Given an infinite sequence of numbers $a_1, a_2, a_3, ...$ such that for each positive integer $k$, there exists positive integer $t$ for which $a_k = a_{k+t} = a_{k+2t} = ....$ Does this sequences must be periodic?

Let $a_{1}, \ldots, a_{n}$ be an infinite sequence of strictly positive integers, so that $a_{k} < a_{k+1}$ for any $k.$ Prove that there exists an infinity of terms $ a_{m},$ which can be written like $a_m = x \cdot a_p + y \cdot a_q$ with $x,y$ strictly positive integers and $p \neq q.$

[b]a)[/b] Let be a sequence $ \left( x_n \right)_{n\ge 1} $ defined by the recursion $ x_{n+1}=\frac{1+x_n}{1-x_n} , $ with $ x_1=2006. $ Calculate $ \lim_{n\to\infty } \frac{x_1+x_2+\cdots +x_n}{n} . $ [b]b)[/b] Prove that if a convergent sequence $ \left( s_n \right)_{n\ge 1} $ verifies $ a_{2^n} =na_n , $ for any natural numbers $ n, $ then $ a_n=0, $ for any natural numbers $ n. $ [i]Cornel Stoicescu[/i]

Determine the general term of the sequence ($a_n$) given by $a_0 =\alpha > 0$ and $a_{n+1} =\frac{a_n}{1+a_n}$ .

Consider the sequence of rational numbers defined by $x_1=\frac{4}{3}$, and $x_{n+1}=\frac{x_n^2}{x_n^2-x_n+1}$. Show that the nu,erator of the lowest term expression of each sum $x_1+x_2+...+x_k$ is a perfect square.

Let $(x_n)$ be an infinite sequence of real numbers from interval $(0, 1)$. An infinite sequence $(a_n)$ of positive integers is defined as follows: $a_1 = 1$, and for $i \ge 1$, $a_{i+1}$ is equal to the smallest positive integer $m$, for which $[x_1 + x_2 + \ldots + x_m] = a_i$. Show that for any indexes $i, j$ holds $a_{i+j} \ge a_i + a_j$. [i]Proposed by Nazar Serdyuk[/i]

Let $k$ be an integer and define a sequence $a_0 , a_1 ,a_2 ,\ldots$ by $$ a_0 =0 , \;\; a_1 =k \;\;\text{and} \;\; a_{n+2} =k^{2}a_{n+1}-a_n \; \text{for} \; n\geq 0.$$ Prove that $a_{n+1} a_n +1$ divides $a_{n+1}^{2} +a_{n}^{2}$ for all $n$.

Let $a_1, a_2, \ldots$ and $b_1, b_2, \ldots$ be sequences such that $a_ib_i - a_i - b_i = 0$ and $a_{i+1} = \frac{2-a_ib_i}{1-b_i}$ for all $i \ge 1$. If $a_1 = 1 + \frac{1}{\sqrt[4]{2}}$, then what is $b_{6}$? [i]Proposed by Andrew Wu[/i]

For every non-negative integer $i$, define the number $M(i)$ as follows: write $i$ down as a binary number, so that we have a string of zeroes and ones, if the number of ones in this string is even, then set $M(i) = 0$, otherwise set $M(i) = 1$. (The first terms of the sequence $M(i)$, $i = 0, 1, 2, ...$ are $0, 1, 1, 0, 1, 0, 0, 1,...$ ) (a) Consider the finite sequence $M(O), M(1), . . . , M(1000) $. Prove that there are at least $320$ terms in this sequence which are equal to their neighbour on the right : $M(i) = M(i + 1 )$ . (b) Consider the finite sequence $M(O), M(1), . . . , M(1000000)$ . Prove that the number of terms $M(i)$ such that $M(i) = M(i +7)$ is at least $450000$. (A Kanel)

Let $r$ be a positive integer, and let $a_0 , a_1 , \cdots $ be an infinite sequence of real numbers. Assume that for all nonnegative integers $m$ and $s$ there exists a positive integer $n \in [m+1, m+r]$ such that \[ a_m + a_{m+1} +\cdots +a_{m+s} = a_n + a_{n+1} +\cdots +a_{n+s} \] Prove that the sequence is periodic, i.e. there exists some $p \ge 1 $ such that $a_{n+p} =a_n $ for all $n \ge 0$.

$a$ and $b$ are fixed real numbers. With $x_n$ we denote the sum of the digits of $an+b$ in the decimal number system. Prove that the sequence $x_n$ contains an infinite constant subsequence.

