Found problems: 963
1998 Belarus Team Selection Test, 3
Let $ R_1,R_2, \ldots$ be the family of finite sequences of positive integers defined by the following rules: $ R_1 \equal{} (1),$ and if $ R_{n - 1} \equal{} (x_1, \ldots, x_s),$ then
\[ R_n \equal{} (1, 2, \ldots, x_1, 1, 2, \ldots, x_2, \ldots, 1, 2, \ldots, x_s, n).\]
For example, $ R_2 \equal{} (1, 2),$ $ R_3 \equal{} (1, 1, 2, 3),$ $ R_4 \equal{} (1, 1, 1, 2, 1, 2, 3, 4).$ Prove that if $ n > 1,$ then the $ k$th term from the left in $ R_n$ is equal to 1 if and only if the $ k$th term from the right in $ R_n$ is different from 1.
2020 Jozsef Wildt International Math Competition, W5
Let $(a_n)_{n\ge1}$ and $(b_n)_{n\ge1}$ be positive real sequences such that
$$\lim_{n\to\infty}\frac{a_{n+1}-a_n}n=a\in\mathbb R^*_+\enspace\text{and}\enspace\lim_{n\to\infty}\frac{b_{n+1}}{nb_n}=b\in\mathbb R^*_+$$
Compute
$$\lim_{n\to\infty}\left(\frac{a_{n+1}}{\sqrt[n+1]{b_{n+1}}}-\frac{a_n}{\sqrt[n]{b_n}}\right)$$
[i]Proposed by D.M. Bătinețu-Giurgiu and Neculai Stanciu[/i]
1993 Romania Team Selection Test, 1
Consider the sequence $z_n = (1+i)(2+i)...(n+i)$.
Prove that the sequence $Im$ $z_n$ contains infinitely many positive and infinitely many negative numbers.
1999 Tournament Of Towns, 5
Tireless Thomas and Jeremy construct a sequence. At the beginning there is one positive integer in the sequence. Then they successively write new numbers in the sequence in the following way: Thomas obtains the next number by adding to the previous number one of its (decimal) digits, while Jeremy obtains the next number by subtracting from the previous number one of its digits. Prove that there is a number in this sequence which will be repeated at least $100$ times.
(A Shapovalov)
1977 IMO Shortlist, 11
Let $n$ be an integer greater than $1$. Define
\[x_1 = n, y_1 = 1, x_{i+1} =\left[ \frac{x_i+y_i}{2}\right] , y_{i+1} = \left[ \frac{n}{x_{i+1}}\right], \qquad \text{for }i = 1, 2, \ldots\ ,\]
where $[z]$ denotes the largest integer less than or equal to $z$. Prove that
\[ \min \{x_1, x_2, \ldots, x_n \} =[ \sqrt n ]\]
2020 Federal Competition For Advanced Students, P2, 3
Let $a$ be a fixed positive integer and $(e_n)$ the sequence, which is defined by $e_0=1$ and
$$ e_n=a + \prod_{k=0}^{n-1} e_k$$
for $n \geq 1$.
Prove that
(a) There exist infinitely many prime numbers that divide one element of the sequence.
(b) There exists one prime number that does not divide an element of the sequence.
(Theresia Eisenkölbl)
1983 Putnam, B4
[b]Problem.[/b] Let $f:\mathbb{R}_0^+\rightarrow\mathbb{R}_0^+$ be a function defined as $$f(n)=n+\lfloor\sqrt{n}\rfloor~\forall~n\in\mathbb{R}_0^+.$$ Prove that for any positive integer $m,$ the sequence $$m,f(m),f(f(m)),f(f(f(m))),\ldots$$ contains a perfect square.
2021 Saudi Arabia JBMO TST, 3
Consider the sequence $a_1, a_2, a_3, ...$ defined by $a_1 = 9$ and
$a_{n + 1} = \frac{(n + 5)a_n + 22}{n + 3}$
for $n \ge 1$.
Find all natural numbers $n$ for which $a_n$ is a perfect square of an integer.
2009 IMAR Test, 4
Given any $n$ positive integers, and a sequence of $2^n$ integers (with terms among them), prove there exists a subsequence made of consecutive terms, such that the product of its terms is a perfect square. Also show that we cannot replace $2^n$ with any lower value (therefore $2^n$ is the threshold value for this property).
2019 Saudi Arabia Pre-TST + Training Tests, 3.3
Define sequence of positive integers $(a_n)$ as $a_1 = a$ and $a_{n+1} = a^2_n + 1$ for $n \ge 1$. Prove that there is no index $n$ for which $$\prod_{k=1}^{n} \left(a^2_k + a_k + 1\right)$$ is a perfect square.
2022 District Olympiad, P3
Let $(x_n)_{n\geq 1}$ be the sequence defined recursively as such: \[x_1=1, \ x_{n+1}=\frac{x_1}{n+1}+\frac{x_2}{n+2}+\cdots+\frac{x_n}{2n} \ \forall n\geq 1.\]Consider the sequence $(y_n)_{n\geq 1}$ such that $y_n=(x_1^2+x_2^2+\cdots x_n^2)/n$ for all $n\geq 1.$ Prove that
[list=a]
[*]$x_{n+1}^2<y_n/2$ and $y_{n+1}<(2n+1)/(2n+2)\cdot y_n$ for all $n\geq 1;$
[*]$\lim_{n\to\infty}x_n=0.$
[/list]
2017 Saudi Arabia IMO TST, 3
Find the greatest positive real number $M$ such that for all positive real sequence $(a_n)$ and for all real number $m < M$, it is possible to find some index $n \ge 1$ that satisfies the inequality $a_1 + a_2 + a_3 + ...+ a_n +a_{n+1} > m a_n$.
