Found problems: 963
2022 Iran MO (3rd Round), 2
For two rational numbers $r,s$ we say:$$r\mid s$$whenever there exists $k\in\mathbb{Z}$ such that:$$s=kr$$
${(a_n)}_{n\in\mathbb{N}}$ is an increasing sequence of pairwise coprime natural numbers and ${(b_n)}_{n\in\mathbb{N}}$ is a sequence of distinct natural numbers. Assume that for all $n\in\mathbb{N}$ we have:
$$\sum_{i=1}^{n}\frac{1}{a_i}\mid\sum_{i=1}^{n}\frac{1}{b_i}$$
Prove that [b]for all[/b] $n\in\mathbb{N}$ we have: $a_n=b_n$.
1985 IMO Longlists, 22
The positive integers $x_1, \cdots , x_n$, $n \geq 3$, satisfy $x_1 < x_2 <\cdots< x_n < 2x_1$. Set $P = x_1x_2 \cdots x_n.$ Prove that if $p$ is a prime number, $k$ a positive integer, and $P$ is divisible by $pk$, then $\frac{P}{p^k} \geq n!.$
1993 Czech And Slovak Olympiad IIIA, 4
The sequence ($a_n$) of natural numbers is defined by $a_1 = 2$ and $a_{n+1}$ equals the sum of tenth powers of the decimal digits of $a_n$ for all $n \ge 1$. Are there numbers which appear twice in the sequence ($a_n$)?
OIFMAT III 2013, 10
Prove that the sequence defined by:
$$ y_ {n + 1} = \frac {1} {2} (3y_ {n} + \sqrt {5y_ {n} ^ {2} -4}) , \,\, \forall n \ge 0$$ with $ y_ {0} = 1$ consists only of integers.
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 $1<\alpha <2$.
2009 Kyiv Mathematical Festival, 5
The sequence of positive integers $\{a_n, n\ge 1\}$ is such that $a_n\le a_{n+1}\le a_n+5$ and $a_n$ is divisible by $n$ for all $n \ge 1$. What are the possible values of $a_1$?
1985 IMO Longlists, 63
Let $x_n = \sqrt[2]{2+\sqrt[3]{3+\cdots+\sqrt[n]{n}}}.$ Prove that
\[x_{n+1}-x_n <\frac{1}{n!} \quad n=2,3,\cdots\]
2021 Ukraine National Mathematical Olympiad, 7
The sequence $a_1,a_2, ..., a_{2n}$ of integers is such that each number occurs in no more than $n$ times. Prove that there are two strictly increasing sequences of indices $b_1,b_2, ..., b_{n}$ and $c_1,c_2, ..., c_{n}$ are such that every positive integer from the set $\{1,2,...,2n\}$ occurs exactly in one of these two sequences, and for each $1\le i \le n$ is true the condition $a_{b_i} \ne a_{c_i}$
.
(Anton Trygub)
2006 QEDMO 3rd, 10
Define a sequence $\left( a_{n}\right) _{n\in\mathbb{N}}$ by $a_{1}=a_{2}=a_{3}=1$ and $a_{n+1}=\dfrac{a_{n}^{2}+a_{n-1}^{2}}{a_{n-2}}$ for every integer $n\geq3$. Show that all elements $a_{i}$ of this sequence are integers.
(L. J. Mordell and apparently Dana Scott, see also http://oeis.org/A064098)
1991 Romania Team Selection Test, 2
The sequence ($a_n$) is defined by $a_1 = a_2 = 1$ and $a_{n+2 }= a_{n+1} +a_n +k$, where $k$ is a positive integer.
Find the least $k$ for which $a_{1991}$ and $1991$ are not coprime.
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]
2004 Switzerland Team Selection Test, 8
Let $m$ be a fixed integer greater than $1$. The sequence $x_0$, $x_1$, $x_2$, $\ldots$ is defined as follows:
\[x_i = \begin{cases}2^i&\text{if }0\leq i \leq m - 1;\\\sum_{j=1}^mx_{i-j}&\text{if }i\geq m.\end{cases}\]
Find the greatest $k$ for which the sequence contains $k$ consecutive terms divisible by $m$ .
[i]Proposed by Marcin Kuczma, Poland[/i]
V Soros Olympiad 1998 - 99 (Russia), 10.6
Find the formula for the general term of the sequence an, for which $a_1 = 1$, $a_2 = 3$, $a_{n+1} = 3a_n-2a_{n-1}$ (you need to express an in terms of $n$).
1994 IMO Shortlist, 1
Let $ a_{0} \equal{} 1994$ and $ a_{n \plus{} 1} \equal{} \frac {a_{n}^{2}}{a_{n} \plus{} 1}$ for each nonnegative integer $ n$. Prove that $ 1994 \minus{} n$ is the greatest integer less than or equal to $ a_{n}$, $ 0 \leq n \leq 998$
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.
