6
votes
0answers
64 views

Problem 6 - IMO 1985

For every real number $x_1$ construct the sequence $x_1,x_2,x_3,\ldots$ by setting $x_{n+1}=x_n(x_n+\frac{1}{n})$ for each $n \ge 1$. Prove that there exists exactly one value of $x_1$ for which $0 ...
1
vote
0answers
54 views

Short list of the IMO 2003

Let $b$ a integer greater than $5$. For each positive integer $n$, consider the number $$x_n=\underbrace{11\ldots1}_{n-1}\ \underbrace{22\ldots 2}_{n}\ 5$$ written in base $b$. Prove that the ...
1
vote
2answers
135 views

Analysis problem from Romanian Contest - 2 sequences which forms another one

Let $a,b$ be 2 real numbers, and the sequences $(a_n)_{n \geq 1}, (b_n)_{n \geq 1}$ defined by $a_{1}=a$, $b_{1}=b$, $a^2+b^2 <1$ and \begin{cases} ...
5
votes
1answer
141 views

Sequence where the sum of digits of all numbers is 7

BdMO 2014 We define a sequence starting with $a_1=7,a_2=16,\ldots,\,$ such that the sum of digits of all numbers of the sequence is $7$ and if $m>n$,then $a_m>a_n$ i.e. all such numbers are ...
1
vote
1answer
60 views

Eliminating numbers from the sequence $1,2,3,4,5,6,7…400$

BdMO 2014 Let us take the sequence $1,2,3,4,5,6,7....400$ .We are going to remove numbers from the sequence such that the sum of any 2 numbers of the remaining sequence is not divisible by 7.What ...
1
vote
1answer
76 views

Sum $\sum_{k=0}^{2013}2^ka_{k}$

let real sequence $a_{0},a_{1},a_{2},\cdots,a_{n}$,such $$a_{0}=2013,a_{n}=-\dfrac{2013}{n}\sum_{k=0}^{n-1}a_{k},n\ge 1$$ How find this sum $$\sum_{k=0}^{2013}2^ka_{k}$$ My idea: since ...
5
votes
1answer
82 views

Maximum value of the lowest sum in a set of numbers

Last year in a maths contest held in Catalonia called Cangur it was posed the following q├╝estion: We write numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10, in a certain order around a circumference. Then ...
1
vote
1answer
65 views

Sum of an infinite geometric series?

BdMO Nationals 12: Each room of the Magic Castle has exactly one exit door.The rooms are designed such that when you can go from one room to the next one through a door, the second room's ...
10
votes
2answers
107 views

Let $a_k=\frac1{\binom{n}k}$, $b_k=2^{k-n}$. Compute $\sum_{k=1}^n\frac{a_k-b_k}k$

Let $a_k=\frac1{\binom{n}k}$, $b_k=2^{k-n}$. Compute $$\sum_{k=1}^n\frac{a_k-b_k}k$$ By computing some partial sums, the answers are 0. It seems an inductive argument is possible.
14
votes
1answer
130 views

Closed form for $\sum_{n=0}^\infty\frac{\operatorname{Li}_{1/2}\left(-2^{-2^{-n}}\right)}{\sqrt{2^n}}$

Let $$S=\sum_{n=0}^\infty\frac{\operatorname{Li}_{1/2}\left(-2^{-2^{-n}}\right)}{\sqrt{2^n}},\tag1$$ where $\operatorname{Li}_a(z)$ is the polylogarithm. For $a=1/2$ it can be represented as ...
5
votes
1answer
117 views

Putnam Series Question

I'm studying for the Putnam Exam and am a bit confused about how to go about solving this problem. Sum the series $$ \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} \frac{m^2n}{3^m(n3^m + m3^n)}. $$ ...
19
votes
1answer
235 views

How to prove $\sum_{n=1}^\infty\operatorname{arccot}\frac{\sqrt[2^n]2+\cos\frac\pi{2^n}}{\sin\frac\pi{2^n}}=\operatorname{arccot}\frac{\ln2}\pi$?

How can I prove the following identity? $$\sum_{n=1}^\infty\operatorname{arccot}\frac{\sqrt[2^n]2+\cos\frac\pi{2^n}}{\sin\frac\pi{2^n}}=\operatorname{arccot}\frac{\ln2}\pi$$
2
votes
1answer
66 views

Is $\sum_{n=1}^{\infty}n/(b_1 + b_2 + \cdots+ b_n)$ convergent?

