Let $a_1 = 1 +\sqrt2$ and for each $n \ge 1$ dene $a_{n+1} = 2 -\frac{1}{a_n}$. Find the greatest integer less than or equal to the product $a_1a_2a_3 ... a_{200}$.

Let $ a_1$, $ a_2$, $ \ldots$, $ a_n$ be distinct positive integers, $ n\ge 3$. Prove that there exist distinct indices $ i$ and $ j$ such that $ a_i \plus{} a_j$ does not divide any of the numbers $ 3a_1$, $ 3a_2$, $ \ldots$, $ 3a_n$. [i]Proposed by Mohsen Jamaali, Iran[/i]

The sequence $ \{a_n\}$ of integers is defined by \[ a_1 \equal{} 2, a_2 \equal{} 7 \] and \[ \minus{} \frac {1}{2} < a_{n \plus{} 1} \minus{} \frac {a^2_n}{a_{n \minus{} 1}} \leq \frac {}{}, n \geq 2. \] Prove that $ a_n$ is odd for all $ n > 1.$

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 $i = 2, 3, \dots, n - 1$.

[b]a)[/b] Provide an example of a sequence $ \left( a_n \right)_{n\ge 1} $ of positive real numbers whose series converges, and has the property that each member (sequence) of the family of sequences $ \left(\left( n^{\alpha } a_n \right)_{n\ge 1}\right)_{\alpha >0} $ is unbounded. [b]b)[/b] Let $ \left( b_n \right)_{n\ge 1} $ be a sequence of positive real numbers, having the property that $$ nb_{n+1}\leqslant b_1+b_2+\cdots +b_n, $$ for any natural number $ n. $ Prove that the following relations are equivalent: $\text{(i)} $ there exists a convergent member (series) of the family of series $ \left( \sum_{i=1}^{\infty } b_i^{\beta } \right)_{\beta >0} $ $ \text{(ii)} $ there exists a member (sequence) of the family of sequences $ \left(\left( n^{\beta } b_n \right)_{n\ge 1}\right)_{\beta >0} $ that is convergent to $ 0. $ [i]Eugen Păltănea[/i]

Let be a natural number $ m\ge 2. $ [b]a)[/b] Let be $ m $ pairwise distinct rational numbers. Prove that there is an ordering of these numbers such that these are terms of an arithmetic progression. [b]b)[/b] Given that for any $ m $ pairwise distinct real numbers there is an ordering of them such that they are terms of an arithmetic sequence, determine the number $ m. $ [i]Bogdan Blaga[/i]

Let $a > 1$ be a positive integer and $d > 1$ be a positive integer coprime to $a$. Let $x_1=1$, and for $k\geq 1$, define $$x_{k+1} = \begin{cases} x_k + d &\text{if } a \text{ does not divide } x_k \\ x_k/a & \text{if } a \text{ divides } x_k \end{cases}$$ Find, in terms of $a$ and $d$, the greatest positive integer $n$ for which there exists an index $k$ such that $x_k$ is divisible by $a^n$.

Determine if there exists an infinite sequence of not necessarily distinct positive integers $a_1, a_2, a_3,\ldots$ such that for any positive integers $m$ and $n$ where $1 \leq m < n$, the number $a_{m+1} + a_{m+2} + \ldots + a_{n}$ is not divisible by $a_1 + a_2 + \ldots + a_m$.

Let $(a_n)_{n\geq 1}$ be a sequence of positive real numbers with the property that $$(a_{n+1})^2 + a_na_{n+2} \leq a_n + a_{n+2}$$ for all positive integers $n$. Show that $a_{2022}\leq 1$.

