Found problems: 5923
Let $f:[0,\infty )\rightarrow\mathbb{R}$ be a periodical function, with period $1$, integrable on $[0,1]$. For a strictly increasing and unbounded sequence $(x_n)_{n\ge 0},\, x_0=0,$ with $\lim_{n\rightarrow\infty} (x_{n+1}-x_n)=0$, we denote $r(n)=\max \{ k\mid x_k\le n\}$.
a) Show that:
\[\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=1}^{r(n)}(x_k-x_{k+1})f(x_k)=\int_0^1 f(x)\, dx\]
b) Show that:
\[ \lim_{n\rightarrow\infty} \frac{1}{\ln n}\sum_{k=1}^{r(n)}\frac{f(\ln k)}{k}=\int_0^1f(x)\, dx\]
For a positive integer $n$, denote $rad(n)$ as product of prime divisors of $n$. And also $rad(1)=1$. Define the sequence $\{a_i\}_{i=1}^{\infty}$ in this way: $a_1 \in \mathbb N$ and for every $n \in \mathbb N$, $a_{n+1}=a_n+rad(a_n)$.
Prove that for every $N \in \mathbb N$, there exist $N$ consecutive terms of this sequence which are in an arithmetic progression.
An integer $N > 1$ is written on the board. Alex writes a sequence of positive integers, obtaining new integers in the following manner: he takes any divisor greater than $1$ of the last number and either adds it to, or subtracts it from the number itself. Is it always (for all $N > 1$) possible for Alex to write the number $2011$ at some point?
The second and fourth terms of a geometric sequence are $ 2$ and $ 6$. Which of the following is a possible first term?
$ \textbf{(A)}\ \minus{}\!\sqrt3 \qquad
\textbf{(B)}\ \minus{}\!\frac{2\sqrt3}{3} \qquad
\textbf{(C)}\ \minus{}\!\frac{\sqrt3}{3} \qquad
\textbf{(D)}\ \sqrt3 \qquad
\textbf{(E)}\ 3$
In a school, every pair of students are either friends or strangers. Friendship is mutual, and no student is friends with themselves. A sequence of (not necessarily distinct) students $A_1, A_2, \dots, A_{2023}$ is called [i]mischievous[/i] if
$\bullet$ Total number of friends of $A_1$ is odd.
$\bullet$ $A_i$ and $A_{i+1}$ are friends for $i=1, 2, \dots, 2022$.
$\bullet$ Total number of friends of $A_{2023}$ is even.
Prove that the total number of [i]mischievous[/i] sequences is even.
Let $n$ be a positive integer and $u_1,u_2,\cdots ,u_n$ be positive integers not larger than $2^k, $ for some integer $k\geq 3.$ A representation of a non-negative integer $t$ is a sequence of non-negative integers $a_1,a_2,\cdots ,a_n$ such that $t=a_1u_1+a_2u_2+\cdots +a_nu_n.$
Prove that if a non-negative integer $t$ has a representation,then it also has a representation where less than $2k$ of numbers $a_1,a_2,\cdots ,a_n$ are non-zero.
Consider the following increasing sequence $1,3,5,7,9,…$ of all positive integers consisting only of odd digits. Find the $2017$ -th term of the above sequence.
Let $ f:[0,1]\longrightarrow [0,1] $ be a nondecreasing function. Prove that the sequence
$$ \left( \int_0^1 \frac{1+f^n(x)}{1+f^{1+n} (x)} \right)_{n\ge 1} $$
is convergent and calculate its limit.
On the blackboard are written the $2003$ integers from $1$ to $2003$. Lucas must delete $90$ numbers. Next, Mauro must choose $37$ from the numbers that remain written. If the $37$ numbers Mauro chooses form an arithmetic progression, Mauro wins. If not, Lucas wins. Decide if Lucas can choose the $90$ numbers he erases so that victory is assured.
For a given positive integer $n$ one has to choose positive integers $a_0, a_1,...$ so that the following conditions hold:
(1) $a_i = a_{i+n}$ for any $i$,
(2) $a_i$ is not divisible by $n$ for any $i$,
(3) $a_{i+a_i}$ is divisible by $a_i$ for any $i$.
For which positive integers $n > 1$ is this possible only if the numbers $a_0, a_1, ...$ are all equal?
$a_1=-5$, $a_2=-6$ and for all $n \geq 2$ the ${(a_n)^\infty}_{n=1}$ sequence defined as,
\[a_{n+1}=a_n+(a_1+1)(2a_2+1)(3a_3+1)\cdots((n-1)a_{n-1}+1)((n^2+n)a_n+2n+1)).\]
If a prime $p$ divides $na_n+1$ for a natural number n, prove that there is a integer $m$ such that $m^2\equiv5(modp)$
Prove that:
[b](a)[/b] If $(a_n)_{n\geq 1}$ is a strictly increasing sequence of positive integers such that $\frac{a_{2n-1}+a_{2n}}{a_n}$ is a constant as $n$ runs through all positive integers, then this constant is an integer greater than or equal to $4$; and
[b](b)[/b] Given an integer $N\geq 4$, there exists a strictly increasing sequene $(a_n)_{n\geq 1}$ of positive integers such that $\frac{a_{2n-1}+a_{2n}}{a_n}=N$ for all indices $n$.
