Found problems: 1239
1982 IMO Longlists, 17
[b](a)[/b] Find the rearrangement $\{a_1, \dots , a_n\}$ of $\{1, 2, \dots, n\}$ that maximizes
\[a_1a_2 + a_2a_3 + \cdots + a_na_1 = Q.\]
[b](b)[/b] Find the rearrangement that minimizes $Q.$
2019 Brazil Team Selection Test, 3
Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.
2012 Estonia Team Selection Test, 2
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?
2008 China Team Selection Test, 2
The sequence $ \{x_{n}\}$ is defined by $ x_{1} \equal{} 2,x_{2} \equal{} 12$, and $ x_{n \plus{} 2} \equal{} 6x_{n \plus{} 1} \minus{} x_{n}$, $ (n \equal{} 1,2,\ldots)$. Let $ p$ be an odd prime number, let $ q$ be a prime divisor of $ x_{p}$. Prove that if $ q\neq2,3,$ then $ q\geq 2p \minus{} 1$.
2012 Tournament of Towns, 1
Given an infinite sequence of numbers $a_1, a_2, a_3,...$ . For each positive integer $k$ there exists a positive integer $t = t(k)$ such that $a_k = a_{k+t} = a_{k+2t} =...$. Is this sequence necessarily periodic? That is, does a positive integer $T$ exist such that $a_k = a_{k+T}$ for each positive integer k?
1982 All Soviet Union Mathematical Olympiad, 328
Every member, starting from the third one, of two sequences $\{a_n\}$ and $\{b_n\}$ equals to the sum of two preceding ones. First members are: $a_1 = 1, a_2 = 2, b_1 = 2, b_2 = 1$. How many natural numbers are encountered in both sequences (may be on the different places)?
2020 Regional Olympiad of Mexico Southeast, 6
Prove that for all $a, b$ and $x_0$ positive integers, in the sequence $x_1, x_2, x_3, \cdots$ defined by
$$x_{n+1}=ax_n+b, n\geq 0$$
Exist an $x_i$ that is not prime for some $i\geq 1$
2014 India PRMO, 2
The first term of a sequence is $2014$. Each succeeding term is the sum of the cubes of the digits of the previous term. What is the $2014$ th term of the sequence?
1979 IMO Longlists, 54
Consider the sequences $(a_n), (b_n)$ defined by
\[a_1=3, \quad b_1=100 , \quad a_{n+1}=3^{a_n} , \quad b_{n+1}=100^{b_n} \]
Find the smallest integer $m$ for which $b_m > a_{100}.$
1973 Dutch Mathematical Olympiad, 5
An infinite sequence of integers $a_1,a_2,a_3, ...$ is given with $a_1 = 0$ and further holds for every natural number $n$ that $a_{n+1} = a_n - n$ if $a_n \ge n$ and $a_{n+1} = a_n + n$ if $a_n < n$ .
(a) Prove that there are infinitely many numbers in the sequence equal to $0$.
(b) Express in terms of $k$ the ordinal number of the $k^e$ number from the sequence, which is equal to $0$.
2016 Miklós Schweitzer, 4
Prove that there exists a sequence $a(1),a(2),\dots,a(n),\dots$ of real numbers such that
\[
a(n+m)\le a(n)+a(m)+\frac{n+m}{\log (n+m)}
\]
for all integers $m,n\ge 1$, and such that the set $\{a(n)/n:n\ge 1\}$ is everywhere dense on the real line.
[i]Remark.[/i] A theorem of de Bruijn and Erdős states that if the inequality above holds with $f(n + m)$ in place of the last term on the right-hand side, where $f(n)\ge 0$ is nondecreasing and $\sum_{n=2}^\infty f(n)/n^2<\infty$, then $a(n)/n$ converges or tends to $(-\infty)$.
2017 Simon Marais Mathematical Competition, A2
Let $a_1,a_2,a_3,\ldots$ be the sequence of real numbers defined by $a_1=1$ and
$$a_m=\frac1{a_1^2+a_2^2+\ldots+a_{m-1}^2}\qquad\text{for }m\ge2.$$
Determine whether there exists a positive integer $N$ such that
$$a_1+a_2+\ldots+a_N>2017^{2017}.$$
2011 IMO Shortlist, 2
Determine all sequences $(x_1,x_2,\ldots,x_{2011})$ of positive integers, such that for every positive integer $n$ there exists an integer $a$ with \[\sum^{2011}_{j=1} j x^n_j = a^{n+1} + 1\]
[i]Proposed by Warut Suksompong, Thailand[/i]
2004 Austrian-Polish Competition, 9
Given are the sequences
\[ (..., a_{-2}, a_{-1}, a_0, a_1, a_2, ...); (..., b_{-2}, b_{-1}, b_0, b_1, b_2, ...); (..., c_{-2}, c_{-1}, c_0, c_1, c_2, ...)\]
of positive real numbers. For each integer $n$ the following inequalities hold:
\[a_n \geq \frac{1}{2} (b_{n+1} + c_{n-1})\]
\[b_n \geq \frac{1}{2} (c_{n+1} + a_{n-1})\]
\[c_n \geq \frac{1}{2} (a_{n+1} + b_{n-1})\]
Determine $a_{2005}$, $b_{2005}$, $c_{2005}$, if $a_0 = 26, b_0 = 6, c_0 = 2004$.
