Found problems: 5923
The sequence of integers $\{ x_{n}\}_{n\ge1}$ is defined as follows: \[x_{1}=1, \;\; x_{n+1}=1+{x_{1}}^{2}+\cdots+{x_{n}}^{2}\;(n=1,2,3 \cdots).\] Prove that there are no squares of natural numbers in this sequence except $x_{1}$.
Find a sequence of maximal length consisting of non-zero integers in which the sum of any seven consecutive terms is positive and that of any eleven consecutive terms is negative.
Let $\{a_n \}_{n=1}^{\infty}$ be a sequence such that $a_1=\frac 12$ and
\[a_n=\biggl( \frac{2n-3}{2n} \biggr) a_{n-1} \qquad \forall n \geq 2.\]
Prove that for every positive integer $n,$ we have $\sum_{k=1}^n a_k <1.$
A real number $a$ is given. The sequence $n_{1}< n_{2}< n_{3}< ...$ consists of all the positive integral $n$ such that $\{na\}< \frac{1}{10}$. Prove that there are at most three different numbers among the numbers $n_{2}-n_{1}$, $n_{3}-n_{2}$, $n_{4}-n_{3}$, $\ldots$.
[i]A corollary of a theorem from ergodic theory[/i]
Let $a_1$, $a_2$, $\ldots\,$, $a_{2019}$ be a sequence of real numbers. For every five indices $i$, $j$, $k$, $\ell$, and $m$ from 1 through 2019, at least two of the numbers $a_i$, $a_j$, $a_k$, $a_\ell$, and $a_m$ have the same absolute value. What is the greatest possible number of distinct real numbers in the given sequence?
An infinite sequence of natural number $\{x_n\}_{n\ge 1}$ is such that $x_{n+1}$ is obtained by adding one of the non-zero digits of $x_n$ to itself. Show this sequence contains an even number.
If $a$, $6$, and $b$, in that order, form an arithmetic sequence, compute $a+b$.
We are given a positive integer $ r$ and a rectangular board $ ABCD$ with dimensions $ AB \equal{} 20, BC \equal{} 12$. The rectangle is divided into a grid of $ 20 \times 12$ unit squares. The following moves are permitted on the board: one can move from one square to another only if the distance between the centers of the two squares is $ \sqrt {r}$. The task is to find a sequence of moves leading from the square with $ A$ as a vertex to the square with $ B$ as a vertex.
(a) Show that the task cannot be done if $ r$ is divisible by 2 or 3.
(b) Prove that the task is possible when $ r \equal{} 73$.
(c) Can the task be done when $ r \equal{} 97$?
Denote by $l(n)$ the largest prime divisor of $n$. Let $a_{n+1} = a_n + l(a_n)$ be a recursively
defined sequence of integers with $a_1 = 2$. Determine all natural numbers $m$ such that there
exists some $i \in \mathbb{N}$ with $a_i = m^2$.
[i]Proposed by Nikola Velov, North Macedonia[/i]
A sequence of positive integers is constructed as follows. If the last digit of $a_n$ is greater than $5$, then $a_{n+1}$ is $9a_n$. If the last digit of $a_n$ is $5$ or less and an has more than one digit, then $a_{n+1}$ is obtained from $a_n$ by deleting the last digit. If $a_n$ has only one digit, which is $5$ or less, then the sequence terminates. Can we choose the first member of the sequence so that it does not terminate?
Let $\sum_{n=1}^\infty a_n$ be a convergent series of positive terms (so $a_i>0$ for all $i$) and set $b_n=\frac1{na_n^2}$ for $n\ge1$. Prove that $\sum_{n=1}^\infty\frac n{b_1+b_2+\ldots+b_n}$ is convergent.
Freddy the Frog is situated at 1 on an infinitely long number line. On day $n$, where $n\ge 1$, Freddy can choose to hop 1 step to the right, stay where he is, or hop $k$ steps to the left, where $k$ is an integer at most $n+1$. After day 5, how many sequences of moves are there such that Freddy has landed on at least one negative number?
[i]Proposed by Andy Xu[/i]
Let $x_1, x_2, \dots, x_n$ be a positive integer sequence such that each term is less than or equal to $2001$ and for every $i\geq 3$, $x_i = |x_{i-1}-x_{i-2}|$. What is the largest possible value of $n$?
$
\textbf{(A)}\ 1000
\qquad\textbf{(B)}\ 2001
\qquad\textbf{(C)}\ 3002
\qquad\textbf{(D)}\ 4003
\qquad\textbf{(E)}\ \text{None of the preceding}
$
Let be the sequence of sets $ \left(\left\{ A\in\mathcal{M}_2\left(\mathbb{R} \right) | A^{n+1} =2007^nA\right\}\right)_{n\ge 1} . $
[b]a)[/b] Prove that each term of the above sequence hasn't a finite cardinal.
