Found problems: 1239
Given is the sequence $(a_n)_{n\geq 0}$ which is defined as follows:$a_0=3$ and $a_{n+1}-a_n=n(a_n-1) \ , \ \forall n\geq 0$.
Determine all positive integers $m$ such that $\gcd (m,a_n)=1 \ , \ \forall n\geq 0$.
The sequence $a_1,a_2,\dots$ of positive integers obeys the following two conditions:
[list]
[*] For all positive integers $m,n$, it happens that $a_m\cdot a_n=a_{mn}$
[*] There exist infinite positive integers $n$ such that $(a_1,a_2,\dots,a_n)$ is a permutation of $(1,2,\dots,n)$
[/list]
Prove that $a_n=n$ for all positive integers $n$.
[i]Proposed by José Alejandro Reyes González[/i]
A sequence $ \left( x_n\right)_{n\ge 0} $ of real numbers satisfies $ x_0>1=x_{n+1}\left( x_n-\left\lfloor x_n\right\rfloor\right) , $ for each $ n\ge 1. $
Prove that if $ \left( x_n\right)_{n\ge 0} $ is periodic, then $ x_0 $ is a root of a quadratic equation. Study the converse.
Let $b$ be an odd positive integer. The sequence $a_1, a_2, a_3, a_4$, is definedin the next way: $a_1$ and $a_2$ are positive integers and for all $k \ge 2$,
$$a_{k+1}= \begin{cases} \frac{a_k + a_{k-1}}{2} \,\,\, if \,\,\, a_k + a_{k-1} \,\,\, is \,\,\, even \\ \frac{a_k + a_{k-1+b}}{2}\,\,\, if \,\,\, a_k + a_{k-1}\,\,\, is \,\,\,odd\end{cases}$$
a) Prove that if $b = 1$, then after a certain term, the sequence will become constant.
b) For each $b \ge 3$ (odd), prove that there exist values of $a_1$ and $a_2$ for which the sequence will become constant after a certain term.
Let $(a_n)_{n \ge 1}$ be a sequence of real numbers satisfying the properties: [list=1]
[*] the sequence $x_n=\sum\limits_{k=1}^n a_k^2$ is convergent;
[*] the sequence $y_n=\sum\limits_{k=1}^n a_k$ is unbounded.
[/list]
Prove that the sequence $(b_n)_{n \ge 1}$ given by $b_n=\{y_n\}$ is divergent.
Note: $\{ x \}$ denotes the fractional part of $x.$
Let $x>1$ be a real number which is not an integer. For each $n\in\mathbb{N}$, let $a_n=\lfloor x^{n+1}\rfloor - x\lfloor x^n\rfloor$. Prove that the sequence $(a_n)$ is not periodic.
[b]2.[/b] Show that the series
$\sum_{n=1}^{\infty}\frac{1}{n}sin(asin(\frac{2n\pi}{N}))e^{bcos(\frac{2n\pi}{N})}$
is convergent for every positive integer N and any real numbers a and b. [b](S. 25)[/b]
Let $k$ be a positive integer. Show that if there exists a sequence $a_0,a_1,\ldots$ of integers satisfying the condition \[a_n=\frac{a_{n-1}+n^k}{n}\text{ for all } n\geq 1,\] then $k-2$ is divisible by $3$.
[i]Proposed by Okan Tekman, Turkey[/i]
Consider the sequence $a_1, a_2, a_3, ...$ defined by $a_1 = 9$ and
$a_{n + 1} = \frac{(n + 5)a_n + 22}{n + 3}$
for $n \ge 1$.
Find all natural numbers $n$ for which $a_n$ is a perfect square of an integer.
The set $ \{a_0, a_1, \ldots, a_n\}$ of real numbers satisfies the following conditions:
[b](i)[/b] $ a_0 \equal{} a_n \equal{} 0,$
[b](ii)[/b] for $ 1 \leq k \leq n \minus{} 1,$ \[ a_k \equal{} c \plus{} \sum^{n\minus{}1}_{i\equal{}k} a_{i\minus{}k} \cdot \left(a_i \plus{} a_{i\plus{}1} \right)\]
Prove that $ c \leq \frac{1}{4n}.$
Esmeralda chooses two distinct positive integers \(a\) and \(b\), with \(b > a\), and writes the equation
\[
x^2 - ax + b = 0
\]
on the board. If the equation has distinct positive integer roots \(c\) and \(d\), with \(d > c\), she writes the equation
\[
x^2 - cx + d = 0
\]
on the board. She repeats the procedure as long as she obtains distinct positive integer roots. If she writes an equation for which this does not occur, she stops.
a) Show that Esmeralda can choose \(a\) and \(b\) such that she will write exactly 2024 equations on the board.
b) What is the maximum number of equations she can write knowing that one of the initially chosen numbers is 2024?
For every real number $x_1$, construct the sequence $x_1,x_2,\ldots$ by setting: \[ x_{n+1}=x_n(x_n+{1\over n}). \] Prove that there exists exactly one value of $x_1$ which gives $0<x_n<x_{n+1}<1$ for all $n$.