2018 Swedish Mathematical Competition, 3
Let m be a positive integer. An $m$-[i]pattern [/i] is a sequence of $m$ symbols of strict inequalities. An $m$-pattern is said to be [i]realized [/i] by a sequence of $m + 1$ real numbers when the numbers meet each of the inequalities in the given order. (For example, the $5$-pattern $ <, <,>, < ,>$ is realized by the sequence of numbers $1, 4, 7, -3, 1, 0$.)
Given $m$, which is the least integer $n$ for which there exists any number sequence $x_1,... , x_n$ such that each $m$-pattern is realized by a subsequence $x_{i_1},... , x_{i_{m + 1}}$ with $1 \le i_1 <... < i_{m + 1} \le n$?
1998 Romania National Olympiad, 2
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 $x.$
2019 Jozsef Wildt International Math Competition, W. 7
If $$\Omega_n=\sum \limits_{k=1}^n \left(\int \limits_{-\frac{1}{k}}^{\frac{1}{k}}(2x^{10} + 3x^8 + 1)\cos^{-1}(kx)dx\right)$$Then find $$\Omega=\lim \limits_{n\to \infty}\left(\Omega_n-\pi H_n\right)$$
KoMaL A Problems 2017/2018, A. 712
We say that a strictly increasing positive real sequence $a_1,a_2,\cdots $ is an [i]elf sequence[/i] if for any $c>0$ we can find an $N$ such that $a_n<cn$ for $n=N,N+1,\cdots$. Furthermore, we say that $a_n$ is a [i]hat[/i] if $a_{n-i}+a_{n+i}<2a_n$ for $\displaystyle 1\le i\le n-1$. Is it true that every elf sequence has infinitely many hats?
1968 All Soviet Union Mathematical Olympiad, 100
The sequence $a_1,a_2,a_3,...$, is constructed according to the rule $$a_1=1, a_2=a_1+1/a_1, ... , a_{n+1}=a_n+1/a_n, ...$$
Prove that $a_{100} > 14$.
2022 Macedonian Mathematical Olympiad, Problem 3
The sequence $(a_n)_{n \ge 1}^\infty$ is given by: $a_1=2$ and $a_{n+1}=a_n^2+a_n$ for all $n \ge 1$.
For an integer $m \ge 2$, $L(m)$ denotes the greatest prime divisor of $m$. Prove that there exists some $k$, for which $L(a_k) > 1000^{1000}$.
[i]Proposed by Nikola Velov[/i]
2006 IMO Shortlist, 1
A sequence of real numbers $ a_{0},\ a_{1},\ a_{2},\dots$ is defined by the formula
\[ a_{i \plus{} 1} \equal{} \left\lfloor a_{i}\right\rfloor\cdot \left\langle a_{i}\right\rangle\qquad\text{for}\quad i\geq 0;
\]here $a_0$ is an arbitrary real number, $\lfloor a_i\rfloor$ denotes the greatest integer not exceeding $a_i$, and $\left\langle a_i\right\rangle=a_i-\lfloor a_i\rfloor$. Prove that $a_i=a_{i+2}$ for $i$ sufficiently large.
[i]Proposed by Harmel Nestra, Estionia[/i]
2019 District Olympiad, 1
Let $(a_n)_{n \ge 1}$ be a sequence of positive real numbers such that the sequence $(a_{n+1}-a_n)_{n \ge 1}$ is convergent to a non-zero real number. Evaluate the limit $$ \lim_{n \to \infty} \left( \frac{a_{n+1}}{a_n} \right)^n.$$
2021 Korea National Olympiad, P5
A real number sequence $a_1, \cdots ,a_{2021}$ satisfies the below conditions.
$$a_1=1, a_2=2, a_{n+2}=\frac{2a_{n+1}^2}{a_n+a_{n+1}} (1\leq n \leq 2019)$$
Let the minimum of $a_1, \cdots ,a_{2021}$ be $m$, and the maximum of $a_1, \cdots ,a_{2021}$ be $M$.
Let a 2021 degree polynomial $$P(x):=(x-a_1)(x-a_2) \cdots (x-a_{2021})$$
$|P(x)|$ is maximum in $[m, M]$ when $x=\alpha$. Show that $1<\alpha <2$.
2016 NZMOC Camp Selection Problems, 8
Two positive integers $r$ and $k$ are given as is an infinite sequence of positive integers $a_1 \le a_2 \le a_3 \le ..$ such that $\frac{r}{a_r}= k + 1$. Prove that there is a positive integer $t$ such that $\frac{t}{a_t}= k$.
2004 IMO Shortlist, 4
Let $k$ be a fixed integer greater than 1, and let ${m=4k^2-5}$. Show that there exist positive integers $a$ and $b$ such that the sequence $(x_n)$ defined by \[x_0=a,\quad x_1=b,\quad x_{n+2}=x_{n+1}+x_n\quad\text{for}\quad n=0,1,2,\dots,\] has all of its terms relatively prime to $m$.
[i]Proposed by Jaroslaw Wroblewski, Poland[/i]