Found problems: 5923
Let a sequence $\{a_n\}$, $n \in \mathbb{N}^{*}$ given, satisfying the condition
\[0 < a_{n+1} - a_n \leq 2001\]
for all $n \in \mathbb{N}^{*}$
Show that there are infinitely many pairs of positive integers $(p, q)$ such that $p < q$ and $a_p$ is divisor of $a_q$.
In a certain sequence of numbers, the first number is $1$, and, for all $n\ge 2$, the product of the first $n$ numbers in the sequence is $n^2$. The sum of the third and the fifth numbers in the sequence is
$\textbf{(A) }\frac{25}{9}\qquad\textbf{(B) }\frac{31}{15}\qquad\textbf{(C) }\frac{61}{16}\qquad\textbf{(D) }\frac{576}{225}\qquad\textbf{(E) }34$
Sequence $ \{a_{n}\}$ is defined by $ a_{1}= 2007,\, a_{n+1}=\frac{a_{n}^{2}}{a_{n}+1}$ for $ n \ge 1.$ Prove that $ [a_{n}] =2007-n$ for $ 0 \le n \le 1004,$ where $ [x]$ denotes the largest integer no larger than $ x.$
Let $c$ be a positive integer. The sequence $\{f_n\}$ is defined as follows:
\[f_1 = 1, f_2 = c, f_{n+1} = 2f_n - f_{n-1} + 2 \quad (n \geq 2).\]
Show that for each $k \in \mathbb N$ there exists $r \in \mathbb N$ such that $f_kf_{k+1}= f_r.$
Each of the six boxes $B_1$, $B_2$, $B_3$, $B_4$, $B_5$, $B_6$ initially contains one coin. The following operations are allowed
Type 1) Choose a non-empty box $B_j$, $1\leq j \leq 5$, remove one coin from $B_j$ and add two coins to $B_{j+1}$;
Type 2) Choose a non-empty box $B_k$, $1\leq k \leq 4$, remove one coin from $B_k$ and swap the contents (maybe empty) of the boxes $B_{k+1}$ and $B_{k+2}$.
Determine if there exists a finite sequence of operations of the allowed types, such that the five boxes $B_1$, $B_2$, $B_3$, $B_4$, $B_5$ become empty, while box $B_6$ contains exactly $2010^{2010^{2010}}$ coins.
[i]Proposed by Hans Zantema, Netherlands[/i]
The function $f$ sends sequences to sequences in the following way: given a sequence $\{a_n\}_{n=0}^{\infty}$ of real numbers, $f$ sends $\{a_n\}_{n=0}^{\infty}$ to the sequence $\{b_n\}_{n=0}^{\infty},$ where $b_n=\sum_{k=0}^n a_k \tbinom{n}{k}$ for all $n \ge 0.$ Let $\{F_n\}_{n=0}^{\infty}$ be the Fibonacci sequence, defined by $F_0=0, F_1=1,$ and $F_{n+2}=F_{n+1}+F_n$ for all $n \ge 0.$ Let $\{c_n\}_{n=0}^{\infty}$ denote the sequence obtained by applying the function $f$ to the sequence $\{F_n\}_{n=0}^{\infty}$ $2022$ times. Find $c_5 \pmod{1000}.$
Sequences $(x_n)_{n\ge1}$, $(y_n)_{n\ge1}$ satisfy the relations $x_n=4x_{n-1}+3y_{n-1}$ and $y_n=2x_{n-1}+3y_{n-1}$ for $n\ge1$. If $x_1=y_1=5$ find $x_n$ and $y_n$.
Calculate $\lim_{n\rightarrow\infty}\frac{x_n}{y_n}$.
Let $ n \geq 3 $ be an integer and let
\begin{align*}
x_1,x_2, \ldots, x_n
\end{align*}
be $ n $ distinct integers. Prove that
\begin{align*}
(x_1 - x_2)^2 + (x_2 - x_3)^2 + \ldots + (x_n - x_1)^2 \geq 4n - 6.
\end{align*}
A sequence $\{x_n\}_{n=1}^{\infty}, 0\leq x_n\leq 1$ is called "Devin" if for any $f\in C[0,1]$
$$ \lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n f(x_i)=\int_0^1 f(x)\,dx $$
Prove that a sequence $\{x_n\}_{n=1}^{\infty}, 0\leq x_n\leq 1$ is "Devin" if and only if for any non-negative integer $k$ it holds
$$\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n x_i^k=\frac{1}{k+1}.$$
[b]Remark[/b]. I left intact the text as it was proposed. Devin is a Bulgarian city and SPA resort, where this competition took place.
The lengths of the three sides of a right triangle form a geometric sequence. The sine of the smallest of the angles in the triangle is $\tfrac{m+\sqrt{n}}{k}$ where $m$, $n$, and $k$ are integers, and $k$ is not divisible by the square of any prime. Find $m + n + k$.
A triangle $\vartriangle A_0A_1A_2$ in the plane has sidelengths $A_0A_1 = 7$,$A_1A_2 = 8$,$A_2A_0 = 9$. For $i \ge 0$, given $\vartriangle A_iA_{i+1}A_{i+2}$, let $A_{i+3}$ be the midpoint of $A_iA_{i+1}$ and let Gi be the centroid of $\vartriangle A_iA_{i+1}A_{i+2}$. Let point $G$ be the limit of the sequence of points $\{G_i\}^{\infty}_{i=0}$. If the distance between $G$ and $G_0$ can be written as $\frac{a\sqrt{b}}{c}$ , where $a, b, c$ are positive integers such that $a$ and $c$ are relatively prime and $b$ is not divisible by the square of any prime, find $a^2 + b^2 + c^2$.