Consider an infinite chessboard whose rows and columns are indexed by positive integers. At most one coin can be put on any cell of the chessboard. Let be given two arbitrary sequences ($a_n$) and ($b_n$) of positive integers ($n \in N$). Assuming that infinitely many coins are available, prove that they can be arranged on the chessboard so that there are $a_n$ coins in the $n$-th row and $b_n$ coins in the $n$-th column for all $n$.
Let $(a_n)^{+\infty}_{n=1}$ be a sequence defined recursively as follows: $a_1=1$ and $$a_{n+1}=1 + \sum\limits_{k=1}^{n}ka_k$$
For every $n > 1$, prove that $\sqrt[n]{a_n} < \frac {n+1}{2}$.
Find all positive integers $n$ such that there exists a sequence of positive integers $a_1$, $a_2$,$\ldots$, $a_n$ satisfying: \[a_{k+1}=\frac{a_k^2+1}{a_{k-1}+1}-1\] for every $k$ with $2\leq k\leq n-1$.
[i]Proposed by North Korea[/i]
Let $a_1,a_2,a_3,\ldots$ be an infinite sequence of positive integers such that $a_{n+2m}$ divides $a_{n}+a_{n+m}$ for all positive integers $n$ and $m.$ Prove that this sequence is eventually periodic, i.e. there exist positive integers $N$ and $d$ such that $a_n=a_{n+d}$ for all $n>N.$
The sequence of real numbers $a_0,a_1,a_2,\ldots$ is defined recursively by \[a_0=-1,\qquad\sum_{k=0}^n\dfrac{a_{n-k}}{k+1}=0\quad\text{for}\quad n\geq 1.\]Show that $ a_{n} > 0$ for all $ n\geq 1$.
[i]Proposed by Mariusz Skalba, Poland[/i]
Let $(a_n)_{n\geq 1}$ be a sequence of positive real numbers with the property that
$$(a_{n+1})^2 + a_na_{n+2} \leq a_n + a_{n+2}$$
for all positive integers $n$. Show that $a_{2022}\leq 1$.
The first two terms of a sequence are $2$ and $3$. Each next term thereafter is the sum of the nearestly previous two terms if their sum is not greather than $10, 0$ otherwise. The $2014$th term is:
(A): $0$, (B): $8$, (C): $6$, (D): $4$, (E) None of the above.
Define the sequence $a_1, a_2,...$ as follows: $a_1 = 1$, and for every $n \ge 2$, $a_n = n - 2$ if $a_{n-1} = 0$ and $a_n = a_{n-1} - 1$, otherwise. Find the number of $1 \le k \le 2016$ such that there are non-negative integers $r, s$ and a positive integer $n$ satisfying $k = r + s$ and $a_{n+r} = a_n + s$.
Denote by $a_n$ the greatest number that is not divisible by $3$ and that divides $n$. Consider the sequence $s_0 = 0, s_n = a_1 +a_2+\cdots+a_n, n \in \mathbb N$. Denote by $A(n)$ the number of all sums $s_k \ (0 \leq k \leq 3^n, k \in \mathbb N_0)$ that are divisible by $3$. Prove the formula
\[A(n) = 3^{n-1} + 2 \cdot 3^{(n/2)-1} \cos \left(\frac{n\pi}{6}\right), \qquad n\in \mathbb N_0.\]
a) Suppose that a sequence of numbers $x_1,x_2,x_3,...$ satisfies the inequality $x_n-2x_{n+1}+x_{n+2} \le 0$ for any $n$ . Moreover $x_o=1,x_{20}=9,x_{200}=6$. What is the maximal value of $x_{2009}$ can be?
b) Suppose that a sequence of numbers $x_1,x_2,x_3,...$ satisfies the inequality $2x_n-3x_{n+1}+x_{n+2} \le 0$ for any $n$. Moreover $x_o=1,x_1=2,x_3=1$. Can $x_{2009}$ be greater then $0,678$ ?
Tireless Thomas and Jeremy construct a sequence. At the beginning there is one positive integer in the sequence. Then they successively write new numbers in the sequence in the following way: Thomas obtains the next number by adding to the previous number one of its (decimal) digits, while Jeremy obtains the next number by subtracting from the previous number one of its digits. Prove that there is a number in this sequence which will be repeated at least $100$ times.
(A Shapovalov)
Let $n$ be the number that is produced by concatenating the numbers $1, 2,... , 4022$,
that is, $n = 1234567891011...40214022$.
a. Show that $n$ is divisible by $3$.
b. Let $a_1 = n^{2011}$, and let $a_i$ be the sum of the digits of $a_{i-1}$ for $i > 1$. Find $a_4$
Determine all sequences of integers $a_1, a_2,. . .,$ such that:
(i) $1 \le a_i \le n$ for all $1 \le i \le n$.
(ii) $| a_i - a_j| = | i - j |$ for any $1 \le i, j \le n$