Given a real sequence $\left \{ x_n \right \}_{n=1}^{\infty}$ with $x_1^2 = 1$. Prove that for each integer $n \ge 2$, $$\sum_{i|n}\sum_{j|n}\frac{x_ix_j}{\textup{lcm} \left ( i,j \right )} \ge \prod_{\mbox{\tiny$\begin{array}{c}
p \: \textup{is prime} \\ p|n \end{array}$} }\left ( 1-\frac{1}{p} \right ). $$
The sequence $\{x_{n}\}$ is defined by \[x_{0}\in [0, 1], \; x_{n+1}=1-\vert 1-2 x_{n}\vert.\] Prove that the sequence is periodic if and only if $x_{0}$ is irrational.
Let a sequence of integers $a_0, a_1, a_2, \cdots, a_{2010}$ such that $a_0 = 1$ and $2011$ divides $a_{k-1}a_k - k$ for all $k = 1, 2, \cdots, 2010$. Prove that $2011$ divides $a_{2010} + 1$.
Define the sequence $s_0$, $s_1$, $s_2$,$ . . .$ by $s_0 = 0$ and $s_n = 3s_{n-1}+2$ for $n \ge 1$. The monic polynomial $f(x)$ defined as $$f(x) =\frac{1}{s_{2023}} \sum^{32}_{k=0} s_{2023+k}x^{32-k}$$ can be factored uniquely (up to permutation) as the product of $16$ monic quadratic polynomials $p_1$, $p_2$, $....$, $p_{16}$ with real coefficients, where $p_i(x) = x^2 + a_ix + b_i$ for $1\le i \le 16$. Compute the integer $N$ that minimizes
$$\left|N - \sum^{16}_{k=1} (a_k + b_k)\right|.$$
A sequence of positive integers $(a_n)_{n \ge 1}$ is of [i]Fibonacci type[/i] if it satisfies the recursive relation $a_{n + 2} = a_{n + 1} + a_n$ for all $n \ge 1$. Is it possible to partition the set of positive integers into an infinite number of Fibonacci type sequences?
[i]Proposed by Ivan Borsenco[/i]
Let $1 + 1/2 + 1/3 +... + 1/n = a_n/b_n$, where $a_n$ and $b_n$ are relatively prime. Show that there exist infinitely many positive integers $n$, such that $b_{n+1} < b_n$. (8)
Let $n$ be a positive integer. A sequence $(a_0,\ldots,a_n)$ of integers is $\textit{acceptable}$ if it satisfies the following conditions:
[list=a]
[*] $0=|a_0|<|a_1|<\cdots<|a_{n-1}|<|a_n|.$
[*]The sets $\{|a_1-a_0|,|a_2-a_1|,\ldots,|a_{n-1}-a_{n-2}|,|a_n-a_{n-1}|\}$ and $\{1,3,9,\ldots,3^{n-1}\}$ are equal.[/list]
Prove that the number of acceptable sequences of integers is $(n+1)!$.
Sequence $(a_n)$ satisfies that $3a_{n+1}+a_n=4(n\geq1),a_1=9$, let $S_n=\sum_{i=1}^{n}a_i$, then the minumum value of $n$ such that $|S_n-n-6|<\frac{1}{125}$ is
$\text{(A)}5\qquad\text{(B)}6\qquad\text{(C)}7\qquad\text{(D)}8$
The increasing sequence $1,3,4,9,10,12,13\cdots$ consists of all those positive integers which are powers of 3 or sums of distinct powers of 3. Find the $100^{\text{th}}$ term of this sequence.
Define a sequence by $a_0=1$, together with the rules $a_{2n+1}=a_n$ and $a_{2n+2}=a_n+a_{n+1}$ for each integer $n\ge0$. Prove that every positive rational number appears in the set $ \left\{ \tfrac {a_{n-1}}{a_n}: n \ge 1 \right\} = \left\{ \tfrac {1}{1}, \tfrac {1}{2}, \tfrac {2}{1}, \tfrac {1}{3}, \tfrac {3}{2}, \cdots \right\} $.
Does there exist a sequence of positive real numbers $\{a_i\}_{i=1}^{\infty}$ satisfying:
\[
\sum_{i=1}^{n} a_i \geq n^2 \quad \text{and} \quad \sum_{i=1}^{n} a_i^2 \leq n^3 + 2025n
\]
for all positive integers $n$.
Let be two distinct natural numbers $ k_1 $ and $ k_2 $ and a sequence $ \left( x_n \right)_{n\ge 0} $ which satisfies
$$ x_nx_m +k_1k_2\le k_1x_n +k_2x_m,\quad\forall m,n\in\{ 0\}\cup\mathbb{N}. $$
Calculate $ \lim_{n\to\infty}\frac{n!\cdot (-1)^{1+n}\cdot x_n^2}{n^n} . $
Let $a,b$ be integers greater than 2. Prove that there exists a positive integer $k$ and a finite sequence $n_1, n_2, \dots, n_k$ of positive integers such that $n_1 = a$, $n_k = b$, and $n_i n_{i+1}$ is divisible by $n_i + n_{i+1}$ for each $i$ ($1 \leq i < k$).