A sequence $a_0, a_1, \dots , a_n, \dots$ of positive integers is constructed as follows: [list] [*] If the last digit of $a_n$ is less than or equal to $5$, then this digit is deleted and $a_{n+1}$ is the number consisting of the remaining digits. (If $a_{n+1}$ contains no digits, the process stops.) [*] Otherwise, $a_{n+1}= 9a_n$. [/list] Can one choose $a_0$ so that this sequence is infinite?

Let $a_1, a_2, a_3, \ldots$ be a sequence of positive real numbers, and $s$ be a positive integer, such that \[a_n = \max \{ a_k + a_{n-k} \mid 1 \leq k \leq n-1 \} \ \textrm{ for all } \ n > s.\] Prove there exist positive integers $\ell \leq s$ and $N$, such that \[a_n = a_{\ell} + a_{n - \ell} \ \textrm{ for all } \ n \geq N.\] [i]Proposed by Morteza Saghafiyan, Iran[/i]

Consider the sequence of integer such that: $ a_1 = 2$ $ a_2 = 5$ $ a_{n + 1} = (2 - n^2)a_n + (2 + n^2)a_{n - 1}, \forall n\ge 2$ Find all triplies $ (x,y,z) \in \mathbb{N}^3$ such that $ a_xa_y = a_z$.

It is given sequence wih length of $2017$ which consists of first $2017$ positive integers in arbitrary order (every number occus exactly once). Let us consider a first term from sequence, let it be $k$. From given sequence we form a new sequence of length 2017, such that first $k$ elements of new sequence are same as first $k$ elements of original sequence, but in reverse order while other elements stay unchanged. Prove that if we continue transforming a sequence, eventually we will have sequence with first element $1$.

Determine all sequences $(x_1,x_2,\ldots,x_{2011})$ of positive integers, such that for every positive integer $n$ there exists an integer $a$ with \[\sum^{2011}_{j=1} j x^n_j = a^{n+1} + 1\] [i]Proposed by Warut Suksompong, Thailand[/i]

Let $a_1,a_2,a_3,\ldots$ be an infinite sequence of positive integers such that $a_{n+2m}$ divides $a_{n}+a_{n+m}$ for all positive integers $n$ and $m.$ Prove that this sequence is eventually periodic, i.e. there exist positive integers $N$ and $d$ such that $a_n=a_{n+d}$ for all $n>N.$

Let $a_0 = 1998$ and $a_{n+1} =\frac{a_n^2}{a_n +1}$ for each nonnegative integer $n$. Prove that $[a_n] = 1994- n$ for $0 \le n \le 1000$

Prove that we can find an infinite set of positive integers of the from $2^n-3$ (where $n$ is a positive integer) every pair of which are relatively prime.

Find the number of sequences $a_1,a_2,\dots,a_{100}$ such that $\text{(i)}$ There exists $i\in\{1,2,\dots,100\}$ such that $a_i=3$, and $\text{(ii)}$ $|a_i-a_{i+1}|\leq 1$ for all $1\leq i<100$.

The sequences $\{u_{n}\}$ and $\{v_{n}\}$ are defined by $u_{0} =u_{1} =1$ ,$u_{n}=2u_{n-1}-3u_{n-2}$ $(n\geq2)$ , $v_{0} =a, v_{1} =b , v_{2}=c$ ,$v_{n}=v_{n-1}-3v_{n-2}+27v_{n-3}$ $(n\geq3)$. There exists a positive integer $N$ such that when $n> N$, we have $u_{n}\mid v_{n}$ . Prove that $3a=2b+c$.

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 $\gcd (m,a_n)=1 \ , \ \forall n\geq 0$.

The sequence $\{x_n\}$ is given by $$x_n=\frac14\left(\left(2+\sqrt3\right)^{2n-1}+\left(2-\sqrt3\right)^{2n-1}\right),\qquad n\in\mathbb N.$$Prove that each $x_n$ is equal to the sum of squares of two consecutive integers.

Define a sequence: $x_0=1$ and for all $n\ge 0$, $x_{2n+1}=x_{n}$ and $x_{2n+2}=x_{n}+x_{n+1}$. Prove that for any relatively prime positive integers $a$ and $b$, there is a non-negative integer $n$ such that $a=x_n$ and $b=x_{n+1}$.

Let $c \ge 1$ be an integer. Define a sequence of positive integers by $a_1 = c$ and \[a_{n+1}=a_n^3-4c\cdot a_n^2+5c^2\cdot a_n+c\] for all $n\ge 1$. Prove that for each integer $n \ge 2$ there exists a prime number $p$ dividing $a_n$ but none of the numbers $a_1 , \ldots , a_{n -1}$ . [i]Proposed by Austria[/i]

Suppose that primes $a_1, a_2, . . . , a_p$ form an increasing arithmetic progression and $a_1 > p$. Prove that if $p$ is a prime, then the difference of the progression is divisible by $p$.

