[b]7.[/b] Let $(z_n)_{n=1}^{\infty}$ be a sequence of complex numbers tending to zero. Prove that there exists a sequence $(\epsilon_n)_{n=1}^{\infty}$ (where $\epsilon_n = +1$ or $-1$) such that the series $\sum_{n=1}^{\infty} \epsilon_n z_n$ is convergente. [b](F. 9)[/b]

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]

Let $a_1,a_2,\dots$ be a sequence of integers satisfying $a_1=2$ and: $$a_n=\begin{cases}a_{n-1}+1, & \text{ if }n\ne a_k \text{ for some }k=1,2,\dots,n-1; \\ a_{n-1}+2, & \text{ if } n=a_k \text{ for some }k=1,2,\dots,n-1. \end{cases}$$ Find the value of $a_{2022!}$.

Each of the integers $a_1,a_2,a_3,\dots,a_{2008}$ is at least $1$ and at most $5$. If $a_n < a_{n+1}$, the pair $(a_n, a_{n+1})$ will be called as an increasing pair. If $a_n > a_{n+1}$, the pair $(a_n, a_{n+1})$ will be called as an decreasing pair. If the sequence contains $103$ increasing pairs, at least how many decreasing pairs are there? $ \textbf{(A)}\ 21 \qquad\textbf{(B)}\ 24 \qquad\textbf{(C)}\ 36 \qquad\textbf{(D)}\ 102 \qquad\textbf{(E)}\ \text{None of the above} $

We define the following operation which will be applied to a row of bars being situated side-by-side on positions $ 1,2,...,N$. Each bar situated at an odd numbered position is left as is, while each bar at an even numbered position is replaced by two bars. After that, all bars will be put side-by-side in such a way that all bars form a new row and are situated on positions $ 1,...,M.$ From an initial number $ a_0>0$ of bars there originates a sequence $ (a_n)_{n \ge 0},$ where $ a_n$ is the number of bars after having applied the operation $ n$ times. $ (a)$ Prove that for no $ n>0$ can we have $ a_n\equal{}1997.$ $ (b)$ Determine all natural numbers that can only occur as $ a_0$ or $ a_1$.

The sequence $(a_n)$ is such that $a_{n+1} = (a_n)^n + n + 1$ for all positive integers $n$, where $a_1$ is some positive integer. Let $k$ be the greatest power of $3$ by which $a_{101}$ is divisible. Find all possible values of $k$. [i]Proposed by Kyrylo Holodnov[/i]

A sequence of real numbers $x_1,x_2,\ldots ,x_n$ is given such that $x_{i+1}=x_i+\frac{1}{30000}\sqrt{1-x_i^2},\ i=1,2,\ldots ,$ and $x_1=0$. Can $n$ be equal to $50000$ if $x_n<1$?

In a given sequence $\{S_1,S_2,...,S_k\}$, for terms $n\ge3$, $S_n=\sum_{i=1}^{n-1} i\cdot S_{n-i}$. For example, if the first two elements are 2 and 3, respectively, the third entry would be $1\cdot3+2\cdot2=7$, and the fourth would be $1\cdot7+2\cdot3+3\cdot2=19$, and so on. Given that a sequence of integers having this form starts with 2, and the 7th element is 68, what is the second element?

Let be two continuous functions $ f:[0,\infty )\longrightarrow\mathbb{R} $ satisfying the following equations: $$ \lim_{x\to\infty } f(x) =\infty =\lim_{x\to\infty } g(x) $$ Prove that there exists a divergent sequence $ \left( k_n \right)_{n\ge 1} $ of nonnegative integers which has the property that each term (function) of the sequence of functions $ \left( h_{n} \right)_{n\ge 1} :[0,\infty )\longrightarrow\mathbb{R} $ defined as $$ h_{n} (x) =f\left( k_n+g(x) -\left\lfloor g(x) \right\rfloor \right) , $$ doesn't have limit at $ \infty . $ [i]Romeo Ilie[/i]

Let $a_1,a_2,a_3,\ldots,a_{250}$ be real numbers such that $a_1=2$ and $$a_{n+1}=a_n+\frac{1}{a_n^2}$$ for every $n=1,2, \ldots, 249$. Let $x$ be the greatest integer which is less than $$\frac{1}{a_1}+\frac{1}{a_2}+\ldots+\frac{1}{a_{250}}$$ How many digits does $x$ have? [i]Proposed by Miroslav Marinov, Bulgaria[/i]

For a given real number $a$ and a positive integer $n$, prove that: i) there exists exactly one sequence of real numbers $x_0,x_1,\ldots,x_n,x_{n+1}$ such that \[\begin{cases} x_0=x_{n+1}=0,\\ \frac{1}{2}(x_i+x_{i+1})=x_i+x_i^3-a^3,\ i=1,2,\ldots,n.\end{cases}\] ii) the sequence $x_0,x_1,\ldots,x_n,x_{n+1}$ in i) satisfies $|x_i|\le |a|$ where $i=0,1,\ldots,n+1$. [i]Liang Yengde[/i]

Let $x$, $y$, and $z$ be nonnegative integers that are less than or equal to 100. Suppose that $x + y + z$, $xy + z$, $x + yz$, and $xyz$ are (in some order) four consecutive terms of an arithmetic sequence. Compute the number of such ordered triples $(x, y, z)$.

A sequence $a_1,a_2,\cdots$ of positive integers contains each positive integer exactly once. Moreover for every pair of distinct positive integer $m$ and $n$, $\frac{1}{1998} < \frac{|a_n- a_m|}{|n-m|} < 1998$, show that $|a_n - n | <2000000$ for all $n$.

Let ${k}$ be a fixed positive integer. A finite sequence of integers ${x_1,x_2, ..., x_n}$ is written on a blackboard. Pepa and Geoff are playing a game that proceeds in rounds as follows. - In each round, Pepa first partitions the sequence that is currently on the blackboard into two or more contiguous subsequences (that is, consisting of numbers appearing consecutively). However, if the number of these subsequences is larger than ${2}$, then the sum of numbers in each of them has to be divisible by ${k}$. - Then Geoff selects one of the subsequences that Pepa has formed and wipes all the other subsequences from the blackboard. The game finishes once there is only one number left on the board. Prove that Pepa may choose his moves so that independently of the moves of Geoff, the game finishes after at most ${3k}$ rounds. (Poland)

Let $\operatorname{rad}(k)$ denote the product of prime divisors of a natural number $k$ (define $\operatorname{rad}(1)=1$). A sequence $(a_n)$ is defined by setting $a_1$ arbitrarily, and $a_{n+1}=a_n+\operatorname{rad}(a_n)$ for $n\ge1$. Prove that the sequence $(a_n)$ contains arithmetic progressions of arbitrary length.

Consider the sequence $(3^{2^n} + 1)_{n\geq 1}$. i) Prove there exist infinitely many primes, none dividing any term of the sequence. ii) Prove there exist infinitely many primes, each dividing some term of the sequence. [i](Dan Schwarz)[/i]