2012 Balkan MO Shortlist, N1
A sequence $(a_n)_{n=1}^{\infty}$ of positive integers satisfies the condition $a_{n+1} = a_n +\tau (n)$ for all positive integers $n$ where $\tau (n)$ is the number of positive integer divisors of $n$. Determine whether two consecutive terms of this sequence can be perfect squares.
Oliforum Contest V 2017, 10
Let $(x_n)_{n\in Z}$ and $(y_n)_{n\in Z}$ be two sequences of integers such that $|x_{n+2} - x_n| \le 2$ and $x_n + x_m = y_{n^2+m^2}$ for all $n, m \in Z$. Show that the sequence of $x_n$s takes at most $6$ distinct values.
(Paolo Leonetti)
2024 Francophone Mathematical Olympiad, 4
Find all integers $n \ge 2$ for which there exists $n$ integers $a_1,a_2,\dots,a_n \ge 2$ such that for all indices $i \ne j$, we have $a_i \mid a_j^2+1$.
2024 Korea Winter Program Practice Test, Q3
Consider any sequence of real numbers $a_0$, $a_1$, $\cdots$. If, for all pairs of nonnegative integers $(m, s)$, there exists some integer $n \in [m+1, m+2024(s+1)]$ satisfying $a_m+a_{m+1}+\cdots+a_{m+s}=a_n+a_{n+1}+\cdots+a_{n+s}$, say that this sequence has [i]repeating sums[/i]. Is a sequence with repeating sums always eventually periodic?
2017 Bundeswettbewerb Mathematik, 1
For which integers $n \geq 4$ is the following procedure possible? Remove one number of the integers $1,2,3,\dots,n+1$ and arrange them in a sequence $a_1,a_2,\dots,a_n$ such that of the $n$ numbers \[ |a_1-a_2|,|a_2-a_3|,\dots,|a_{n-1}-a_n|,|a_n-a_1| \] no two are equal.
2020 AMC 10, 21
There exists a unique strictly increasing sequence of nonnegative integers $a_1 < a_2 < \dots < a_k$ such that \[\frac{2^{289}+1}{2^{17}+1} = 2^{a_1} + 2^{a_2} + \dots + 2^{a_k}.\] What is $k?$
$\textbf{(A) } 117 \qquad \textbf{(B) } 136 \qquad \textbf{(C) } 137 \qquad \textbf{(D) } 273 \qquad \textbf{(E) } 306$
2008 Germany Team Selection Test, 1
Let $ n > 1$ be an integer. Find all sequences $ a_1, a_2, \ldots a_{n^2 \plus{} n}$ satisfying the following conditions:
\[ \text{ (a) } a_i \in \left\{0,1\right\} \text{ for all } 1 \leq i \leq n^2 \plus{} n;
\]
\[ \text{ (b) } a_{i \plus{} 1} \plus{} a_{i \plus{} 2} \plus{} \ldots \plus{} a_{i \plus{} n} < a_{i \plus{} n \plus{} 1} \plus{} a_{i \plus{} n \plus{} 2} \plus{} \ldots \plus{} a_{i \plus{} 2n} \text{ for all } 0 \leq i \leq n^2 \minus{} n.
\]
[i]Author: Dusan Dukic, Serbia[/i]
1996 Czech And Slovak Olympiad IIIA, 1
A sequence $(G_n)_{n=0}^{\infty}$ satisfies $G(0) = 0$ and $G(n) = n-G(G(n-1))$ for each $n \in N$. Show that
(a) $G(k) \ge G(k -1)$ for every $k \in N$;
(b) there is no integer $k$ for which $G(k -1) = G(k) = G(k +1)$.
2017 USA TSTST, 2
Ana and Banana are playing a game. First Ana picks a word, which is defined to be a nonempty sequence of capital English letters. (The word does not need to be a valid English word.) Then Banana picks a nonnegative integer $k$ and challenges Ana to supply a word with exactly $k$ subsequences which are equal to Ana's word. Ana wins if she is able to supply such a word, otherwise she loses.
For example, if Ana picks the word "TST", and Banana chooses $k=4$, then Ana can supply the word "TSTST" which has 4 subsequences which are equal to Ana's word.
Which words can Ana pick so that she wins no matter what value of $k$ Banana chooses?
(The subsequences of a string of length $n$ are the $2^n$ strings which are formed by deleting some of its characters, possibly all or none, while preserving the order of the remaining characters.)
[i]Proposed by Kevin Sun
1980 IMO Shortlist, 13
Given three infinite arithmetic progressions of natural numbers such that each of the numbers 1,2,3,4,5,6,7 and 8 belongs to at least one of them, prove that the number 1980 also belongs to at least one of them.