Let $\sum_{n=1}^{\infty}a_n$ be a convergent series of positive terms (so $a_i > 0$ for all $i$) and set $b_n = 1/(na_n^2)$ for $n\ge1$. Is $\sum_{n=1}^{\infty}n/(b_1 + b_2 + \cdots + b_n)$ ...
6
votes
1answer
124 views

Define a sequence by $a_1 = 1, a_2 = 1/2$, and $a_{n+2} = a_{n+1} - a_na_{n+1}/2$ for $n$ a positive integer.

Define a sequence by $a_1 = 1, a_2 = 1/2$, and $$a_{n+2} = a_{n+1} - a_na_{n+1}/2$$ for $n$ a positive integer. Find $$\lim_{n\to\infty}na_n$$ if it exists. Well, we can deduce that $\lim a_n=0$ by ...
5
votes
2answers
103 views

$a_1=1,a_{n+1}=\frac{n}{a_n}+\frac{a_n}{n}$. Prove that for $n\ge4$, $\lfloor{a_n^2}\rfloor=n$

Define a sequence $\left\lbrace a_{n}\right\rbrace$ by $\displaystyle{a_{1} = 1\,,\ a_{n + 1} = {n \over a_n} + {a_n \over n}.\quad}$ Prove that for $n \geq 4,\,\,\left\lfloor ...
1
vote
1answer
84 views

Suppose for all $n$, $a_{n+1}\le a_n + \frac1{n^p}$. Find all positive $p$ such that we can guarantee $\{a_n\}$ always converge.

Let $\{a_n\}$ be any sequence of positive real numbers. Suppose for all $n$, $a_{n+1}\le a_n + \frac1{n^p}$. Find all positive $p$ such that we can guarantee $\{a_n\}$ always converge. For example, ...
1
vote
1answer
63 views

Prove $1 + \sum_{i=0}^n(\frac1{x_i}\prod_{j\neq i}(1+\frac1{x_j-x_i}))=\prod_{i=0}^n(1+\frac1{x_i})$

Prove the identity $$1 + \sum_{i=0}^n \left(\frac1{x_i}\prod_{j\neq i} \left(1+\frac1{x_j-x_i} \right) \right)=\prod_{i=0}^n \left(1+\frac1{x_i} \right)$$ and hence deduce the inequality in Problem ...
4
votes
2answers
350 views

Compute $\sum_{j=1}^k\cos^n(j\pi/k)\sin(nj\pi/k)$

Compute the series $\sum_{j=1}^k\cos^n(j\pi/k)\sin(nj\pi/k)$ Hint: the answer is in fact 0
2
votes
1answer
186 views

These two sequences have the same limit

Let $a_1$ and $b_1$ be any two positive numbers, and define $\{ a_n\}$ and $\{ b_n\}$ by $$a_n = \frac{2a_{n-1}b_{n-1}}{a_{n-1}+b_{n-1}},$$ $$b_n = \sqrt{a_{n-1}b_{n-1} }.$$ Prove that the ...
0
votes
1answer
50 views

On a infinite series problem of IMC

In the solution 2 of problem of 2 of IMC 1999 I want to ask why $$\sum_{n=1}^{\infty}\frac{\pi (n)}{n^2}= \sum_{n=1}^{\infty}(\pi (1)+ \pi(2)+\cdots + \pi(n))\left( ...
0
votes
1answer
90 views

Infinite Series (Telescoping?)

$$\sum_{n=0}^\infty \frac{\tan(a/2^n)}{2^n},$$ where $a$ isn't a multiple of $\pi$. I've been going through several telescoping questions, and It seems I have hit a brick wall with this one, any ...
5
votes
1answer
106 views

Existence of $j$ with strange sequence.

I define a sequence $(a_n)$ $$a_n= \begin{cases} 0 &\text{if $\cos{\left ( \dfrac{2^n\pi}{q}\right )}<-\dfrac12$} \\\\ 1 &\text{if $\cos{\left ( \dfrac{2^n\pi}{q}\right )}>-\dfrac12$} ...
4
votes
1answer
118 views

Probability that the first digit of $2^{n}$ is 1

Let $a_{n}$ be the number of terms in the sequence $2^{1},2^{2},\cdots ,2^{n}$ which begins with digit 1. Prove that $$\log2 -\frac{1}{n}<\frac{a_{n}}{n}<\log2\text{ (log base is 10)}$$ ...
2
votes
3answers
77 views

Summation of a finite series

Let $$f(n) = \frac{1}{1} + \frac{1}{2} + \frac{1}{3}+ \frac{1}{4}+...+ \frac{1}{2^n-1} \forall \ n \ \epsilon \ \mathbb{N} $$ If it cannot be summed , are there any approximations to the series ?
1
vote
2answers
65 views

Is $u_n\le(1-a)^n\forall n\in\mathbb{N}$?