Let $r$ be a positive integer, and let $a_0 , a_1 , \cdots $ be an infinite sequence of real numbers. Assume that for all nonnegative integers $m$ and $s$ there exists a positive integer $n \in [m+1, m+r]$ such that \[ a_m + a_{m+1} +\cdots +a_{m+s} = a_n + a_{n+1} +\cdots +a_{n+s} \] Prove that the sequence is periodic, i.e. there exists some $p \ge 1 $ such that $a_{n+p} =a_n $ for all $n \ge 0$.

Define the sequences $(a_n),(b_n)$ by \begin{align*} & a_n, b_n > 0, \forall n\in\mathbb{N_+} \\ & a_{n+1} = a_n - \frac{1}{1+\sum_{i=1}^n\frac{1}{a_i}} \\ & b_{n+1} = b_n + \frac{1}{1+\sum_{i=1}^n\frac{1}{b_i}} \end{align*} 1) If $a_{100}b_{100} = a_{101}b_{101}$, find the value of $a_1-b_1$; 2) If $a_{100} = b_{99}$, determine which is larger between $a_{100}+b_{100}$ and $a_{101}+b_{101}$.

A sequence $(a_n)_{n \in N}$ of squares of nonzero integers is such that for each $n$ the difference $a_{n+1} - a_n$ is a prime or the square of a prime. Show that all such sequences are finite and determine the longest sequence.

A circle $K_1$ of radius $r_1 = 1\slash 2$ is inscribed in a semi-circle $H$ with diameter $AB$ and radius $1.$ A sequence of different circles $K_2, K_3, \ldots$ with radii $r_2, r_3, \ldots$ respectively are drawn so that for each $n\geq 1$, the circle $K_{n+1}$ is tangent to $H$, $K_n$ and $AB.$ Prove that $a_n = 1\slash r_n$ is an integer for each $n$, and that it is a perfect square for $n$ even and twice a perfect square for $n$ odd.

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 $a_{2022}\leq 1$.

The sequence $(a_n)$ is defined as: $$a_1=1007$$ $$a_{i+1}\geq a_i+1$$ Prove the inequality: $$\frac{1}{2016}>\sum_{i=1}^{2016}\frac{1}{a_{i+1}^{2}+a_{i+2}^2}$$

Consider the sequence $(a_n)_{n\geqslant 1}$ defined by $a_1=1/2$ and $2n\cdot a_{n+1}=(n+1)a_n.$[list=a] [*]Determine the general formula for $a_n.$ [*]Let $b_n=a_1+a_2+\cdots+a_n.$ Prove that $\{b_n\}-\{b_{n+1}\}\neq \{b_{n+1}\}-\{b_{n+2}\}.$ [/list]

Consider the sequences $(a_n), (b_n)$ defined by \[a_1=3, \quad b_1=100 , \quad a_{n+1}=3^{a_n} , \quad b_{n+1}=100^{b_n} \] Find the smallest integer $m$ for which $b_m > a_{100}.$

For a positive integer $k$, let $p(k)$ be the smallest prime that does not divide $k$. Given a positive integer $a$, define the infinite sequence $a_0, a_1, \ldots$ by $a_0 = a$ and, for $n > 0$, $a_n$ is the smallest positive integer with the following properties: • $a_n$ has not yet appeared in the sequence, that is, $a_n \neq a_i$ for $0 \leq i < n$; • $(a_{n-1})^{a_n} - 1$ is a multiple of $p(a_{n-1})$. Prove that every positive integer appears as a term in the sequence, that is, for every positive integer $m$ there is $n$ such that $a_n = m$.

Let $c,d \geq 2$ be naturals. Let $\{a_n\}$ be the sequence satisfying $a_1 = c, a_{n+1} = a_n^d + c$ for $n = 1,2,\cdots$. Prove that for any $n \geq 2$, there exists a prime number $p$ such that $p|a_n$ and $p \not | a_i$ for $i = 1,2,\cdots n-1$.

Determine how many $100$-positive integer sequences satisfy the two conditions following: - At least one term of the sequence is equal to $4$ or $5$. - Any two adjacent terms differ as a maximum in $2$.

For which positive integers $m$ does there exist an infinite arithmetic sequence of integers $a_1, a_2, . . .$ and an infinite geometric sequence of integers $g_1, g_2, . . .$ satisfying the following properties? [list] [*] $a_n - g_n$ is divisible by $m$ for all integers $n \ge 1$; [*] $a_2 - a_1$ is not divisible by $m$. [/list] [i]Holden Mui[/i]

Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.

Let a sequence $(a_n)$ satisfy: $a_1=5,a_2=13$ and $a_{n+1}=5a_n-6a_{n-1},\forall n\ge2$ a) Prove that $(a_n, a_{n+1})=1,\forall n\ge1$ b) Prove that: $2^{k+1}|p-1\forall k\in\mathbb{N}$, if p is a prime factor of $a_{2^k}$