Found problems: 5923
Let $p_n$ be a bounded sequence of integers which satisfies the recursion
$$p_n =\frac{p_{n-1} +p_{n-2} + p_{n-3}p _{n-4}}{p_{n-1} p_{n-2}+ p_{n-3} +p_{n-4}}.$$
Show that the sequence eventually becomes periodic.
Each term of an infinite sequence of natural numbers is obtained from the previous term by adding to it one of its nonzero digits. Prove that this sequence contains an even number.
Decide whether for every arrangement of the numbers $1,2,3, . . . ,15$ in a sequence one can color these numbers with at most four different colors in such a way that the numbers of each color form a monotone subsequence.
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 $a_{2018}-a_{2017}$.
Prove that for any integer $n$, $n\geq 3$, there exist $n$ positive integers $a_1,a_2,\ldots,a_n$ in arithmetic progression, and $n$ positive integers in geometric progression $b_1,b_2,\ldots,b_n$ such that
\[ b_1 < a_1 < b_2 < a_2 <\cdots < b_n < a_n . \]
Give an example of two such progressions having at least five terms.
[i]Mihai Baluna[/i]
Let $a_1, a_2,...,a_{2018}$ be a sequence of numbers such that all its elements are elements of a set $\{-1,1\}$. Sum
$$S=\sum \limits_{1 \leq i < j \leq 2018} a_i a_j$$ can be negative and can also be positive. Find the minimal value of this sum
Let be a natural number $ k $ and let be two infinite sequences $ \left( x_n \right)_{n\ge 1} ,\left( y_n \right)_{n\ge 1} $ such that
$$ \{1\}\cap\{ x_1,x_2,\ldots ,x_k\}=\{1\}\cap\{ y_1,y_2,\ldots ,y_k\} =\{ x_1,x_2,\ldots ,x_k\}\cap\{ y_1,y_2,\ldots ,y_k\} =\emptyset , $$
and defined by the following recurrence relations:
$$ x_{n+k}=\frac{y_n}{x_n} ,\quad y_{n+k} =\frac{y_n-1}{x_n-1} $$
Prove that $ \left( x_n \right)_{n\ge 1} $ and $ \left( y_n \right)_{n\ge 1} $ are periodic.
[i]Dumitru Acu[/i]
Draw on the plane $(p, q)$ all points with coordinates $(p,q)$, for which the equation $\sin^2x+p\sin x+q=0$ has solutions and all its positive solutions form an arithmetic progression.
Consider the sequence of numbers $(a_n)$ ($n = 1, 2, \ldots$) defined as follows: $ a_1\in (1, 2)$, $ a_{k + 1} = a_k + \frac{k}{a_k}$ ($k = 1, 2, \ldots$). Prove that there exists at most one pair of distinct positive integers $(i, j)$ such that $a_i + a_j$ is an integer.
Consider a sequence $\{a_n\}$ of integers, satisfying $a_1=1, a_2=2$ and $a_{n+1}$ is the largest prime divisor of $a_1+a_2+\ldots+a_n$. Find $a_{100}$.
Let $n$ be a positive integer. Define a sequence by setting $a_{1}= n$ and, for each $k > 1$, letting $a_{k}$ be the unique integer in the range $0\leq a_{k}\leq k-1$ for which $a_{1}+a_{2}+...+a_{k}$ is divisible by $k$. For instance, when $n = 9$ the obtained sequence is $9,1,2,0,3,3,3,...$. Prove that for any $n$ the sequence $a_{1},a_{2},...$ eventually becomes constant.
[list]
$(a)$ A sequence $x_1,x_2,\dots$ of real numbers satisfies
\[x_{n+1}=x_n \cos x_n \textrm{ for all } n\geq 1.\]
Does it follows that this sequence converges for all initial values $x_1?$ (5 points)
$(b)$ A sequence $y_1,y_2,\dots$ of real numbers satisfies
\[y_{n+1}=y_n \sin y_n \textrm{ for all } n\geq 1.\]
Does it follows that this sequence converges for all initial values $y_1?$ (5 points)[/list]
How many finite sequences $x_1,x_2,\ldots,x_m$ are there such that $x_i=1$ or $2$ and $\sum_{i=1}^mx_i=10$?
$\textbf{(A)}~89$
$\textbf{(B)}~73$
$\textbf{(C)}~107$
$\textbf{(D)}~119$
A sequence $(a_n)$ positive integers is determined by equalities $a_1=20,a_2=22$ and $a_{n+1}=4a_n^2+5a_{n-1}^3$ for all $n \geq 2$.
Find the maximum power of two which divides $a_{2023}$.
How many sets of two or more consecutive positive integers have a sum of 15?