Consider the sequence $\{u_n\}$ where $u_0=1,u_1=1-a$ for some $0< a < 1/4$, and $u_{n+2} = u_{n+1}-au_n$. Is $u_n\le(1-a)^n\forall n\in\mathbb{N}$?
5
votes
1answer
55 views

An infinite fraction

Here's a problem I saw on the AoPS twitter. I thought I might as well post it so that it could be discussed and a solution recorded. What is the value of the following? $$\cfrac{4}{{1 + ...
5
votes
1answer
71 views

increasing sequence specific properties

I am doing some olympiad exercises and have difficulties with the following one: Consider a sequence $a_1,a_2,...$ which is strictly monotonically increasing and $a_1,a_2,...\in\mathbb N$ Now I know ...
7
votes
4answers
390 views

Given that $f(1)= 2013,$ find the value of $f(2013)$?

Suppose that $f$ is a function defined on the set of natural numbers such that $$f(1)+ 2^2f(2)+ 3^2f(3)+...+n^2f(n) = n^3f(n)$$ for all positive integers $n$. Given that $f(1)= 2013$, find the value ...
6
votes
1answer
67 views

question about limit and series

consider following hypotheses $ m\in\mathbb N$ $c\in \mathbb C\,$ ,$\, \; a_j\in \mathbb C$ $a_j\in \mathbb C\;$ , $\;|a_j|=1,\;\forall\;1\le j\le m$ if ...
0
votes
3answers
206 views

Find the function that satisfies the following

Let $f: \mathbb{R} \to \mathbb{R}$ inconstant so that $\exists \lim_{x \to +\infty} f(x) $ and for any arithmetical progression $(a_n)$ the sequence $(f(a_n))$ is an arithmetical progression. ...
5
votes
3answers
136 views

Find $\lim_{n\to \infty}\frac{1}{\ln n}\sum_{j,k=1}^{n}\frac{j+k}{j^3+k^3}.$

Find $$\lim_{n\to \infty}\frac{1}{\ln n}\sum_{j,k=1}^{n}\frac{j+k}{j^3+k^3}\;.$$
11
votes
1answer
256 views

Prove that $\frac{\pi}{4}\le\sum_{n=1}^{\infty} \arcsin\left(\frac{\sqrt{n+1}-\sqrt{n}}{n+1}\right)$

Prove that $$\frac{\pi}{4}\le\sum_{n=1}^{\infty} \arcsin\left(\frac{\sqrt{n+1}-\sqrt{n}}{n+1}\right)$$ EDIT: inspired by Michael Hardy's suggestion I got that $$\arcsin ...
3
votes
4answers
178 views

how prove $\sum_{n=1}^\infty\frac{a_n}{b_n+a_n} $is convergent?

Let$a_n,b_n\in\mathbb R$ and $(a_n+b_n)b_n\neq 0\quad \forall n\in \mathbb{N}$. The series $\sum_{n=1}^\infty\frac{a_n}{b_n} $ and $\sum_{n=1}^\infty(\frac{a_n}{b_n})^2 $ are convergent. How to prove ...
1
vote
2answers
138 views

$\limsup\left(\frac{a_1+a_{n+1}}{a_n}\right)^n\ge c$

Let $a_n>0,n\in\mathbb{N}$ be a sequence of positive real numbers. There exists a positive real number $c$ such that $\limsup\left(\frac{a_1+a_{n+1}}{a_n}\right)^n\ge c$ as $n\to\infty$ for all ...
0
votes
1answer
113 views

Convergence of series with modified denominator

Suppose the series with positive terms $\sum_{n=1}^{\infty} a_n$ converges. Let $r_n=\sum_{k=n}^{\infty}a_k$. Prove or disprove that $\sum_{n=1}^{\infty}\frac{a_n}{r_n}$ diverges, and prove or ...
10
votes
2answers
148 views

Does there exist a sequence of real numbers $\{a_n\}$ such that $\sum_na_n^k$ converges for $k=1$ but diverges for every other odd positive integer?

Does there exist a sequence of real numbers $\{a_n\}$ such that $\sum_na_n^k$ converges for $k=1$ but diverges for every other odd positive integer?
15
votes
1answer
240 views

How to compute the series $\sum\limits_{x=0}^\infty\sum\limits_{y=0}^\infty\sum\limits_{z=0}^\infty\frac{1}{2^x(2^{x+y}+2^{x+z}+2^{z+y})}$

How to compute the series $\displaystyle\sum_{x=0}^\infty\sum_{y=0}^\infty\sum_{z=0}^\infty\frac{1}{2^x(2^{x+y}+2^{x+z}+2^{z+y})}$ ? Thanks in advance.
4
votes
1answer
86 views

* if $\sum_{n=1}^\infty u_{n}$ be divergent then $\sum_{n=1}^\infty n u_{n}$ is convergent or divergent *

let $\sum_{n=1}^\infty u_{n}$ be divergent $\sum_{n=1}^\infty n u_{n}$ this series is divergent or convergent? thanks in advance
25
votes
1answer
740 views

A question about series with a strange property.

