Found problems: 1239
1971 IMO Shortlist, 10
Prove that we can find an infinite set of positive integers of the from $2^n-3$ (where $n$ is a positive integer) every pair of which are relatively prime.
1961 All-Soviet Union Olympiad, 1
Prove that for any three infinite sequences of natural numbers $(a_n)_{n\ge 1}$, $(b_n)_{n\ge 1}$, $(c_n)_{n\ge 1}$, there exist numbers $p$ and $q$ such that $a_p\ge a_q$, $b_p\ge b_q$ and $c_p\ge c_q$.
2020 EGMO, 6
Let $m > 1$ be an integer. A sequence $a_1, a_2, a_3, \ldots$ is defined by $a_1 = a_2 = 1$, $a_3 = 4$, and for all $n \ge 4$, $$a_n = m(a_{n - 1} + a_{n - 2}) - a_{n - 3}.$$
Determine all integers $m$ such that every term of the sequence is a square.
1985 Tournament Of Towns, (084) T5
Every member of a given sequence, beginning with the second , is equal to the sum of the preceding one and the sum of its digits . The first member equals $1$ . Is there, among the members of this sequence, a number equal to $123456$ ?
(S. Fomin , Leningrad)
2019 Jozsef Wildt International Math Competition, W. 33
Let $0 < \frac{1}{q} \leq \frac{1}{p} < 1$ and $\frac{1}{p}+\frac{1}{q}=1$. Let $u_k$, $v_k$, $a_k$ and $b_k$ be non-negative real sequences such as $u^2_k > a^p_k$ and $v_k > b^q_k$, where $k = 1, 2,\cdots , n$. If $0 < m_1\leq u_k \leq M_1$ and $0 < m_2 \leq v_k \leq M_2$ , then $$\left(\sum \limits_{k=1}^n\left(l^p\left(u_k+v_k\right)^2-\left(a_k+b_k\right)^p\right)\right)^{\frac{1}{p}}\geq \left(\sum \limits_{k=1}^n\left(u_k^2-a_k^p\right)\right)^{\frac{1}{p}}\left(\sum \limits_{k=1}^n\left(v_k^2-b_k^p\right)\right)^{\frac{1}{p}}$$where $$l=\frac{M_1M_2+m_1m_2}{2\sqrt{m_1M_1m_2M_2}}$$
1961 Putnam, B7
Given a sequence $(a_n)$ of non-negative real numbers such that $a_{n+m}\leq a_{n} a_{m} $ for all pairs of positive integers $m$ and $n,$ prove that the sequence $(\sqrt[n]{a_n })$ converges.
2010 Dutch IMO TST, 1
Consider sequences $a_1, a_2, a_3,...$ of positive integers. Determine the smallest possible value of $a_{2010}$ if
(i) $a_n < a_{n+1}$ for all $n\ge 1$,
(ii) $a_i + a_l > a_j + a_k$ for all quadruples $ (i, j, k, l)$ which satisfy $1 \le i < j \le k < l$.
1995 North Macedonia National Olympiad, 1
Let $ a_0 $ be a real number. The sequence $ \{a_n \} $ is given by $ a_ {n + 1} = 3 ^ n-5a_n $, $ n = 0,1,2, \ldots $.
a) Express the general member $ a_n $ through $ a_0 $ and $ n. $
b) Find such $ a_0, $ that $ a_ {n + 1}> a_n, $ for every $ n. $
2013 Abels Math Contest (Norwegian MO) Final, 1b
The sequence $a_1, a_2, a_3,...$ is defined so that $a_1 = 1$ and $a_{n+1} =\frac{a_1 + a_2 + ...+ a_n}{n}+1$ for $n \ge 1$. Show that for every positive real number $b$ we can find $a_k$ so that $a_k < bk$.
2009 VJIMC, Problem 4
Let $(a_n)_{n=1}^\infty$ be a sequence of real numbers. We say that the sequence $(a_n)_{n=1}^\infty$ covers the set of
positive integers if for any positive integer $m$ there exists a positive integer $k$ such that $\sum_{n=1}^\infty a_n^k=m$.
a) Does there exist a sequence of real positive numbers which covers the set of positive integers?
b) Does there exist a sequence of real numbers which covers the set of positive integers?
1995 Singapore Team Selection Test, 1
Let $f(x) = \frac{1}{1+x}$ where $x$ is a positive real number, and for any positive integer $n$,
let $g_n(x) = x + f(x) + f(f(x)) + ... + f(f(... f(x)))$, the last term being $f$ composed with itself $n$ times. Prove that
(i) $g_n(x) > g_n(y)$ if $x > y > 0$.
(ii) $g_n(1) = \frac{F_1}{F_2}+\frac{F_2}{F_3}+...+\frac{F_{n+1}}{F_{n+2}}$ , where $F_1 = F_2 = 1$ and $F_{n+2} = F_{n+1} +F_n$ for $n \ge 1$.
2009 Germany Team Selection Test, 2
Let $ a_1$, $ a_2$, $ \ldots$, $ a_n$ be distinct positive integers, $ n\ge 3$. Prove that there exist distinct indices $ i$ and $ j$ such that $ a_i \plus{} a_j$ does not divide any of the numbers $ 3a_1$, $ 3a_2$, $ \ldots$, $ 3a_n$.
[i]Proposed by Mohsen Jamaali, Iran[/i]
2018 Irish Math Olympiad, 9
The sequence of positive integers $a_1, a_2, a_3, ...$ satisfies $a_{n+1} = a^2_{n} + 2018$ for $n \ge 1$.
Prove that there exists at most one $n$ for which $a_n$ is the cube of an integer.
