Found problems: 1239
Let $(x_{n})_{n=1}^{+\infty}$ be a sequence defined recursively with $x_{n+1} = x_{n}(x_{n}-2)$ and $x_{1} = \frac{7}{2}$. Let $x_{2021} = \frac{a}{b}$, where $a,b \in \mathbb{N}$ are coprime. Show that if $p$ is a prime divisor of $a$, then either $3|p-1$ or $p=3$.
[i]Authored by Nikola Velov[/i]
A sequence of positive integers is given such that the sum of any $6$ consecutive terms does not exceed $11$.
Prove that for any positive integer $a$ in the sequence one can find consecutive terms with sum $a$
Let $n$ be a positive integer, and let $f(n)$ denote the last nonzero digit in the decimal expansion of $n!$.
$(\text a)$ Show that if $a_1,a_2,\ldots,a_k$ are distinct nonnegative integers, then $f(5^{a_1}+5^{a_2}+\ldots+5^{a_k})$ depends only on the sum $a_1+a_2+\ldots+a_k$.
$(\text b)$ Assuming part $(\text a)$, we can define
$$g(s)=f(5^{a_1}+5^{a_2}+\ldots+5^{a_k}),$$where $s=a_1+a_2+\ldots+a_k$. Find the least positive integer $p$ for which
$$g(s)=g(s+p),\enspace\text{for all }s\ge1,$$or show that no such $p$ exists.
For positive integer $n$ we define $f(n)$ as sum of all of its positive integer divisors (including $1$ and $n$). Find all positive integers $c$ such that there exists strictly increasing infinite sequence of positive integers $n_1, n_2,n_3,...$ such that for all $i \in \mathbb{N}$ holds $f(n_i)-n_i=c$
For each integer $a_0 > 1$, define the sequence $a_0, a_1, a_2, \ldots$ for $n \geq 0$ as
$$a_{n+1} =
\begin{cases}
\sqrt{a_n} & \text{if } \sqrt{a_n} \text{ is an integer,} \\
a_n + 3 & \text{otherwise.}
\end{cases}
$$
Determine all values of $a_0$ such that there exists a number $A$ such that $a_n = A$ for infinitely many values of $n$.
[i]Proposed by Stephan Wagner, South Africa[/i]
Let $ \left( a_n\right)_{n\ge 1} $ be a sequence defined by $ a_1>0 $ and $ \frac{a_{n+1}}{a}=\frac{a_n}{a}+\frac{a}{a_n} , $ with $ a>0. $ Calculate $ \lim_{n\to\infty} \frac{a_n}{\sqrt{n+a}} . $
[i]Florin Rotaru[/i]
Find all integers $n \geq 3$ for which there exist real numbers $a_1, a_2, \dots a_{n + 2}$ satisfying $a_{n + 1} = a_1$, $a_{n + 2} = a_2$ and
$$a_ia_{i + 1} + 1 = a_{i + 2},$$
for $i = 1, 2, \dots, n$.
[i]Proposed by Patrik Bak, Slovakia[/i]
A sequence \(\{a_n\}\) is defined as follows: \(a_1 = 1\), \(a_2 = \frac{1}{3}\), and for all \(n \geq 1,\) \(\frac{(1+a_n)(1+a_{n+2})}{(1+a_n+1)^2} = \frac{a_na_{n+2}}{a_{n+1}^2}\).
Prove that, for all \(n \geq 1\), \(a_1 + a_2 + ... + a_n < \frac{34}{21}\).
The sequence $a_1,a_2, ..., a_{2n}$ of integers is such that each number occurs in no more than $n$ times. Prove that there are two strictly increasing sequences of indices $b_1,b_2, ..., b_{n}$ and $c_1,c_2, ..., c_{n}$ are such that every positive integer from the set $\{1,2,...,2n\}$ occurs exactly in one of these two sequences, and for each $1\le i \le n$ is true the condition $a_{b_i} \ne a_{c_i}$
.
(Anton Trygub)
Determine if there exists an infinite sequence of positive integers $a_1,a_2, a_3, ...$ such that
(i) each positive integer occurs exactly once in the sequence, and
(ii) each positive integer occurs exactly once in the sequence $ |a_1 - a_2|, |a_2 - a_3|, ..., |a+k - a_{k+1}|, ...$
Consider the sequence in which $a_1 = 1$ and $a_n$ is obtained by juxtaposing the decimal representation of $n$ at the end of the decimal representation of $a_{n-1}$. That is, $a_1 = 1$, $a_2 = 12$, $a_3 = 123$, $\dots$ , $a_9 = 123456789$, $a_{10} = 12345678910$ and so on. Prove that infinitely many numbers of this sequence are multiples of $7$.
Do there exist two bounded sequences $a_1, a_2,\ldots$ and $b_1, b_2,\ldots$ such that for each positive integers $n$ and $m>n$ at least one of the two inequalities $|a_m-a_n|>1/\sqrt{n},$ and $|b_m-b_n|>1/\sqrt{n}$ holds?
Let $k\geq 0$ an integer. The sequence $a_0,\ a_1,\ a_2, \ a_3, \ldots$ is defined as follows:
[LIST]
[*] $a_0=k$ [/*]
[*] For $n\geq 1$, we have that $a_n$ is the smallest integer greater than $a_{n-1}$ so that $a_n+a_{n-1}$ is a perfect square. [/*]
[/LIST]
Prove that there are exactly $\left \lfloor{\sqrt{2k}} \right \rfloor$ positive integers that cannot be written as the difference of two elements of such a sequence.