Does there exist a sequence $\left(a_n\right)_{n\ge1}$ with $a_n < a_{n+1}+a_{n^2}, \forall n=1,2,3,\ldots$ such that the series $\displaystyle{\sum_{n=1}^{\infty}a_n}$ converges? This is the ...
2
votes
1answer
75 views

Sequence $a_k=1-\frac{\lambda^2}{4a_{k-1}},\ k=2,3,\ldots,n$.

Consider the sequence $a_1, a_2,\ldots,a_n$ with $a_1=1$ and defined recursively by $$a_k=1-\frac{\lambda^2}{4a_{k-1}},\ k=2,3,\ldots,n.$$ Find $\lambda>1$ such that $a_n=0$. The answer is ...
21
votes
1answer
596 views

Does there exist a sequence $\{a_n\}_{n\ge1}$ with $a_n < a_{n+1}+a_{n^2}$ such that $\sum_{n=1}^{\infty}a_n$ converges?

Does there exist a sequence $\{a_n\}_{n\ge1}$ with $a_n < a_{n+1}+a_{n^2}$ such that $\sum_{n=1}^{\infty}a_n$ converges? Does there exist a sequence with the same property but with each term ...
7
votes
4answers
412 views

Evaluate the sum $\sum_{k=0}^{\infty}\frac{1}{(4k+1)(4k+2)(4k+3)(4k+4)}$?

Evaluate the series $$\sum_{k=0}^{\infty}\frac{1}{(4k+1)(4k+2)(4k+3)(4k+4)}=?$$ Can you help me ? This is a past contest problem.
66
votes
6answers
2k views

Contest problem about convergent series

The following is probably a math contest problem. I have been unable to locate the original source. Suppose that $\{a_i\}$ is a set of positive real numbers and the series $$\sum_{n = 1}^\infty ...
1
vote
1answer
82 views

Contest problem - region in which a sequence converges

(Please do not offer a solution! I am seeking a hint; more on that below.) Problem: For any pair $(x, y)$ of real numbers, define the sequence $a_n (x, y)$ as follows: $$a_0 (x,y) = x$$ ...
2
votes
2answers
193 views

A question with the sequence $e_{n}=\left(1+\frac{1}{n}\right)^{n}$

Prove that $a$) the following sequence is increasing $$e_{n}=\left(1+\frac{1}{n}\right)^{n},\quad n\ge1;$$ $b$) the inequality below holds $$e_{n} \leq3,\quad n\ge1.$$
11
votes
0answers
508 views

Quadratic Recurrence Relation

The following sequence appeared in IMC 2012 (a math competition): $$a_1 = \frac{1}{2}, \qquad a_{n+1} = \frac{n a_n^2}{1+(n+1)a_n}$$ I am trying to find an explicit formula for the sequence. It ...
4
votes
1answer
152 views

An infinite series of a product of three logarithms

I was told this very interesting question today, and despite my efforts I did not manage to get very far. Evaluate $$\sum_{n=1}^\infty \log \left(1+\frac{1}{n}\right)\log ...
2
votes
1answer
142 views

Explicit formula for sequence with parity-based recursion

How do we find an explicit formula for the sequence $(a_i)_{i=1}^\infty$ in terms of $a_1=C$ if $$a_{i+1}=\begin{cases} a_i-13 & i \text{ even}, \\ 2a_i & i \text{ odd}.\end{cases}\quad i\ge2 ...
1
vote
1answer
278 views

Just wondering what this imo problem is asking and how to solve

Just wondering what this imo problem is asking(it looks simple but i don't understand what's important in the question) and how to solve: Suppose that $s_1,s_2,s_3,\ldots$ is a strictly increasing ...
0
votes
1answer
255 views

How can I find all increasing sequences $\{a_i\}_{i=1}^{\infty}$ such that $d(x_1+x_2+\cdots+x_k)=d(a_{x_{1}}+a_{x_{2}}+\cdots + a_{x_{k}})$?

How can one find all increasing sequences $\{a_i\}_{i=1}^{\infty}$ such that $$d(x_1+x_2+\cdots+x_k)=d(a_{x_{1}}+a_{x_{2}}+\cdots + a_{x_{k}}),$$ holds for all $k$-tuples $(x_1,x_2,\cdots,x_k)$ of ...