Let $(a_n)_{n \geq 0}$ and $(b_n)_{n \geq 0}$ be sequences of real numbers such that $ a_0>\frac{1}{2}$ , $a_{n+1} \geq a_n$ and $b_{n+1}=a_n(b_n+b_{n+2})$ for all non-negative integers $n$ . Show that the sequence $(b_n)_{n \geq 0}$ is bounded .
Consider a sequence of positive integers $a_1,a_2,a_3,...$ such that $a_1>1$ and
$$a_{n+1}=\frac{a_n}{p}+p,$$
where $p$ is the greatest prime factor of $a_n$. Prove that for any choice of $a_1$, the sequence $a_1,a_2,a_3,...$ has an infinite terms that are equal between them.
Given distinct positive integer $ a_1,a_2,…,a_{2020} $. For $ n \ge 2021 $, $a_n$ is the smallest number different from $a_1,a_2,…,a_{n-1}$ which doesn't divide $a_{n-2020}...a_{n-2}a_{n-1}$. Proof that every number large enough appears in the sequence.
For each real number $x$, let $\lfloor x \rfloor$ denote the largest integer not exceeding $x$.
A sequence $\{a_n \}_{n=1}^{\infty}$ is defined by $a_n = \frac{1}{4^{\lfloor -\log_4 n \rfloor}}, \forall n \geq 1.$ Let $b_n = \frac{1}{n^2} \left( \sum_{k=1}^n a_k - \frac{1}{a_1+a_2} \right), \forall n \geq 1.$
a) Find a polynomial $P(x)$ with real coefficients such that $b_n = P \left( \frac{a_n}{n} \right), \forall n \geq 1$.
b) Prove that there exists a strictly increasing sequence $\{n_k \}_{k=1}^{\infty}$ of positive integers such that $$\lim_{k \to \infty} b_{n_k} = \frac{2024}{2025}.$$
Let $a_n = 2^{n-1}$ for $n > 0$. Let
\[
b_n = \sum\limits_{r+s \leq n} a_ra_s
\]
Find $b_n-b_{n-1}$, $b_n-2b_{n-1}$ and $b_n$.
Let $a_n$ and $b_n$ to be two sequences defined as below:
$i)$ $a_1 = 1$
$ii)$ $a_n + b_n = 6n - 1$
$iii)$ $a_{n+1}$ is the least positive integer different of $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n$.
Determine $a_{2009}$.
A sequence of real numbers $ x_0, x_1, x_2, \ldots$ is defined as follows: $ x_0 \equal{} 1989$ and for each $ n \geq 1$
\[ x_n \equal{} \minus{} \frac{1989}{n} \sum^{n\minus{}1}_{k\equal{}0} x_k.\]
Calculate the value of $ \sum^{1989}_{n\equal{}0} 2^n x_n.$
Prove that for every positive integer $d > 1$ and $m$ the sequence $a_n=2^{2^n}+d$ contains two terms $a_k$ and $a_l$ ($k \neq l$) such that their greatest common divisor is greater than $m$.
[i]Proposed by T. Hakobyan[/i]
Define the polymonial sequence $\left \{ f_n\left ( x \right ) \right \}_{n\ge 1}$ with $f_1\left ( x \right )=1$, $$f_{2n}\left ( x \right )=xf_n\left ( x \right ), \; f_{2n+1}\left ( x \right ) = f_n\left ( x \right )+ f_{n+1} \left ( x \right ), \; n\ge 1.$$ Look for all the rational number $a$ which is a root of certain $f_n\left ( x \right ).$
A sequence $(a_n)_0^N$ of real numbers is called concave if $2a_n\ge a_{n-1} + a_{n+1}$ for all integers $n, 1 \le n \le N - 1$.
$(a)$ Prove that there exists a constant $C >0$ such that
\[\left(\displaystyle\sum_{n=0}^{N}a_n\right)^2\ge C(N - 1)\displaystyle\sum_{n=0}^{N}a_n^2\:\:\:\:\:(1)\]
for all concave positive sequences $(a_n)^N_0$
$(b)$ Prove that $(1)$ holds with $C = \frac{3}{4}$ and that this constant is best
possible.
For every integer $n\ge 3$ find all sequences of real numbers $(x_1,x_2,\ldots ,x_n)$ such that $\sum_{i=1}^{n}x_i=n$ and $\sum_{i=1}^{n} (x_{i-1}-x_i+x_{i+1})^2=n$, where $x_0=x_n$ and $x_{n+1}=x_1$.
Given are positive integers $r$ and $k$ and an infi nite sequence of positive integers $a_1 \le a_2 \le ...$ such that $\frac{r}{a_r}= k + 1$. Prove that there is a $t$ satisfying $\frac{t}{a_t}=k$.
Let be two nonnegative real numbers $ a,b $ with $ b>a, $ and a sequence $ \left( x_n \right)_{n\ge 1} $ of real numbers such that the sequence $ \left( \frac{x_1+x_2+\cdots +x_n}{n^a} \right)_{n\ge 1} $ is bounded.
Show that the sequence $ \left( x_1+\frac{x_2}{2^b} +\frac{x_3}{3^b} +\cdots +\frac{x_n}{n^b} \right)_{n\ge 1} $ is convergent.
Let $k$ and $m$ be positive integers. For a positive integer $n$, let $f(n)$ be the number of integer sequences $x_1,\,\ldots,\,x_k,\,y_1,\,\ldots,\,y_m,\,z$ satisfying $1\leq x_1\leq\cdots\leq x_k\leq z\leq n$ and $1\leq y_1\leq\cdots\leq y_m\leq z\leq n$. Show that $f(n)$ can be expressed as a polynomial in $n$ with nonnegative coefficients.