Let $k$ be a positive integer. Show that if there exists a sequence $a_0,a_1,\ldots$ of integers satisfying the condition \[a_n=\frac{a_{n-1}+n^k}{n}\text{ for all } n\geq 1,\] then $k-2$ is divisible by $3$. [i]Proposed by Okan Tekman, Turkey[/i]

For an integer $n \ge 1$ we consider sequences of $2n$ numbers, each equal to $0, -1$ or $1$. The [i]sum product value[/i] of such a sequence is calculated by first multiplying each pair of numbers from the sequence, and then adding all the results together. For example, if we take $n = 2$ and the sequence $0,1, 1, -1$, then we find the products $0\cdot 1, 0\cdot 1, 0\cdot -1, 1\cdot 1, 1\cdot -1, 1\cdot -1$. Adding these six results gives the sum product value of this sequence: $0+0+0+1+(-1)+(-1) = -1$. The sum product value of this sequence is therefore smaller than the sum product value of the sequence $0, 0, 0, 0$, which equals $0$. Determine for each integer $n \ge 1$ the smallest sum product value that such a sequence of $2n$ numbers could have. [i]Attention: you are required to prove that a smaller sum product value is impossible.[/i]

Let $a_1 <a_2 < \ldots$ be an increasing sequence of positive integers. Let the series $$\sum_{i=1}^{\infty} \frac{1}{a_i }$$ be convergent. For any real number $x$, let $k(x)$ be the number of the $a_i$ which do not exceed $x$. Show that $\lim_{x\to \infty} \frac{k(x)}{x}=0.$

Let $f(x) = \frac{1}{1+x}$ where $x$ is a positive real number, and for any positive integer $n$, let $g_n(x) = x + f(x) + f(f(x)) + ... + f(f(... f(x)))$, the last term being $f$ composed with itself $n$ times. Prove that (i) $g_n(x) > g_n(y)$ if $x > y > 0$. (ii) $g_n(1) = \frac{F_1}{F_2}+\frac{F_2}{F_3}+...+\frac{F_{n+1}}{F_{n+2}}$ , where $F_1 = F_2 = 1$ and $F_{n+2} = F_{n+1} +F_n$ for $n \ge 1$.

Let $ a_0$, $ a_1$, $ a_2$, $ \ldots$ be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, $ \gcd (a_i, a_{i \plus{} 1}) > a_{i \minus{} 1}$. Prove that $ a_n\ge 2^n$ for all $ n\ge 0$. [i]Proposed by Morteza Saghafian, Iran[/i]

Consider all binary sequences of length $n$. In a sequence that allows the interchange of positions of an arbitrary set of $k$ adjacent numbers, ($k < n$), two sequences are said to be [i]equivalent [/i] if they can be transformed from one sequence to another by a finite number of transitions as above. Find the number of sequences that are not equivalent.

A four-term sequence is formed by adding each term of a four-term arithmetic sequence of positive integers to the corresponding term of a four-term geometric sequence of positive integers. The first three terms of the resulting four-term sequence are 57, 60, and 91. What is the fourth term of this sequence? $\textbf{(A) }190\qquad\textbf{(B) }194\qquad\textbf{(C) }198\qquad\textbf{(D) }202\qquad\textbf{(E) }206$

Let $a_1,a_2,\cdots, a_{31} ;b_1,b_2, \cdots, b_{31}$ be positive integers such that $a_1< a_2<\cdots< a_{31}\leq2015$ , $ b_1< b_2<\cdots<b_{31}\leq2015$ and $a_1+a_2+\cdots+a_{31}=b_1+b_2+\cdots+b_{31}.$ Find the maximum value of $S=|a_1-b_1|+|a_2-b_2|+\cdots+|a_{31}-b_{31}|.$

Find the formula for the general term of the sequence an, for which $a_1 = 1$, $a_2 = 3$, $a_{n+1} = 3a_n-2a_{n-1}$ (you need to express an in terms of $n$).

Let $a > 1$ be a positive integer and $d > 1$ be a positive integer coprime to $a$. Let $x_1=1$, and for $k\geq 1$, define $$x_{k+1} = \begin{cases} x_k + d &\text{if } a \text{ does not divide } x_k \\ x_k/a & \text{if } a \text{ divides } x_k \end{cases}$$ Find, in terms of $a$ and $d$, the greatest positive integer $n$ for which there exists an index $k$ such that $x_k$ is divisible by $a^n$.

The sequence ($a_n$) is defined by $a_1 = a_2 = 1$ and $a_{n+2 }= a_{n+1} +a_n +k$, where $k$ is a positive integer. Find the least $k$ for which $a_{1991}$ and $1991$ are not coprime.

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?

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.

