Prove that exists sequences $ (a_n)_{n\ge 0}$ with $ a_n\in \{\minus{}1,\plus{}1\}$, for any $ n\in \mathbb{N}$, such that: \[ \lim_{n\rightarrow \infty}\left(\sqrt{n\plus{}a_1}\plus{}\sqrt{n\plus{}a_2}\plus{}...\plus{}\sqrt{n\plus{}a_n}\minus{}n\sqrt{n\plus{}a_0}\right)\equal{}\frac{1}{2}\]

Let $P(x, y)$ and $Q(x, y)$ be polynomials in two variables with integer coefficients. The sequences of integers $a_0, a_1,\ldots$ and $b_0, b_1,\ldots$ satisfy \[a_{n+1}=P(a_n,b_n),\quad b_{n+1}=Q(a_n,b_n)\]for all $n\geqslant 0$. Let $m_n$ be the number of integer points of the coordinate plane, lying strictly inside the segment with endpoints $(a_n,b_n)$ and $(a_{n+1},b_{n+1})$. Prove that the sequence $m_0,m_1,\ldots$ is non-decreasing.

Let an odd prime $p$ be a given number satisfying $2^h \neq 1 \pmod{p}$ for all $h < p-1, h \in \mathbb{N}^{*},$ and an even integer $a \in \left(\frac{p}{2},p \right).$ Let us consider the sequence $\{a_n\}^{\infty}_{n=0}$ defined by $a_0 = a$ and $a_{n+1} = p - b_n$ for $n = 0, 1, 2, \ldots$, where $b_n$ is the greatest odd divisor of $a_n.$ Show that $\{a_n\}$ is periodical and find its least positive period.

Raina the frog is playing a game in a circular pond with six lilypads around its perimeter numbered clockwise from $1$ to $6$ (so that pad $1$ is adjacent to pad $6$). She starts at pad $1$, and when she is on pad i, she may jump to one of its two adjacent pads, or any pad labeled with $j$ for which $j - i$ is even. How many jump sequences enable Raina to hop to each pad exactly once?

Let $ a$, $ b$ and $ k$ be three positive integers. We define two sequences $ \left( a_{n}\right)$ and $ \left( b_{n}\right)$ by the starting values $ a_{1}\equal{}a$ and $ b_{1}\equal{}b$ and the recurrent equations $ a_{n\plus{}1}\equal{}ka_{n}\plus{}b_{n}$ and $ b_{n\plus{}1}\equal{}kb_{n}\plus{}a_{n}$ for each positive integer $ n$. Prove that if $ a_{1}\perp b_{1}$, $ a_{2}\perp b_{2}$ and $ a_{3}\perp b_{3}$ hold, then $ a_{n}\perp b_{n}$ holds for every positive integer $ n$. Here, the abbreviation $ x\perp y$ stands for "the numbers $ x$ and $ y$ are coprime".

For a positive integer $n$, let $S_n$ be the set of positive integers that do not exceed $n$ and are coprime to $n$. Define $f(n)$ as the smallest positive integer that allows $S_n$ to be partitioned into $f(n)$ disjoint subsets, each forming an arithmetic progression. Prove that there exist infinitely many pairs $(a, b)$ satisfying $a, b > 2025$, $a \mid b$, and $f(a) \nmid f(b)$.

Let $r$, $s$ be real numbers, find maximum $t$ so that if $a_1, a_2, \ldots$ is a sequence of positive real numbers satisfying \[ a_1^r + a_2^r + \cdots + a_n^r \le 2023 \cdot n^t \] for all $n \ge 2023$ then the sum \[ b_n = \frac 1{a_1^s} + \cdots + \frac 1{a_n^s} \] is unbounded, i.e for all positive reals $M$ there is an $n$ such that $b_n > M$.

A set $S$ consists of $k$ sequences of $0,1,2$ of length $n$. For any two sequences $(a_i),(b_i)\in S$ we can construct a new sequence $(c_i)$ such that $c_i=\left\lfloor\frac{a_i+b_i+1}2\right\rfloor$ and include it in $S$. Assume that after performing finitely many such operations we obtain all the $3n$ sequences of $0,1,2$ of length $n$. Find the least possible value of $k$.

Let $x_{0}$, $x_{1}$, $x_{2}$, $\cdots$ be a sequence of numbers, where each $x_{k}$ is either $0$ or $1$. For each positive integer $n$, define \[S_{n} = \displaystyle\sum^{n-1}_{k=0}{x_{k}2^{k}}\] Suppose $7S_{n} \equiv 1\pmod {2^{n}}$ for all $n\geq 1$. What is the value of the sum \[x_{2019}+2x_{2020}+4x_{2021}+8x_{2022}?\] $ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 15$

Let $x_1$ be a given positive integer. A sequence $\{x_n\}_ {n\geq 1}$ of positive integers is such that $x_n$, for $n \geq 2$, is obtained from $x_ {n-1}$ by adding some nonzero digit of $x_ {n-1}$. Prove that a) the sequence contains an even term; b) the sequence contains infinitely many even terms.

The number of apples growing on each of six apple trees form an arithmetic sequence where the greatest number of apples growing on any of the six trees is double the least number of apples growing on any of the six trees. The total number of apples growing on all six trees is $990$. Find the greatest number of apples growing on any of the six trees.

Consider the sequence $ (x_n)_{n\ge 0}$ where $ x_n\equal{}2^{n}\minus{}1\ ,\ n\in \mathbb{N}$. Determine all the natural numbers $ p$ for which: \[ s_p\equal{}x_0\plus{}x_1\plus{}x_2\plus{}...\plus{}x_p\] is a power with natural exponent of $ 2$.

Chameleons of five colors live on the island. When one chameleon bites another, the color of bitten chameleon changes to one of these five colors according to some rule, and the new color depends only on the color of the bitten and the color of the bitting. It is known that $2023$ red chameleons can agree on a sequence of bites between themselves, after which they will all turn blue. What is the smallest $k$ that can guarantee that $k$ red chameleons, biting only each other, can turn blue? (For example, the rules might be: if a red chameleon bites a green one, the bitten one changes color to blue; if a green one bites a red one, the bitten one remains red, that is, "changes color to red"; if red bites red, the bitten one changes color to yellow, etc. The rules for changing colors may be different.)

By $rad(x)$ we denote the product of all distinct prime factors of a positive integer $n$. Given $a \in N$, a sequence $(a_n)$ is defined by $a_0 = a$ and $a_{n+1} = a_n+rad(a_n)$ for all $n \ge 0$. Prove that there exists an index $n$ for which $\frac{a_n}{rad(a_n)} = 2022$

Let the [i]square decomposition[/i] of a number be defined as the sequence of numbers given by the following algorithm. Given a positive integer $n$, add the largest possible perfect square that is less than or equal to $n$ to a sequence, and then subtract that number from $n$. Repeat as many times as necessary until your current $n$ is $0$. So for example, the square decomposition of $60$ would be $49, 9, 1, 1$. Define the size of a square decomposition to be the number of numbers in the sequence. Say that the maximal size of a square decomposition of a number in the range $[1, 2020]$ is $m$. Find the largest number in the range $[1, 2020]$ that has a square decomposition of size $m$. [i]Proposed by Timothy Qian[/i]

Given $a_0 > 1$, the sequence $a_0, a_1, a_2, ...$ is such that for all $k > 0$, $a_k$ is the smallest integer greater than $a_{k-1}$ which is relatively prime to all the earlier terms in the sequence. Find all $a_0$ for which all terms of the sequence are primes or prime powers.