[i]Note.[/i] If $x$ is a real number, $\left \lfloor{x} \right \rfloor$ denotes the greatest integer smaller or equal than $x$.
Let $a_{1}, \ldots, a_{n}$ be an infinite sequence of strictly positive integers, so that $a_{k} < a_{k+1}$ for any $k.$ Prove that there exists an infinity of terms $ a_{m},$ which can be written like $a_m = x \cdot a_p + y \cdot a_q$ with $x,y$ strictly positive integers and $p \neq q.$
The sequence $(a_n)$ is defined with the recursion $a_{n + 1} = 5a^6_n + 3a^3_{n-1} + a^2_{n-2}$ for $n\ge 2$ and the set of initial values $\{a_0, a_1, a_2\} = \{2013, 2014, 2015\}$. (That is, the initial values are these three numbers in any order.)
Show that the sequence contains no sixth power of a natural number.
The infinite sequence $a_0, a_1, a_2, a_3,... $ is defined by $a_0 = 2$ and
$$a_n =\frac{2a_{n-1} + 1}{a_{n-1} + 2}$$ , $n = 1, 2, 3, ...$ Prove that $1 < a_n < 1 + \frac{1}{3^n}$ for all $n = 1, 2, 3, . .$
Let $n$ be a positive integer, and consider a sequence $a_1 , a_2 , \dotsc , a_n $ of positive integers. Extend it periodically to an infinite sequence $a_1 , a_2 , \dotsc $ by defining $a_{n+i} = a_i $ for all $i \ge 1$. If \[a_1 \le a_2 \le \dots \le a_n \le a_1 +n \] and \[a_{a_i } \le n+i-1 \quad\text{for}\quad i=1,2,\dotsc, n, \] prove that \[a_1 + \dots +a_n \le n^2. \]
The sequences $(a_n)$ and (c_n) are given by $a_0 =\frac12$, $c_0=4$ , and for $n \ge 0$ , $a_{n+1}=\frac{2a_n}{1+a_n^2}$, $c_{n+1}=c_n^2-2c_n+2$
Prove that for all $n\ge 1$, $a_n=\frac{2c_0c_1...c_{n-1}}{c_n}$
The sequence $(a_n)$ is given by $a_1 = x \in \mathbb{R}$ and $3a_{n+1} = a_n+1$ for $n \geq 1$. Set
$A = \sum_{n=1}^\infty \Big[ a_n - \frac{1}{6}\Big]$, $B = \sum_{n=1}^\infty \Big[ a_n + \frac{1}{6}\Big]$.
Compute the sum $A+B$ in terms of $x$.
Find the greatest natural number $n$ such there exist natural numbers $x_{1}, x_{2}, \ldots, x_{n}$ and natural $a_{1}< a_{2}< \ldots < a_{n-1}$ satisfying the following equations for $i =1,2,\ldots,n-1$: \[x_{1}x_{2}\ldots x_{n}= 1980 \quad \text{and}\quad x_{i}+\frac{1980}{x_{i}}= a_{i}.\]
$\boxed{A3}$The sequence $a_1,a_2,a_3,...$ is defined by $a_1=a_2=1,a_{2n+1}=2a_{2n}-a_n$ and $a_{2n+2}=2a_{2n+1}$ for $n\in{N}.$Prove that if $n>3$ and $n-3$ is divisible by $8$ then $a_n$ is divisible by $5$
Consider the sequence $(a_n)_{n\geqslant 1}$ defined by $a_1=1/2$ and $2n\cdot a_{n+1}=(n+1)a_n.$[list=a]
[*]Determine the general formula for $a_n.$
[*]Let $b_n=a_1+a_2+\cdots+a_n.$ Prove that $\{b_n\}-\{b_{n+1}\}\neq \{b_{n+1}\}-\{b_{n+2}\}.$
[/list]
For any sequence of real numbers $(a_n), n \in N$, define a new sequence $(b_n)$ as $b_n =a_{n+2}+sa_{n+1}+ta_{n}$, where $s,t$ are given real numbers.
Find all ordered pairs $(s,t)$ satisfying the following property: any sequence $(a_n)$ converges as soon as the sequence $(b_n)$ converges.
A sequence $ \left( a_n \right)_{n\ge 1} $ has the property that it´s nondecreasing, nonconstant and, for every natural $ n, a_n\big| n^2. $ Show that at least one of the following affirmations are true.
$ \text{(i)} $ There exists an index $ n_1 $ such that $ a_n=n, $ for all $ n\ge n_1. $
$ \text{(ii)} $ There exists an index $ n_2 $ such that $ a_n=n^2, $ for all $ n\ge n_2. $
Let $r$ be a positive integer, and let $a_0 , a_1 , \cdots $ be an infinite sequence of real numbers. Assume that for all nonnegative integers $m$ and $s$ there exists a positive integer $n \in [m+1, m+r]$ such that
\[ a_m + a_{m+1} +\cdots +a_{m+s} = a_n + a_{n+1} +\cdots +a_{n+s} \]
Prove that the sequence is periodic, i.e. there exists some $p \ge 1 $ such that $a_{n+p} =a_n $ for all $n \ge 0$.