Found problems: 5923
Consider a sequence of words each consisting of two letters, $A$ and $B$ . The first word is "$A$" , while the second word is "$B$" . The $k$-th word is obtained from the ($k -2$)-nd by writing after it the ($k -1$)th one. (So the first few elements of the sequence are "$A$" , "$B$" ,"$AB$" , "$BAB$" , "$ABBAB$" . ) Does there exist in this sequence a "periodical" word, i.e. a word of the form $P P P ... P$ , where $P$ is a word , repeated at least once?
(Remark: For instance, the word $BABBBABB$ is of the form $PP$ , in which $P$ is repeated exactly once . )
(A. Andjans, Riga)
Let $a>0$ be a real number. Find the minimal constant $C_a$ for which the inequality$$\displaystyle
C_a\sum_{k=1}^n \frac1{x_k-x_{k-1}} >\sum_{k=1}^n \frac{k+a}{x_k}$$holds for any positive integer $n$ and any sequence $0=x_0<x_1<\cdots <x_n$ of real numbers.
Let be two real numbers $ \alpha ,\beta $ and two sequences $ \left(x_n \right)_{n\ge 1} ,\left(y_n \right)_{n\ge 1} $ whose smallest periods are $ p,q, $ respectively. Prove that the sequence $ \left( \alpha x_n+\beta y_n\right)_{n\ge 1} $ is periodic if $ \text{gcd}^2 (p,q) | \text{lcm} (p,q) , $ and in this case find its smallest period.
How few numbers is it possible to cross out from the sequence
$$1, 2,3,..., 2023$$
so that among those left no number is the product of any two (distinct) other numbers?
The sequence $x_0, x_1, x_2, ...$ is defined by $x_0 = 0$, $x_{k+1} = [(n - \sum_0^k x_i)/2]$. Show that $x_k = 0$ for all sufficiently large $k$ and that the sum of the non-zero terms $x_k$ is $n-1$.
Denote by $S_n$ the group of permutations of the sequence $(1,2,\dots,n).$ Suppose that $G$ is a subgroup of $S_n,$ such that for every $\pi\in G\setminus\{e\}$ there exists a unique $k\in \{1,2,\dots,n\}$ for which $\pi(k)=k.$ (Here $e$ is the unit element of the group $S_n.$) Show that this $k$ is the same for all $\pi \in G\setminus \{e\}.$
Let be a number $ a\in \left[ 1,\infty \right) $ and a function $ f\in\mathcal{C}^2(-a,a) . $ Show that the sequence
$$ \left( \sum_{k=1}^n f\left( \frac{k}{n^2} \right) \right)_{n\ge 1} $$
is convergent, and determine its limit.
Let $k>1$ be an integer. The sequence $a_1,a_2, \cdots$ is defined as:
$a_1=1, a_2=k$ and for all $n>1$ we have: $a_{n+1}-(k+1)a_n+a_{n-1}=0$
Find all positive integers $n$ such that $a_n$ is a power of $k$.
[i]Proposed by Amirhossein Pooya[/i]
Let $0<\alpha<1$ be a fixed number. On a lake shaped like a convex polygon, at some point there is a duck and at another point a water lily grows. If the duck is at point $X{}$, then in one move it can swim towards one of the vertices $Y$ of the polygon a distance equal to a $\alpha\cdot XY$. Find all $\alpha{}$ for which, regardless of the shape of the lake and the initial positions of the duck and the lily, after a sequence of adequate moves, the distance between the duck and the lily will be at most one meter.
Let $a_1 = 1, a_2, a_3,...$: be a sequence of positive integers such that $$a_k < 1 + a_1 + a_2 +... + a_{k-1}$$ for all $k > 1$. Prove that every positive integer can be expressed as a sum of $a_i$s.
Let $\mathcal{S}$ be a set consisting of $n \ge 3$ positive integers, none of which is a sum of two other distinct members of $\mathcal{S}$. Prove that the elements of $\mathcal{S}$ may be ordered as $a_1, a_2, \dots, a_n$ so that $a_i$ does not divide $a_{i - 1} + a_{i + 1}$ for all $i = 2, 3, \dots, n - 1$.
Let $d_i$ be the first decimal digit of $2^i$ for every non-negative integer $i$. Prove that for each positive integer $n$ there exists a decimal digit other than $0$ which can be found in the sequence $d_0, d_1, \dots, d_{n-1}$ strictly less than $\frac{n}{17}$ times.
