For any odd prime $p$ and any integer $n,$ let $d_p (n) \in \{ 0,1, \dots, p-1 \}$ denote the remainder when $n$ is divided by $p.$ We say that $(a_0, a_1, a_2, \dots)$ is a [i]p-sequence[/i], if $a_0$ is a positive integer coprime to $p,$ and $a_{n+1} =a_n + d_p (a_n)$ for $n \geqslant 0.$ (a) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $(a_0, a_1, a_2, \dots)$ and $(b_0, b_1, b_2, \dots)$ such that $a_n >b_n$ for infinitely many $n,$ and $b_n > a_n$ for infinitely many $n?$ (b) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $(a_0, a_1, a_2, \dots)$ and $(b_0, b_1, b_2, \dots)$ such that $a_0 <b_0,$ but $a_n >b_n$ for all $n \geqslant 1?$ [I]United Kingdom[/i]

Let $(x_{n})_{n=1}^{+\infty}$ be a sequence defined recursively with $x_{n+1} = x_{n}(x_{n}-2)$ and $x_{1} = \frac{7}{2}$. Let $x_{2021} = \frac{a}{b}$, where $a,b \in \mathbb{N}$ are coprime. Show that if $p$ is a prime divisor of $a$, then either $3|p-1$ or $p=3$. [i]Authored by Nikola Velov[/i]

Let a sequence $(a_i)_{i=10}^{\infty}$ be defined as follows: [list=a] [*] $a_{10}$ is some positive integer, which can of course be written in base 10. [*] For $i \geq 10$ if $a_i > 0$, let $b_i$ be the positive integer whose base-$(i + 1)$ representation is the same as $a_i$'s base-$i$ representation. Then let $a_{i + 1} = b_i - 1$. If $a_i = 0$, $a_{i + 1} = 0$. [/list] For example, if $a_{10} = 11$, then $b_{10} = 11_{11} (= 12_{10})$; $a_{11} = 11_{11} - 1 = 10_{11} (= 11_{10})$; $b_{11} = 10_{12} (= 12_{10})$; $a_{12} = 11$. Does there exist $a_{10}$ such that $a_i$ is strictly positive for all $i \geq 10$?

The sequence $1,2,3,4,0,9,6,9,4,8,7,\ldots$ is formed so that each term, starting from the fifth, is the units digit of the sum of the previous four. (a) Do the digits $2,0,0,4$ occur in the sequence in this order? (b) Will the initial digits $1,2,3,4$ ever occur again in this order?

Let ${k}$ be a fixed positive integer. A finite sequence of integers ${x_1,x_2, ..., x_n}$ is written on a blackboard. Pepa and Geoff are playing a game that proceeds in rounds as follows. - In each round, Pepa first partitions the sequence that is currently on the blackboard into two or more contiguous subsequences (that is, consisting of numbers appearing consecutively). However, if the number of these subsequences is larger than ${2}$, then the sum of numbers in each of them has to be divisible by ${k}$. - Then Geoff selects one of the subsequences that Pepa has formed and wipes all the other subsequences from the blackboard. The game finishes once there is only one number left on the board. Prove that Pepa may choose his moves so that independently of the moves of Geoff, the game finishes after at most ${3k}$ rounds. (Poland)

Define the sequence $A_1, A_2, A_3, \dots$ by $A_1 = 1$ and for $n=1,2,3,\dots$ $$A_{n+1}=\frac{A_n+2}{A_n +1}.$$ Define the sequences $B_1, B_2, B_3,\dots$ by $B_1=1$ and for $n=1,2,3,\dots$ $$B_{n+1}=\frac{B_n^2 +2}{2B_n}.$$ Prove that $B_{n+1}=A_{2^n}$ for all non-negative integers $n$.

Consider all possible functions defined for $x = 1, 2, ..., M$ and taking values $​​y = 1, 2, ..., n$. We denote the set of such functions by $T.$ By $T_0$ we denote the subset of $T$ consisting of functions whose value changes exactly by $ 1$ (in one direction or another) when the argument changes by $1$. Prove that if $M\ge 2n-4$, then among the functions from of the set $T$, there is a function that coincides at least at one point with any function from $T_0$. Specify at least one such function. Prove that if $M <2n-4$, then there is no such function.

a) Prove that for each positive integer $n$ there is a unique positive integer $a_n$ such that $$(1 + \sqrt5)^n =\sqrt{a_n} + \sqrt{a_n+4^n} . $$ b) Prove that $a_{2010}$ is divisible by $5\cdot 4^{2009}$ and find the quotient

Integer sequence $(x_{n})$ is defined as follows; $x_{1} = 1$, and for each integer $n \geq 1$, $x_{n+1}$ is equal to the largest number that can be obtained by permutation of the digits of $x_{n}+2$. Find the smallest $n$ for which the decimal representation of $x_{n}$ contains exactly $2022$ digits

Given three numbers $x, y, z$ denote the absolute values of the differences of each pair by $x_1,y_1, z_1$. From $x_1, y_1, z_1$ form in the same fashion the numbers $x_2, y_2, z_2$, etc. It is known that $x_n = x,y_n = y, z_n = z$ for some $n$. Find $y$ and $z$ if $x = 1$.

