Let $a_1\in\mathbb{Z}$, $a_2=a_1^2-a_1-1$, $\dots$ ,$a_{n+1}=a_n^2-a_n-1$. Prove that $a_{n+1}$ and $2n+1$ are coprime.

Let $a_1,a_2,...$ be a sequence of positive integers with $a_1=2$. For each $n \ge 1$, $a_{n+1}$ is the biggest prime divisor of $a_1a_2...a_n+1$. Prove that the sequence does not contain numbers $5$ and $11$.

Define the sequence $a_1 = 1, a_2, a_3, ...$ by $$a_{n+1} = a_1^2 + a_2 ^2 + a_3^2 + ... + a_n^2 + n$$ Show that $1$ is the only square in the sequence.

The persons $P_1, P_2, . . . , P_{n-1}, P_n$ sit around a table, in this order, and each one of them has a number of coins. In the start, $P_1$ has one coin more than $P_2, P_2$ has one coin more than $P_3$, etc., up to $P_{n-1}$ who has one coin more than $P_n$. Now $P_1$ gives one coin to $P_2$, who in turn gives two coins to $P_3 $ etc., up to $ Pn$ who gives n coins to $ P_1$. Now the process continues in the same way: $P_1$ gives $n+ 1$ coins to $P_2$, $P_2$ gives $n+2$ coins to $P_3$; in this way the transactions go on until someone has not enough coins, i.e. a person no more can give away one coin more than he just received. At the moment when the process comes to an end in this manner, it turns out that there are two neighbours at the table such that one of them has exactly five times as many coins as the other. Determine the number of persons and the number of coins circulating around the table.

Let $a_{0},a_{1},a_{2},\ldots $ be an increasing sequence of nonnegative integers such that every nonnegative integer can be expressed uniquely in the form $a_{i}+2a_{j}+4a_{k}$, where $i,j$ and $k$ are not necessarily distinct. Determine $a_{1998}$.

A sequence $a_1,a_2,...,a_{2007}$ where $a_i \in\{2,3\}$ for $i = 1,2,...,2007$ and an integer sequence $x_1,x_2,...,x_{2007}$ satisfies the following: $a_ix_i + x_{i+2 }\equiv 0$ ($mod 5$) , where the indices are taken modulo $2007$. Prove that $x_1,x_2,...,x_{2007}$ are all multiples of $5$.

Call a sequence of positive integers $(a_n)_{n \ge 1}$ a "CGMO sequence" if $(a_n)_{n \ge 1}$ strictly increases, and for all integers $n \ge 2022$, $a_n$ is the smallest integer such that there exists a non-empty subset of $\{a_{1}, a_{2}, \cdots, a_{n-1} \}$ $A_n$ where $a_n \cdot \prod\limits_{a \in A_n} a$ is a perfect square. Proof: there exists $c_1, c_2 \in \mathbb{R}^{+}$ s.t. for any "CGMO sequence" $(a_n)_{n \ge 1}$ , there is a positive integer $N$ that satisfies any $n \ge N$, $c_1 \cdot n^2 \le a_n \le c_2 \cdot n^2$.

Determine whether there exists an infinite sequence $a_1, a_2, a_3, \dots$ of positive integers which satisfies the equality \[a_{n+2}=a_{n+1}+\sqrt{a_{n+1}+a_{n}} \] for every positive integer $n$.

Let $k$ be a fixed natural number. In the infinite number of real line, each integer is colored with color ..., red, green, blue, red, green, blue, ... and so on. A number of flea settles at first at integer points. On each turn, a flea will jump over the other tick so that the distance $k$ is the original distance. Formally, we may choose $2$ tails $A, B$ that are spaced $n$ and move $A$ to the different side of $B$ so the current distance is $kn$. Some fleas may occupy the same point because we consider the size of fleas very small. Determine all the values of $k$ so that, whatever the initial position of the ticks, we always get a position where all ticks land on the same color.

Consider infinite sequences $a_1,a_2,\dots$ of positive integers satisfying $a_1=1$ and $$a_n \mid a_k+a_{k+1}+\dots+a_{k+n-1}$$ for all positive integers $k$ and $n.$ For a given positive integer $m,$ find the maximum possible value of $a_{2m}.$ [i]Proposed by Krit Boonsiriseth[/i]

A capricious mathematician writes a book with pages numbered from $2$ to $400$. The pages are to be read in the following order. Take the last unread page ($400$), then read (in the usual order) all pages which are not relatively prime to it and which have not been read before. Repeat until all pages are read. So, the order would be $2, 4, 5, ... , 400, 3, 7, 9, ... , 399, ...$. What is the last page to be read?

$\boxed{A5}$Let $n\in{N},n>2$,and suppose $a_1,a_2,...,a_{2n}$ is a permutation of the numbers $1,2,...,2n$ such that $a_1<a_3<...<a_{2n-1}$ and $a_2>a_4>...>a_{2n}.$Prove that \[(a_1-a_2)^2+(a_3-a_4)^2+...+(a_{2n-1}-a_{2n})^2>n^3\]

Find all sequences of positive integers $a_1,a_2,\dots$ such that $$(n^2+1)a_n = n(a_{n^2}+1)$$ for all positive integers $n$.

Let $(a_n)$ be sequnce of positive integers such that first $k$ members $a_1,a_2,...,a_k$ are distinct positive integers, and for each $n>k$, number $a_n$ is the smallest positive integer that can't be represented as a sum of several (possibly one) of the numbers $a_1,a_2,...,a_{n-1}$. Prove that $a_n=2a_{n-1}$ for all sufficently large $n$.