Let an be a sequence such that $a_0 = 0$ and: $a_{3n+1} = a_{3n} + 1 = a_n + 1$ $a_{3n+2} = a_{3n} + 2 = a_n + 2$ for all natural numbers $n$. How many $n$ less than $2012$ have the property that $a_n = 7$?

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.

For a positive integer $n$, define $f(n)$ to be the number of sequences $(a_1,a_2,\dots,a_k)$ such that $a_1a_2\cdots a_k=n$ where $a_i\geq 2$ and $k\ge 0$ is arbitrary. Also we define $f(1)=1$. Now let $\alpha>1$ be the unique real number satisfying $\zeta(\alpha)=2$, i.e $ \sum_{n=1}^{\infty}\frac{1}{n^\alpha}=2 $ Prove that [list] (a) \[ \sum_{j=1}^{n}f(j)=\mathcal{O}(n^\alpha) \] (b) There is no real number $\beta<\alpha$ such that \[ \sum_{j=1}^{n}f(j)=\mathcal{O}(n^\beta) \] [/list]

The sum of $n$ terms of an arithmetic progression is $153$, and the common difference is $2$. If the first interm is an integer, and $n>1$, then the number of possible values for $n$ is: $ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6 $

Let $A$ be the sequence of zeroes and ones (binary sequence). The sequence can be modified by the following operation: we may pick a block or a contiguous subsequence where there are an unequal number of zeroes and ones, and then flip their order within the block (so block $a_1, a_2, \ldots, a_r$ becomes $a_r, a_{r-1}, \ldots, a_1$). As an example, let $A$ be the sequence $1,1,0,0,1$. We can pick block $1,0,0$ and flip it, so the sequence $1,\boxed{1,0,0},1$ becomes $1,\boxed{0,0,1},1$. However, we cannot pick block $1,1,0,0$ and flip their order since they contain the same number of $1$s and $0$s. Two sequences $A$ and $B$ are called [i]related[/i] if $A$ can be transformed into $B$ using a finite number the operation mentioned above. Determine the largest natural number $n$ for which there exists $n$ different sequences $A_1, A_2, \ldots, A_n$ where each sequence consists of 2022 digits, and for every index $i \neq j$, the sequence $A_i$ is not related to $A_j$.

$a$ and $b$ are natural numbers such that $b > a > 1$, and $a$ does not divide $b$. The sequence of natural numbers $\{b_n\}_{n=1}^\infty$ satisfies $b_{n + 1} \geq 2b_n \forall n \in \mathbb{N}$. Does there exist a sequence $\{a_n\}_{n=1}^\infty$ of natural numbers such that for all $n \in \mathbb{N}$, $a_{n + 1} - a_n \in \{a, b\}$, and for all $m, l \in \mathbb{N}$ ($m$ may be equal to $l$), $a_m + a_l \not\in \{b_n\}_{n=1}^\infty$?

Given are positive integers $r$ and $k$ and an infinite sequence of positive integers $a_1 \le a_2 \le ...$ such that $\frac{r}{a_r}= k + 1$. Prove that there is a $t$ satisfying $\frac{t}{a_t}=k$.

The sequence $ \{x_{n}\}_{n \ge 1}$ is defined by \[ x_{1} \equal{} 2, x_{n \plus{} 1} \equal{} \frac {2 \plus{} x_{n}}{1 \minus{} 2x_{n}}\;\; (n \in \mathbb{N}). \] Prove that a) $ x_{n}\not \equal{} 0$ for all $ n \in \mathbb{N}$, b) $ \{x_{n}\}_{n \ge 1}$ is not periodic.

For each positive integer $n$, let $S(n)$ be the sum of the digits of $n^2 +1$. A sequence $\{a_n\}$ is defined, with $a_0$ an arbitrary positive integer and $a_{n+1} = S(a_n)$. Prove that the sequence $\{a_n\}$ is eventually periodic with period three.

