cdot (x \minus{} a_2) \cdots (x \minus{} a_n) \leq x^n \minus{} a^n_1\] for all $ x > a_1.$

A natural number is called a [i]prime power[/i] if that number can be expressed as $p^n$ for some prime $p$ and natural number $n$. Determine the largest possible $n$ such that there exists a sequence of prime powers $a_1, a_2, \dots, a_n$ such that $a_i = a_{i - 1} + a_{i - 2}$ for all $3 \le i \le n$.

The sequence $a_1, a_2, ... , a_k, ...$ is constructed according to the rules: $$a_{2n} = a_n,a_{4n+1} = 1,a_{4n+3} = 0$$Prove that it is non-periodical sequence.