[b]b)[/b] Determine the intersection of the fourth element of the above sequence with the $ 2007\text{th} $ element.
[i]Gheorghe Iurea[/i]
[hide=Note]Similar with [url]https://artofproblemsolving.com/community/c7h1928039p13233629[/url].[/hide]
The sequence $a_1,a_2,\ldots$ is defined by $a_1=2019$, $a_2=2020$, $a_3=2021$, $a_{n+3}=a_n(a_{n+1}a_{n+2}+1)$ for $n\ge 1$. Determine the value of the infinite sum
\[\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}+\cdots \]
Let the sequence of real numbers $(a_n),n=1,2,3...$ with $a_1=2$ and $a_n=\left(\frac{n+1}{n-1} \right)\left(a_1+a_2+...+a_{n-1} \right),n\geq 2$.
Find the term $a_{2013}$.
The integer sequence $(s_i)$ "having pattern 2016'" is defined as follows:
$\circ$ The first member $s_1$ is 2.
$\circ$ The second member $s_2$ is the least positive integer exceeding $s_1$ and having digit 0 in its decimal notation.
$\circ$ The third member $s_3$ is the least positive integer exceeding $s_2$ and having digit 1 in its decimal notation.
$\circ$ The third member $s_3$ is the least positive integer exceeding $s_2$ and having digit 6 in its decimal notation.
The following members are defined in the same way. The required digits change periodically: $2 \rightarrow 0 \rightarrow 1 \rightarrow 6 \rightarrow 2 \rightarrow 0 \rightarrow \ldots$. The first members of this sequence are the following: $2; 10; 11; 16; 20; 30; 31; 36; 42; 50$.\\
Does this sequence contain a) 2001, b) 2006?
We have a set of ${221}$ real numbers whose sum is ${110721}$. It is deemed that the numbers form a rectangular table such that every row as well as the first and last columns are arithmetic progressions of more than one element. Prove that the sum of the elements in the four corners is equal to ${2004}$.
For a sequence $\{a_i\}_{i\ge1}$ consisting of only positive integers, prove that if for all different positive integers $i$ and $j$, we have $a_i\nmid a_j$, then
$$\{p\mid p\text{ is a prime and }p\mid a_i\text{ for some }i\}$$is a infinite set.
Find all integers $n$ with $n \geq 4$ for which there exists a sequence of distinct real numbers $x_1, \ldots, x_n$ such that each of the sets $$\{x_1, x_2, x_3\}, \{x_2, x_3, x_4\},\ldots,\{x_{n-2}, x_{n-1}, x_n\}, \{x_{n-1}, x_n, x_1\},\text{ and } \{x_n, x_1, x_2\}$$ forms a 3-term arithmetic progression when arranged in increasing order.
[color=darkblue]The sequence $ (a_n)_{n \in \mathbb{N}}$ is defined as follows:
\[ a_n \equal{} \dfrac{2}{3 \plus{} 1} \plus{} \dfrac{2^2}{3^2 \plus{} 1} \plus{} \dfrac{2^3}{3^4 \plus{} 1} \plus{} \ldots \plus{} \dfrac{2^{n \plus{} 1}}{3^{2^n} \plus{} 1}
\]
Prove that $ a_n < 1$ for any $ n \in \mathbb{N}$[/color]
Find the number of positive integers $j\leq 3^{2013}$ such that \[j=\sum_{k=0}^m\left((-1)^k\cdot 3^{a_k}\right)\] for some strictly increasing sequence of nonnegative integers $\{a_k\}$. For example, we may write $3=3^1$ and $55=3^0-3^3+3^4$, but $4$ cannot be written in this form.
There is graph $ G_0$ on vertices $ A_1, A_2, \ldots, A_n$. Graph $ G_{n \plus{} 1}$ on vertices $ A_1, A_2, \ldots, A_n$ is constructed by the rule: $ A_i$ and $ A_j$ are joined only if in graph $ G_n$ there is a vertices $ A_k\neq A_i, A_j$ such that $ A_k$ is joined with both $ A_i$ and $ A_j$. Prove that the sequence $ \{G_n\}_{n\in\mathbb{N}}$ is periodic after some term with period $ T \le 2^n$.
Let $a_0,a_1,\ldots$ be an infinite sequence of positive numbers. Prove that the inequality $1+a_n>\sqrt[n]2a_{n-1}$ holds for infinitely many positive integers $n$.
Let $a_1<a_2<a_3<a_4<\cdots$ be an infinite sequence of real numbers in the interval $(0,1)$. Show that there exists a number that occurs exactly once in the sequence
\[ \frac{a_1}{1},\frac{a_2}{2},\frac{a_3}{3},\frac{a_4}{4},\ldots.\]
[i]Merlijn Staps[/i]