Consider the following sequence of sets: $ \{ 1,2\} ,\{ 3,4,5\}, \{ 6,7,8,9\} ,... $
[b]a)[/b] Find the samllest element of the $ 100\text{-th} $ term.
[b]b)[/b] Is $ 2015 $ the largest element of one of these sets?
Prove that from any sequence of $1996$ real numbers $a_1$, $a_2$,$...$, $a_{1996}$ one can choose one or several numbers standing successively one after another so that their sum differs from an integer by less than $0.001$.
(A Kanel)
An arithmetic progression is formed by $9$ positive integers such that the product of these $9$ terms is a multiple of $3$. Prove that said product is also multiple of $81$.
Do there exist a sequence $a_{1}, a_{2}, a_{3}, \ldots$ of real numbers and a non-constant polynomial $P(x)$ such that $a_{m}+a_{n}=P(mn)$ for every positive integral $m$ and $n?$
[i]Proposed by A. Golovanov[/i]
$n$ is a given positive integer, such that it’s possible to weigh out the mass of any product weighing $1,2,3,\cdots ,ng$ with a counter balance without sliding poise and $k$ counterweights, which weigh $x_ig(i=1,2,\cdots ,k),$ respectively, where $x_i\in \mathbb{N}^*$ for any $i \in \{ 1,2,\cdots ,k\}$ and $x_1\leq x_2\leq\cdots \leq x_k.$
$(1)$Let $f(n)$ be the least possible number of $k$. Find $f(n)$ in terms of $n.$
$(2)$Find all possible number of $n,$ such that sequence $x_1,x_2,\cdots ,x_{f(n)}$ is uniquely determined.
Find constants $A > B$ such that $\frac{f\left( \frac{1}{1+2x}\right) }{f(x)}$ is independent of $x$, where $f(x) = \frac{1 + Ax}{1 + Bx}$ for all real $x \ne - \frac{1}{B}$. Put $a_0 = 1$, $a_{n+1} = \frac{1}{1 + 2a_n}$. Find an expression for an by considering $f(a_0), f(a_1), ...$.
We call a sequence of length $n$ of zeros and ones a "string of length $n$" and the elements of the same sequence "bits". Let $m,n$ be two positive integers so that $m<2^n$. Arik holds $m$ strings of length $n$. Giora wants to find a new string of length $n$ different from all those Arik holds. For this Giora may ask Arik questions of the form:
"What is the value of bit number $i$ in string number $j$?"
where $1\leq i\leq n$ and $1\leq j\leq m$.
What is the smallest number of questions needed for Giora to complete his task when:
[b]a)[/b] $m=n$?
[b]b)[/b] $m=n+1$?
Find the greatest positive integer $m$ with the following property:
For every permutation $a_1, a_2, \cdots, a_n,\cdots$ of the set of positive integers, there exists positive integers $i_1<i_2<\cdots <i_m$ such that $a_{i_1}, a_{i_2}, \cdots, a_{i_m}$ is an arithmetic progression with an odd common difference.
For $n=2^m$ (m is a positive integer) consider the set $M(n) = \{ 1,2,...,n\}$ of natural numbers.
Prove that there exists an order $a_1, a_2, ..., a_n$ of the elements of M(n), so that for all $1\leq i < j < k \leq n$ holds: $a_j - a_i \neq a_k - a_j$.
A sequence of integers $(x_n)_{n=1}^{\infty}$ satisfies $x_1 = 1$ and $x_n < x_{n+1} \le 2n$ for all $n$.
Show that for every positive integer $k$ there exist indices $r, s$ such that $x_r-x_s = k$.
Two players $A$ and $B$ play the following game taking alternate moves. In each move, a player writes one digit on the blackboard. Each new digit is written either to the right or left of the sequence of digits already written on the blackboard. Suppose that $A$ begins the game and initially the blackboard was empty. $B$ wins the game if ,after some move of $B$, the sequence of digits written in the blackboard represents a perfect square. Prove that $A$ can prevent $B$ from winning.
Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.
The sequence $(x_n)$ is determined by the conditions:
$x_0=1992,x_n=-\frac{1992}{n} \cdot \sum_{k=0}^{n-1} x_k$ for $n \geq 1$.
Find $\sum_{n=0}^{1992} 2^nx_n$.