2006 IMO Shortlist, 3
The sequence $c_{0}, c_{1}, . . . , c_{n}, . . .$ is defined by $c_{0}= 1, c_{1}= 0$, and $c_{n+2}= c_{n+1}+c_{n}$ for $n \geq 0$. Consider the set $S$ of ordered pairs $(x, y)$ for which there is a finite set $J$ of positive integers such that $x=\textstyle\sum_{j \in J}{c_{j}}$, $y=\textstyle\sum_{j \in J}{c_{j-1}}$. Prove that there exist real numbers $\alpha$, $\beta$, and $M$ with the following property: An ordered pair of nonnegative integers $(x, y)$ satisfies the inequality \[m < \alpha x+\beta y < M\] if and only if $(x, y) \in S$.
[i]Remark:[/i] A sum over the elements of the empty set is assumed to be $0$.
2004 Kazakhstan National Olympiad, 6
The sequence of integers $ a_1 $, $ a_2 $, $ \dots $ is defined as follows:
$ a_1 = 1 $ and $ n> 1 $, $ a_ {n + 1} $ is the smallest integer greater than $ a_n $ and such, that $ a_i + a_j \neq 3a_k $ for any $ i, j $ and $ k $ from $ \{1, 2, \dots, n + 1 \} $ are not necessarily different.
Define $ a_ {2004} $.
2016 IFYM, Sozopol, 7
We are given a non-infinite sequence $a_1,a_2…a_n$ of natural numbers. While it is possible, on each turn are chosen two arbitrary indexes $i<j$ such that $a_i \nmid a_j$, and then $a_i$ and $a_j$ are changed with their $gcd$ and $lcm$. Prove that this process is non-infinite and the created sequence doesn’t depend on the made choices.
2014 Taiwan TST Round 3, 5
Let $n$ be a positive integer, and consider a sequence $a_1 , a_2 , \dotsc , a_n $ of positive integers. Extend it periodically to an infinite sequence $a_1 , a_2 , \dotsc $ by defining $a_{n+i} = a_i $ for all $i \ge 1$. If \[a_1 \le a_2 \le \dots \le a_n \le a_1 +n \] and \[a_{a_i } \le n+i-1 \quad\text{for}\quad i=1,2,\dotsc, n, \] prove that \[a_1 + \dots +a_n \le n^2. \]
2019 Bulgaria EGMO TST, 2
The sequence of real numbers $(a_n)_{n\geq 0}$ is such that $a_0 = 1$, $a_1 = a > 2$ and $\displaystyle a_{n+1} = \left(\left(\frac{a_n}{a_{n-1}}\right)^2 -2\right)a_n$ for every positive integer $n$. Prove that $\displaystyle \sum_{i=0}^k \frac{1}{a_i} < \frac{2+a-\sqrt{a^2-4}}{2}$ for every positive integer $k$.
2014 Balkan MO Shortlist, A3
$\boxed{A3}$The sequence $a_1,a_2,a_3,...$ is defined by $a_1=a_2=1,a_{2n+1}=2a_{2n}-a_n$ and $a_{2n+2}=2a_{2n+1}$ for $n\in{N}.$Prove that if $n>3$ and $n-3$ is divisible by $8$ then $a_n$ is divisible by $5$
2024 IMC, 4
Let $g$ and $h$ be two distinct elements of a group $G$, and let $n$ be a positive integer. Consider a sequence $w=(w_1,w_2,\dots)$ which is not eventually periodic and where each $w_i$ is either $g$ or $h$. Denote by $H$ the subgroup of $G$ generated by all elements of the form $w_kw_{k+1}\dotsc w_{k+n-1}$ with $k \ge 1$. Prove that $H$ does not depend on the choice of the sequence $w$ (but may depend on $n$).
1987 Bulgaria National Olympiad, Problem 4
The sequence $(x_n)_{n\in\mathbb N}$ is defined by $x_1=x_2=1$, $x_{n+2}=14x_{n+1}-x_n-4$ for each $n\in\mathbb N$. Prove that all terms of this sequence are perfect squares.
2021 Serbia National Math Olympiad, 6
A finite sequence of natural numbers $a_1, a_2, \dots, a_n$ is given. A sub-sequence $a_{k+1}, a_{k+2}, \dots, a_l$ will be called a [i]repetition[/i] if there exists a natural number $p\leq \frac{l-k}2$ such that $a_i=a_{i+p}$ for $k+1\leq i\leq l-p$, but $a_i\neq a_{i+p}$ for $i=k$ (if $k>0$) and $i=l-p+1$ (if $l<n$).
Show that the sequence contains less than $n$ repetitions.
2019 Simon Marais Mathematical Competition, A4
Suppose $x_1,x_2,x_3,\dotsc$ is a strictly decreasing sequence of positive real numbers such that the series $x_1+x_2+x_3+\cdots$ diverges.
Is it necessary true that the series $\sum_{n=2}^{\infty}{\min \left\{ x_n,\frac{1}{n\log (n)}\right\} }$ diverges?
2022 IMO Shortlist, N3
Let $a > 1$ be a positive integer and $d > 1$ be a positive integer coprime to $a$. Let $x_1=1$, and for $k\geq 1$, define
$$x_{k+1} = \begin{cases}
x_k + d &\text{if } a \text{ does not divide } x_k \\
x_k/a & \text{if } a \text{ divides } x_k
\end{cases}$$
Find, in terms of $a$ and $d$, the greatest positive integer $n$ for which there exists an index $k$ such that $x_k$ is divisible by $a^n$.
1990 Tournament Of Towns, (271) 5
The numerical sequence $\{x_n\}$ satisfies the condition $$x_{n+1}=|x_n|-x_{n-1}$$ for all $n > 1$. Prove that the sequence is periodic with period $9$, i.e. for any $n > 1$ we have $x_n = x_{n+9}$.
(M Kontsevich, Moscow)