Define a sequence $<x_n>$ by $x_0 = 0$ and $$\large x_n = \left\{ \begin{array}{ll} x_{n-1} + \frac{3^r-1}{2} & if \,\,n = 3^{r-1}(3k + 1)\\ & \\ x_{n-1} - \frac{3^r+1}{2} & if \,\, n = 3^{r-1}(3k + 2)\\ \end{array} \right. $$ where $k, r$ are integers. Prove that every integer occurs exactly once in the sequence.

Let $a\in\left[ \tfrac{1}{2},\ \tfrac{3}{2}\right]$ be a real number. Sequences $(u_n),\ (v_n)$ are defined as follows: $$u_n=\frac{3}{2^{n+1}}\cdot (-1)^{\lfloor2^{n+1}a\rfloor},\ v_n=\frac{3}{2^{n+1}}\cdot (-1)^{n+\lfloor 2^{n+1}a\rfloor}.$$ a. Prove that $${{({{u}_{0}}+{{u}_{1}}+\cdots +{{u}_{2018}})}^{2}}+{{({{v}_{0}}+{{v}_{1}}+\cdots +{{v}_{2018}})}^{2}}\le 72{{a}^{2}}-48a+10+\frac{2}{{{4}^{2019}}}.$$ b. Find all values of $a$ in the equality case.

The sequence $a_1,a_2, ..., a_{2n}$ of integers is such that each number occurs in no more than $n$ times. Prove that there are two strictly increasing sequences of indices $b_1,b_2, ..., b_{n}$ and $c_1,c_2, ..., c_{n}$ are such that every positive integer from the set $\{1,2,...,2n\}$ occurs exactly in one of these two sequences, and for each $1\le i \le n$ is true the condition $a_{b_i} \ne a_{c_i}$ . (Anton Trygub)

We define a sequence of positive integers $a_1,a_2,a_3,\dots$ as follows: Let $a_1=1$ and iteratively, for $k =2,3,\dots$ let $a_k$ be the largest prime factor of $1+a_1a_2\cdots a_{k-1}$. Show that the number $11$ is not an element of this sequence.

Real numbers $ a_{1}$, $ a_{2}$, $ \ldots$, $ a_{n}$ are given. For each $ i$, $ (1 \leq i \leq n )$, define \[ d_{i} \equal{} \max \{ a_{j}\mid 1 \leq j \leq i \} \minus{} \min \{ a_{j}\mid i \leq j \leq n \} \] and let $ d \equal{} \max \{d_{i}\mid 1 \leq i \leq n \}$. (a) Prove that, for any real numbers $ x_{1}\leq x_{2}\leq \cdots \leq x_{n}$, \[ \max \{ |x_{i} \minus{} a_{i}| \mid 1 \leq i \leq n \}\geq \frac {d}{2}. \quad \quad (*) \] (b) Show that there are real numbers $ x_{1}\leq x_{2}\leq \cdots \leq x_{n}$ such that the equality holds in (*). [i]Author: Michael Albert, New Zealand[/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.

Given a real number $a$ and a sequence $(x_n)_{n=1}^\infty$ defined by: $$\left\{\begin{matrix} x_1=1 \\ x_2=0 \\ x_{n+2}=\frac{x_n^2+x_{n+1}^2}{4}+a\end{matrix}\right.$$ for all positive integers $n$. 1. For $a=0$, prove that $(x_n)$ converges. 2. Determine the largest possible value of $a$ such that $(x_n)$ converges.

Let $B$ be a set of $k$ sequences each having $n$ terms equal to $1$ or $-1$. The product of two such sequences $(a_1, a_2, \ldots , a_n)$ and $(b_1, b_2, \ldots , b_n)$ is defined as $(a_1b_1, a_2b_2, \ldots , a_nb_n)$. Prove that there exists a sequence $(c_1, c_2, \ldots , c_n)$ such that the intersection of $B$ and the set containing all sequences from $B$ multiplied by $(c_1, c_2, \ldots , c_n)$ contains at most $\frac{k^2}{2^n}$ sequences.

Find all the real number sequences $\{b_n\}_{n \geq 1}$ and $\{c_n\}_{n \geq 1}$ that satisfy the following conditions: (i) For any positive integer $n$, $b_n \leq c_n$; (ii) For any positive integer $n$, $b_{n+1}$ and $c_{n+1}$ is the two roots of the equation $x^2+b_nx+c_n=0$.

Let $(a_n)_n\geq 0$ and $a_{m+n}+a_{m-n}=\frac{1}{2}(a_{2m}+a_{2n})$ for every $m\geq n\geq0.$ If $a_1=1,$ then find the value of $a_{2007}.$

The sequence $x_{n}$ is defined by: $x_{0}=1, x_{1}=0, x_{2}=1,x_{3}=1, x_{n+3}=\frac{(n^2+n+1)(n+1)}{n}x_{n+2}+(n^2+n+1)x_{n+1}-\frac{n+1}{n}x_{n} (n=1,2,3..)$ Prove that all members of the sequence are perfect squares.

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