Given a non negative real $a$ and a sequence $(u_n)$ defined by \[ \begin{cases} u_1=3\\ u_{n+1}=\frac{u_n}{2}+\frac{n^2}{4n^2+a}\sqrt{u_n^2+3} \end{cases} \] a) Prove that for $a=0$, the sequence is convergent and find its limit. b) For $a\in [0,1]$, prove that the sequence if convergent.

A sequence $(a_n)$ of real numbers is defined by $a_0=1$, $a_1=2015$ and for all $n\geq1$, we have $$a_{n+1}=\frac{n-1}{n+1}a_n-\frac{n-2}{n^2+n}a_{n-1}.$$ Calculate the value of $\frac{a_1}{a_2}-\frac{a_2}{a_3}+\frac{a_3}{a_4}-\frac{a_4}{a_5}+\ldots+\frac{a_{2013}}{a_{2014}}-\frac{a_{2014}}{a_{2015}}$.

Let $ n \geq 2$ be a positive integer and $ \lambda$ a positive real number. Initially there are $ n$ fleas on a horizontal line, not all at the same point. We define a move as choosing two fleas at some points $ A$ and $ B$, with $ A$ to the left of $ B$, and letting the flea from $ A$ jump over the flea from $ B$ to the point $ C$ so that $ \frac {BC}{AB} \equal{} \lambda$. Determine all values of $ \lambda$ such that, for any point $ M$ on the line and for any initial position of the $ n$ fleas, there exists a sequence of moves that will take them all to the position right of $ M$.

Let $x_0, x_1, x_2, \ldots$ be the sequence defined by $x_i= 2^i$ if $0 \leq i \leq 2003$ $x_i=\sum_{j=1}^{2004} x_{i-j}$ if $i \geq 2004$ Find the greatest $k$ for which the sequence contains $k$ consecutive terms divisible by 2004.

A $7\times 7$ board has a lamp on each of its $49$ squares, which can be on or off. The allowed operation is to choose $3$ consecutive cells of a row or a column that have two lamps neighboring each other on and the other off, and change the state of all three. Namely [img]https://cdn.artofproblemsolving.com/attachments/e/b/28737b19c940ff5e1c98d05533c77069e990f5.png[/img] Give a configuration of exactly $8$ lit lamps located in the first $4$ rows of the board such that, through a succession of permitted operations, a single lamp is lit on the board and that it is located in the last row. Show the sequence of operations used to achieve the goal.

The numbers from $1$ to $2025$ are arranged in some order in the cells of the $1 \times 2025$ strip. Let's call a [i]flip[/i] an operation that takes two arbitrary cells of a strip and swaps the numbers written in them, but only if the larger of these numbers is located to the left of the smaller one. A [i]flop[/i] is a set of several flips that do not contain common cells that are executed simultaneously. (For example, a simultaneous flip between the 2nd and 8th cells and a flip between the 5th and 101st cells.) Prove that there exists a sequence of $66$ flops such that for any initial arrangement, applying this sequence of flops to it will result in the numbers being ordered from left to right in ascending order.

Let $n\ge 2$ be an integer. A sequence $\alpha = (a_1, a_2,..., a_n)$ of $n$ integers is called [i]Lima [/i] if $\gcd \{a_i - a_j \text{ such that } a_i> a_j \text{ and } 1\le i, j\le n\} = 1$, that is, if the greatest common divisor of all the differences $a_i - a_j$ with $a_i> a_j$ is $1$. One operation consists of choosing two elements $a_k$ and $a_{\ell}$ from a sequence, with $k\ne \ell $ , and replacing $a_{\ell}$ by $a'_{\ell} = 2a_k - a_{\ell}$ . Show that, given a collection of $2^n - 1$ Lima sequences, each one formed by $n$ integers, there are two of them, say $\beta$ and $\gamma$, such that it is possible to transform $\beta$ into $\gamma$ through a finite number of operations. Notes. The sequences $(1,2,2,7)$ and $(2,7,2,1)$ have the same elements but are different. If all the elements of a sequence are equal, then that sequence is not Lima.