[b]a)[/b] Find the number of infinite sequences of integers $ \left( a_n \right)_{n\ge 1} $ that have the property that $ a_na_{n+2}a_{n+3}=-1, $ for any natural number $ n. $ [b]b)[/b] Prove that there is no infinite sequence of integers $ \left( b_n \right)_{n\ge 1} $ that have the property that $ b_nb_{n+2}b_{n+3}=2005, $ for any natural number $ n. $

Let $m$ be a fixed integer greater than $1$. The sequence $x_0$, $x_1$, $x_2$, $\ldots$ is defined as follows: \[x_i = \begin{cases}2^i&\text{if }0\leq i \leq m - 1;\\\sum_{j=1}^mx_{i-j}&\text{if }i\geq m.\end{cases}\] Find the greatest $k$ for which the sequence contains $k$ consecutive terms divisible by $m$ . [i]Proposed by Marcin Kuczma, Poland[/i]

Periodic sequences $(a_n),(b_n),(c_n)$ and $(d_n)$ satisfy the following conditions: $$a_{n+1}=a_n+b_n,\enspace\enspace b_{n+1}=b_n+c_n,$$ $$c_{n+1}=c_n+d_n,\enspace\enspace d_{n+1}=d_n+a_n,$$ for $n=1,2,\ldots$. Prove that $a_2=b_2=c_2=d_2=0$.

Consider the infinite, strictly increasing sequence of positive integer $(a_n)$ such that i. All terms of sequences are pairwise coprime. ii. The sum $\frac{1}{\sqrt{a_1a_2}} +\frac{1}{\sqrt{a_2a_3}}+ \frac{1}{\sqrt{a_3a_4}} + ..$ is unbounded. Prove that this sequence contains infinitely many primes.

The sequences $(x_n),(y_n),(z_n),n\in\mathbb N$, are defined by the relations $$x_{n+1}=\frac{2x_n}{x_n^2-1},\qquad y_{n+1}=\frac{2y_n}{y_n^2-1},\qquad z_{n+1}=\frac{2z_n}{z_n^2-1},$$where $x_1=2$, $y_1=4$, and $x_1y_1z_1=x_1+y_1+z_1$. (a) Show that $x_n^2\ne1$, $y_n^2\ne1$, $z_n^2\ne1$ for all $n$; (b) Does there exist a $k\in\mathbb N$ for which $x_k+y_k+z_k=0$?

The sequence $a_1,a_2,...,a_n$ satisfies the following conditions: $$a_1=0, |a_2|=|a_1+1|, ..., |a_n|=|a_{n-1}+1|.$$ Prove that $$(a_1+a_2+...+a_n)/n \ge -1/2$$

$a_1,a_2,\ldots,a_n$ is a sequence of positive integers that has at least $\frac {2n}{3}+1$ distinct numbers and each positive integer has occurred at most three times in it. Prove that there exists a permutation  $b_1,b_2,\ldots,b_n$ of $a_i $'s such that all the $n$ sums $b_i+b_{i+1}$ are distinct ($1\le i\le n $ , $b_{n+1}\equiv b_1 $) [i]Proposed by Mohsen Jamali[/i]

There exists a unique strictly increasing sequence of nonnegative integers $a_1 < a_2 < \dots < a_k$ such that \[\frac{2^{289}+1}{2^{17}+1} = 2^{a_1} + 2^{a_2} + \dots + 2^{a_k}.\] What is $k?$ $\textbf{(A) } 117 \qquad \textbf{(B) } 136 \qquad \textbf{(C) } 137 \qquad \textbf{(D) } 273 \qquad \textbf{(E) } 306$

[b]a)[/b] Let be a continuous function $ f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R}_{>0} . $ Show that there exists a natural number $ n_0 $ and a sequence of positive real numbers $ \left( x_n \right)_{n>n_0} $ that satisfy the following relation. $$ n\int_0^{x_n} f(t)dt=1,\quad n_0<\forall n\in\mathbb{N} $$ [b]b)[/b] Prove that the sequence $ \left( nx_n \right)_{n> n_0} $ is convergent and find its limit.

Given is the sequence $(a_n)_{n\geq 0}$ which is defined as follows:$a_0=3$ and $a_{n+1}-a_n=n(a_n-1) \ , \ \forall n\geq 0$. Determine all positive integers $m$ such that $\gcd (m,a_n)=1 \ , \ \forall n\geq 0$.