$ \textbf{(A) } 1\qquad \textbf{(B) } 2\qquad \textbf{(C) } 3\qquad \textbf{(D) } 4\qquad \textbf{(E) } 5$
Let $a_n$ be a sequence such that $a_1=1$, $a_2=1$, and $a_{n+2}=\tfrac{a_{n+1}a_n}{a_{n+1}+a_n}$. Find the value of \[\sum_{n=1}^\infty \frac{1}{a_n3^n}.\]
Consider two finite sequences of real numbers \( a_1, a_2, \dots, a_n \) and \( b_1, b_2, \dots, b_n \). Let \( \alpha(x) = \#\{i | a_i = x \} \) and \( \beta(x) = \#\{i | b_i = -x \} \). Prove that there exists a permutation \( \sigma \in S_n \) (the symmetric group of \( n \) elements) such that \( a_{\sigma(i)} + b_i \neq 0 \) for all \( i = 1, \dots, n \) if and only if \( \alpha(x) + \beta(x) \leq n \) for all \( x \in \mathbb{R} \).
All the integers from $1$ to $100$ are arranged in a $10 \times 10$ table as shown below. Prove that if some ten numbers are removed from the table, the remaining $90$ numbers contain 10 numbers in Arithmetic Progression.
$1 \,\,\,\,2\,\, \,\,3 \,\,\,\,... \,\,10$
$11 \,\,12 \,\,13 \,\,... \,\,20$
$\,\,.\,\,\,\,.\,\,\,.$
$\,\,.\,\,\,\,.\,\,\,\,.$
$91 \,\,92 \,\,93\,\, ... \,\,100$
A number is called a palindromic number if its decimal representation read from the left to the right is the same as read from the right to the left. Let $(x_n)$ be the increasing sequence of all palindromic numbers. Determine all primes, which are divisors of at least one of the differences $x_{n+1} - x_n$.
Let $n\ge 2$ be a natural number. Let $a_1\le a_2\le a_3\le \cdots \le a_n$ be real numbers such that $a_1+a_2+\cdots +a_n>0$ and $n(a_1^2+a_2^2+\cdots +a_n^2)=2(a_1+a_2+\cdots +a_n)^2.$ If $m=\lfloor n/2\rfloor+1$, the smallest integer larger than $n/2$, then show that $a_m>0.$
There are $ n$ markers, each with one side white and the other side black. In the beginning, these $ n$ markers are aligned in a row so that their white sides are all up. In each step, if possible, we choose a marker whose white side is up (but not one of the outermost markers), remove it, and reverse the closest marker to the left of it and also reverse the closest marker to the right of it. Prove that, by a finite sequence of such steps, one can achieve a state with only two markers remaining if and only if $ n \minus{} 1$ is not divisible by $ 3$.
[i]Proposed by Dusan Dukic, Serbia[/i]
Given: (i) $a$, $b > 0$; (ii) $a$, $A_1$, $A_2$, $b$ is an arithmetic progression; (iii) $a$, $G_1$, $G_2$, $b$ is a geometric progression. Show that
\[A_1 A_2 \ge G_1 G_2.\]
Let $a_0,a_1,\ldots$ be a sequence of positive integers with $a_0=1$, $a_1=2$ and
\[a_n = a_{n-1}^{a_{n-1}a_{n-2}}-1\]
for all $n\geq 2$. Show that if $p$ is a prime less than $2^k$ for some positive integer $k$, then there exists $n\leq k+1$ such that $p\mid a_n$.
Let $k\geq4$ be an integer. Sunny and Ming play a game with strings. A string is a sequence that every element of it is an integer between $1$ and $k$, inclusive. At first, Sunny chooses two positive integers $N,L\geq2$ and write down $N$ strings, each having length $L$. Then Ming mark at most $\frac{N}{2}$ strings. Then Sunny chooses an unmarked string $s$ and calculate the biggest integer $n$ such that there exists another string satisfying its first $n$ element is the same as the first $n$ element of $s$. Then Sunny burn down all strings which first $n$ element if different from the first $n$ element of $s$, leaving only the ones which have the same first $n$ element of $s$. Finally, Ming chooses an integer $d$ between $1$ and $k$, inclusive, and remove all strings which $(n+1)$th element is $d$. Sunny's score would be the number of strings left. Find the maximum score that Sunny can guarantee to get.
[i]Proposed by USJL[/i]
Let $\{a_n\}$ be a sequence of positive integers such that $a_{n+1} = a_n^2+1$ for all $n \geq 1$. Prove that there is no positive integer $N$ such that $$\prod_{k=1}^N(a_k^2+a_k+1)$$ is a perfect square.