2022 Brazil Team Selection Test, 3
Let $p$ be an odd prime number and suppose that $2^h \not \equiv 1 \text{ (mod } p\text{)}$ for all integer $1 \leq h \leq p-2$. Let $a$ be an even number such that $\frac{p}{2} < a < p$. Define the sequence $a_0, a_1, a_2, \ldots$ as $$a_0 = a, \qquad a_{n+1} = p -b_n, \qquad n = 0,1,2, \ldots,$$ where $b_n$ is the greatest odd divisor of $a_n$. Show that the sequence is periodic and determine its period.
2022 Kazakhstan National Olympiad, 6
Given an infinite positive integer sequence $\{x_i\}$ such that $$x_{n+2}=x_nx_{n+1}+1$$ Prove that for any positive integer $i$ there exists a positive integer $j$ such that $x_j^j$ is divisible by $x_i^i$.
[i]Remark: Unfortunately, there was a mistake in the problem statement during the contest itself. In the last sentence, it should say "for any positive integer $i>1$ ..."[/i]
1985 IMO Shortlist, 7
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 $\frac{P}{p^k} \geq n!.$
2023 Switzerland Team Selection Test, 10
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 $a^n$.
1979 All Soviet Union Mathematical Olympiad, 273
For every $n$, the decreasing sequence $\{x_k\}$ satisfies a condition $$x_1+x_4/2+x_9/3+...+x_n^2/n \le 1$$
Prove that for every $n$, it also satisfies $$x_1+x_2/2+x_3/3+...+x_n/n\le 3$$
1974 Swedish Mathematical Competition, 1
Let $a_n = 2^{n-1}$ for $n > 0$. Let
\[
b_n = \sum\limits_{r+s \leq n} a_ra_s
\]
Find $b_n-b_{n-1}$, $b_n-2b_{n-1}$ and $b_n$.
2023 Belarusian National Olympiad, 11.5
A sequence of positive integers is given such that the sum of any $6$ consecutive terms does not exceed $11$.
Prove that for any positive integer $a$ in the sequence one can find consecutive terms with sum $a$
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$.
1992 IMO Longlists, 49
Given real numbers $x_i \ (i = 1, 2, \cdots, 4k + 2)$ such that
\[\sum_{i=1}^{4k +2} (-1)^{i+1} x_ix_{i+1} = 4m \qquad ( \ x_1=x_{4k+3} \ )\]
prove that it is possible to choose numbers $x_{k_{1}}, \cdots, x_{k_{6}}$ such that
\[\sum_{i=1}^{6} (-1)^{i} k_i k_{i+1} > m \qquad ( \ x_{k_{1}} = x_{k_{7}} \ )\]
2007 IMO, 1
Real numbers $ a_{1}$, $ a_{2}$, $ \ldots$, $ a_{n}$ are given. For each $ i$, $ (1 \leq i \leq n )$, define
\[ d_{i} \equal{} \max \{ a_{j}\mid 1 \leq j \leq i \} \minus{} \min \{ a_{j}\mid i \leq j \leq n \}
\]
and let $ d \equal{} \max \{d_{i}\mid 1 \leq i \leq n \}$.
(a) Prove that, for any real numbers $ x_{1}\leq x_{2}\leq \cdots \leq x_{n}$,
\[ \max \{ |x_{i} \minus{} a_{i}| \mid 1 \leq i \leq n \}\geq \frac {d}{2}. \quad \quad (*)
\]
(b) Show that there are real numbers $ x_{1}\leq x_{2}\leq \cdots \leq x_{n}$ such that the equality holds in (*).
[i]Author: Michael Albert, New Zealand[/i]
1992 French Mathematical Olympiad, Problem 4
Given $u_0,u_1$ with $0<u_0,u_1<1$, define the sequence $(u_n)$ recurrently by the formula
$$u_{n+2}=\frac12\left(\sqrt{u_{n+1}}+\sqrt{u_n}\right).$$(a) Prove that the sequence $u_n$ is convergent and find its limit.
(b) Prove that, starting from some index $n_0$, the sequence $u_n$ is monotonous.
2022 BMT, 3
Suppose we have four real numbers $a,b,c,d$ such that $a$ is nonzero, $a,b,c$ form a geometric sequence, in that order, and $b,c,d$ form an arithmetic sequence, in that order. Compute the smallest possible value of $\frac{d}{a}.$ (A geometric sequence is one where every succeeding term can be written as the previous term multiplied by a constant, and an arithmetic sequence is one where every succeeeding term can be written as the previous term added to a constant.)
2015 Romania Team Selection Tests, 3
Define a sequence of integers by $a_0=1$ , and $a_n=\sum_{k=0}^{n-1} \binom{n}{k}a_k$ , $n \geq 1$ . Let $m$ be a positive integer , let $p$ be a prime , and let $q$ and $r$ be non-negative integers . Prove that :
$$a_{p^mq+r} \equiv a_{p^{m-1}q+r} \pmod{p^m}$$