An infinite sequence $p_1, p_2, p_3, \ldots$ of natural numbers in the decimal system has the following property: For every $i \in \mathbb{N}$ the last digit of $p_{i+1}$ is different from $9$, and by omitting this digit one obtains number $p_i$. Prove that this sequence contains infinitely many composite numbers.

A sequence $\{an\}$ of positive integers is defined by \[a_n=\left[ n +\sqrt n + \frac 12 \right] , \qquad \forall n \in \mathbb N\] Determine the positive integers that occur in the sequence.

From the sequence $1,\frac12, \frac13, ...$ can one choose (a) a subsequence of $100$ different numbers, (b) an infinite subsequence such that each number (beginning from the third) is equal to the difference between the two preceding numbers ($a_k=a_{k-2}-a_{k-1}$)? (SI Tokarev)

Let $n$ be a positive integer. Given a sequence $\varepsilon_1$, $\dots$, $\varepsilon_{n - 1}$ with $\varepsilon_i = 0$ or $\varepsilon_i = 1$ for each $i = 1$, $\dots$, $n - 1$, the sequences $a_0$, $\dots$, $a_n$ and $b_0$, $\dots$, $b_n$ are constructed by the following rules: \[a_0 = b_0 = 1, \quad a_1 = b_1 = 7,\] \[\begin{array}{lll} a_{i+1} = \begin{cases} 2a_{i-1} + 3a_i, \\ 3a_{i-1} + a_i, \end{cases} & \begin{array}{l} \text{if } \varepsilon_i = 0, \\ \text{if } \varepsilon_i = 1, \end{array} & \text{for each } i = 1, \dots, n - 1, \\[15pt] b_{i+1}= \begin{cases} 2b_{i-1} + 3b_i, \\ 3b_{i-1} + b_i, \end{cases} & \begin{array}{l} \text{if } \varepsilon_{n-i} = 0, \\ \text{if } \varepsilon_{n-i} = 1, \end{array} & \text{for each } i = 1, \dots, n - 1. \end{array}\] Prove that $a_n = b_n$. [i]Proposed by Ilya Bogdanov, Russia[/i]

Let $(a_n)_{n \geq 1}$ be a sequence of real numbers. We define a sequence of real functions $(f_n)_{n \geq 0}$ such that for all $x \in \mathbb{R}$, the following holds: \[ f_0(x) = 1 \quad \text{and} \quad f_n(x) = \int_{a_n}^{x} f_{n-1}(t) \, dt \quad \text{for } n \geq 1. \] Find all possible sequences $(a_n)_{n \geq 1}$ such that $f_n(0) = 0$ for all $n \geq 2$.\\ [b]Note:[/b] It is not necessarily true that $f_1(0) = 0$.

The sequences $a_n$, $b_n$ and $c_n$ are defined recursively in the following way: $a_0 = 1/6$, $b_0 = 1/2$, $c_0 = 1/3,$ $$a_{n+1}= \frac{(a_n + b_n)(a_n + c_n)}{(a_n - b_n)(a_n - c_n)},\,\, b_{n+1}= \frac{(b_n + a_n)(b_n + c_n)}{(b_n - a_n)(b_n - c_n)},\,\, c_{n+1}= \frac{(c_n + a_n)(c_n + b_n)}{(c_n - a_n)(c_n - b_n)}$$ For each natural number $N$, the following polynomials are defined: $A_n(x) =a_o+a_1 x+ ...+ a_{2N}x^{2N}$ $B_n(x) =b_o+a_1 x+ ...+ a_{2N}x^{2N}$ $C_n(x) =a_o+a_1 x+ ...+ a_{2N}x^{2N}$ Assume the sequences are well defined. Show that there is no real $c$ such that $A_N(c) = B_N(c) = C_N(c) = 0$.