The sequence $c_{0}, c_{1}, . . . , c_{n}, . . .$ is defined by $c_{0}= 1, c_{1}= 0$, and $c_{n+2}= c_{n+1}+c_{n}$ for $n \geq 0$. Consider the set $S$ of ordered pairs $(x, y)$ for which there is a finite set $J$ of positive integers such that $x=\textstyle\sum_{j \in J}{c_{j}}$, $y=\textstyle\sum_{j \in J}{c_{j-1}}$. Prove that there exist real numbers $\alpha$, $\beta$, and $M$ with the following property: An ordered pair of nonnegative integers $(x, y)$ satisfies the inequality \[m < \alpha x+\beta y < M\] if and only if $(x, y) \in S$. [i]Remark:[/i] A sum over the elements of the empty set is assumed to be $0$.

Alice and Bob play a game. Bob starts by picking a set $S$ consisting of $M$ vectors of length $n$ with entries either $0$ or $1$. Alice picks a sequence of numbers $y_1\le y_2\le\dots\le y_n$ from the interval $[0,1]$, and a choice of real numbers $x_1,x_2\dots,x_n\in \mathbb{R}$. Bob wins if he can pick a vector $(z_1,z_2,\dots,z_n)\in S$ such that $$\sum_{i=1}^n x_iy_i\le \sum_{i=1}^n x_iz_i,$$otherwise Alice wins. Determine the minimum value of $M$ so that Bob can guarantee a win. [i]Proposed by DVDthe1st[/i]

Define a sequence $(a_n)$ by $a_1 =1, a_2 =2,$ and $a_{k+2}=2a_{k+1}+a_k$ for all positive integers $k$. Determine all real numbers $\beta >0$ which satisfy the following conditions: (A) There are infinitely pairs of positive integers $(p,q)$ such that $\left| \frac{p}{q}- \sqrt{2} \, \right| < \frac{\beta}{q^2 }.$ (B) There are only finitely many pairs of positive integers $(p,q)$ with $\left| \frac{p}{q}- \sqrt{2} \,\right| < \frac{\beta}{q^2 }$ for which there is no index $k$ with $q=a_k.$

The sequences $(a_n)$ and $(b_n)$ are defined by $a_1 = b_1 = 1$ and $a_{n+1} = a_n +b_n, b_{n+1} = a_nb_n$ for $n = 1,2,...$ Show that every two distinct terms of the sequence $(a_n)$ are coprime

Let $\{fn\}$ be the Fibonacci sequence $\{1, 1, 2, 3, 5, \dots.\}. $ (a) Find all pairs $(a, b)$ of real numbers such that for each $n$, $af_n +bf_{n+1}$ is a member of the sequence. (b) Find all pairs $(u, v)$ of positive real numbers such that for each $n$, $uf_n^2 +vf_{n+1}^2$ is a member of the sequence.

A sequence of $N$ consecutive positive integers is called [i]good [/i] if it is possible to choose two of these numbers so that their product is divisible by the sum of the other $N-2$ numbers. For which $N$ do there exist infinitely many [i]good [/i] sequences?

Let be a $ \beta >1. $ Calculate $ \lim_{n\to\infty} \frac{k(n)}{n} ,$ where $ k(n) $ is the smallest natural number that satisfies the inequality $ (1+n)^k\ge n^k\beta . $ [i]Neculai Hârţan[/i]

Let $m, n$ be positive integers. Consider a sequence of positive integers $a_1, a_2, ... , a_n$ that satisfies $m = a_1 \ge a_2\ge ... \ge a_n \ge 1$. Then define, for $1\le  i\le  m$, $b_i =$ # $\{ j \in \{1, 2, ... , n\}: a_j \ge i\}$, so $b_i$ is the number of terms $a_j $ of the given sequence for which $a_j  \ge i$. Similarly, we define, for $1\le   j \le  n$, $c_j=$ # $\{ i \in \{1, 2, ... , m\}: b_i \ge j\}$ , thus $c_j$ is the number of terms bi in the given sequence for which $b_i \ge j$. E.g.: If $a$ is the sequence $5, 3, 3, 2, 1, 1$ then $b$ is the sequence $6, 4, 3, 1, 1$. (a) Prove that $a_j = c_j $ for $1  \le j  \le n$. (b) Prove that for $1\le  k \le m$: $\sum_{i=1}^{k} b_i = k \cdot b_k + \sum_{j=b_{k+1}}^{n} a_j$.

Consider the sequence $(a_n)_{n\ge 1}$ such that $a_1=1$ and $a_{n+1}=\sqrt{a_n+n^2}$, $\forall n\ge 1$. $\textbf{(a)}$ Prove that there is exactly one rational number among the numbers $a_1,a_2,a_3,\dots$. $\textbf{(b)}$ Consider the sequence $(S_n)_{n\ge 1}$ such that $$S_n=\sum_{i=1}^n\frac{4}{\left (\left \lfloor a_{i+1}^2\right \rfloor-\left \lfloor a_i^2\right \rfloor\right)\left(\left \lfloor a_{i+2}^2\right \rfloor-\left \lfloor a_{i+1}^2\right \rfloor\right)}.$$ Prove that there exists an integer $N$ such that $S_n>0.9$, $\forall n>N$. [i] (Stefan Obadă)[/i]

Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \cdots$ defined by \[ a_0 = M + \frac{1}{2} \qquad \textrm{and} \qquad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots \] contains at least one integer term.