Ana and Banana are playing a game. First Ana picks a word, which is defined to be a nonempty sequence of capital English letters. (The word does not need to be a valid English word.) Then Banana picks a nonnegative integer $k$ and challenges Ana to supply a word with exactly $k$ subsequences which are equal to Ana's word. Ana wins if she is able to supply such a word, otherwise she loses. For example, if Ana picks the word "TST", and Banana chooses $k=4$, then Ana can supply the word "TSTST" which has 4 subsequences which are equal to Ana's word. Which words can Ana pick so that she wins no matter what value of $k$ Banana chooses? (The subsequences of a string of length $n$ are the $2^n$ strings which are formed by deleting some of its characters, possibly all or none, while preserving the order of the remaining characters.) [i]Proposed by Kevin Sun

Let $A$ be a set of $2025$ non-negative integers and $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ be a function with the following two properties: 1) For every two distinct positive integers $x,y$ there exists $a\in A$, such that $x-y$ divides $f(x+a) - f(y+a)$. 2) For every positive integer $N$ there exists a positive integer $t$ such that $f(x) \neq f(y)$ whenever $x,y \in [t, t+N]$ are distinct. Prove that there are infinitely many primes $p$ such that $p$ divides $f(x)$ for some positive integer $x$.

Define the sequence ($x_n$) as follows: the first term is $1$, the next two are $2,4$, the next three are $5,7,9$, the next four are $10,12,14,16$, and so on. Express $x_n$ as a function of $n$.

The sequence $\{a_n\}$ is defined as follows: \begin{align*} a_n = \sqrt{1 + \left(1 + \frac 1n \right)^2} + \sqrt{1 + \left(1 - \frac 1n \right)^2}. \end{align*} What is the value of the expression given below? \begin{align*} \frac{4}{a_1} + \frac{4}{a_2} + \dots + \frac{4}{a_{96}}.\end{align*} $\textbf{(A)}\ \sqrt{18241} \qquad\textbf{(B)}\ \sqrt{18625} - 1 \qquad\textbf{(C)}\ \sqrt{18625} \qquad\textbf{(D)}\ \sqrt{19013} - 1\qquad\textbf{(E)}\ \sqrt{19013}$

[b]a)[/b] Determine all sequences of real numbers $ \left( x_n\right)_{n\in\mathbb{N}\cup\{ 0\}} $ that satisfy $ x_{n+2}+x_{n+1}=x_n, $ for any nonnegative integer $ n. $ [b]b)[/b] If $ y_k>0 $ and $ y_k^k=y_k+k, $ for all naturals $ k, $ calculate $ \lim_{n\to\infty }\frac{\ln n}{n\left( x_n-1\right)} . $

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 $p \neq q.$

Let $a_0,a_1,a_2...$ be a sequence of natural numbers with the following property: $a_n^2$ divides $a_{n-1} a_{n+1}$ for $\forall$ $n\in \mathbb{N}$. Prove that, if for some natural $k\geq 2$ the numbers $a_1$ and $a_k$ are coprime, then $a_1$ divides $a_0$.

Let $(x_n)_{n=1}^\infty$ be a sequence defined recursively with: $x_1=2$ and $x_{n+1}=\frac{x_n(x_n+n)}{n+1}$ for all $n \ge 1$. Prove that $$n(n+1) >\frac{(x_1+x_2+ \ldots +x_n)^2}{x_{n+1}}.$$ [i]Proposed by Nikola Velov[/i]

Let $(a_i)_{i\in \mathbb{N}}$ be a sequence with $a_1=\frac{3}2$ such that $$a_{n+1}=1+\frac{n}{a_n}$$ Find $n$ such that $2020\le a_n <2021$

Let $ n > 1$ be an integer. Find all sequences $ a_1, a_2, \ldots a_{n^2 \plus{} n}$ satisfying the following conditions: \[ \text{ (a) } a_i \in \left\{0,1\right\} \text{ for all } 1 \leq i \leq n^2 \plus{} n; \] \[ \text{ (b) } a_{i \plus{} 1} \plus{} a_{i \plus{} 2} \plus{} \ldots \plus{} a_{i \plus{} n} < a_{i \plus{} n \plus{} 1} \plus{} a_{i \plus{} n \plus{} 2} \plus{} \ldots \plus{} a_{i \plus{} 2n} \text{ for all } 0 \leq i \leq n^2 \minus{} n. \] [i]Author: Dusan Dukic, Serbia[/i]

Let $n$ be a positive integer, and consider a sequence $a_1 , a_2 , \dotsc , a_n $ of positive integers. Extend it periodically to an infinite sequence $a_1 , a_2 , \dotsc $ by defining $a_{n+i} = a_i $ for all $i \ge 1$. If \[a_1 \le a_2 \le \dots \le a_n \le a_1 +n \] and \[a_{a_i } \le n+i-1 \quad\text{for}\quad i=1,2,\dotsc, n, \] prove that \[a_1 + \dots +a_n \le n^2. \]

Given $7$ distinct positive integers, prove that there is an infinite arithmetic progression of positive integers $a, a + d, a + 2d,..$ with $a < d$, that contains exactly $3$ or $4$ of the $7$ given integers.

For a positive integer $x$, define a sequence $a_0, a_1, a_2, . . .$ according to the following rules: $a_0 = 1$, $a_1 = x + 1$ and $$a_{n+2} = xa_{n+1} - a_n$$ for all $n \ge 0$. Prove that there exist infinitely many positive integers x such that this sequence does not contain a prime number.

Do there exist two bounded sequences $a_1, a_2,\ldots$ and $b_1, b_2,\ldots$ such that for each positive integers $n$ and $m>n$ at least one of the two inequalities $|a_m-a_n|>1/\sqrt{n},$ and $|b_m-b_n|>1/\sqrt{n}$ holds?

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 $u_n = v_n.$