Found problems: 1239
2000 Saint Petersburg Mathematical Olympiad, 11.3
Every month a forester Ermolay has planted 2000 trees along a fence. On every tree, he has written how many oaks there are among itself and trees at his right and left. This way a sequence of 2000 numbers was created. How many distinct sequences could the forester Ermolay get? (oak is a certain type of tree)
[I]Proposed by A. Khrabrov, D.Rostovski[/i]
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 :
$$a_{p^mq+r} \equiv a_{p^{m-1}q+r} \pmod{p^m}$$
2005 Morocco TST, 3
Let $a_1,a_2,\ldots$ be an infinite sequence of real numbers, for which there exists a real number $c$ with $0\leq a_i\leq c$ for all $i$, such that \[\left\lvert a_i-a_j \right\rvert\geq \frac{1}{i+j} \quad \text{for all }i,\ j \text{ with } i \neq j. \] Prove that $c\geq1$.
2020 Romania EGMO TST, P1
Let $a$ be a positive integer and $(a_n)_{n\geqslant 1}$ be a sequence of positive integers satisfying $a_n<a_{n+1}\leqslant a_n+a$ for all $n\geqslant 1$. Prove that there are infinitely many primes which divide at least one term of the sequence.
[i]Moldavia Olympiad, 1994[/i]
2020 June Advanced Contest, 4
Let \(c\) be a positive real number. Alice wishes to pick an integer \(n\) and a sequence \(a_1\), \(a_2\), \(\ldots\) of distinct positive integers such that \(a_{i} \leq ci\) for all positive integers \(i\) and \[n, \qquad n + a_1, \qquad n + a_1 - a_2, \qquad n + a_1 - a_2 + a_3, \qquad \cdots\] is a sequence of distinct nonnegative numbers. Find all \(c\) such that Alice can fulfil her wish.
Revenge EL(S)MO 2024, 3
Fix a positive integer $n$. Define sequences $a, b, c \in \mathbb{Q}^{n+1}$ by $(a_0, b_0, c_0) = (0, 0, 1)$ and
\[
a_k = (n-k+1) \cdot c_{k-1}, \quad
b_k = \binom nk - c_k - a_k, \quad \text{and} \quad
c_k = \frac{b_{k-1}}{k}
\]
for each integer $1 \leq k \leq n$.
$ $ $ $ $ $ $ $ $ $ Determine for which $n$ it happens that $a, b, c \in \mathbb{Z}^{n+1}$.
Proposed by [i]Jonathan Du[/i]
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.
1980 IMO Longlists, 14
Let $\{x_n\}$ be a sequence of natural numbers such that \[(a) 1 = x_1 < x_2 < x_3 < \ldots; \quad (b) x_{2n+1} \leq 2n \quad \forall n.\] Prove that, for every natural number $k$, there exist terms $x_r$ and $x_s$ such that $x_r - x_s = k.$
2023 Durer Math Competition Finals, 5
$a_1,a_2,\dots,a_k$ and $b_1,b_2,\dots,a_n$ are integer sequences with $a_1=b_1=0$, $a_k=26,b_n=25$, and $k+n=23$. It is known that $a_{i+1}-a_i=2\enspace\text{or}\enspace3$ and $b_{j+1}-b_j=2\enspace\text{or}\enspace3$ for all applicable $i$ and $j$, and the numbers $a_2,\dots,a_k,b_2,\dots,b_n$ are pairwise different. Determine the total number of such pairs of sequences.
2008 Estonia Team Selection Test, 4
Sequence $(G_n)$ is defined by $G_0 = 0, G_1 = 1$ and $G_n = G_{n-1} + G_{n-2} + 1$ for every $n \ge2$. Prove that for every positive integer $m$ there exist two consecutive terms in the sequence that are both divisible by $m$.
2006 Grigore Moisil Urziceni, 1
[b]a)[/b] $ \lim_{n\to\infty } \sum_{j=1}^n\frac{n}{n^2+n+j} =1 $
[b]b)[/b] $ \lim_{n\to\infty } \left( n- \sum_{j=1}^n\frac{n^2}{n^2+n+j} \right) =3/2 $
[i]Cristinel Mortici[/i]
1992 Romania Team Selection Test, 4
Let $A$ be the set of all ordered sequences $(a_1,a_2,...,a_{11})$ of zeros and ones. The elements of $A$ are ordered as follows: The first element is $(0,0,...,0)$, and the $n + 1$−th is obtained from the $n$−th by changing the first component from the right such that the newly obtained sequence was not obtained before. Find the $1992$−th term of the ordered set $A$
1982 All Soviet Union Mathematical Olympiad, 344
Given a sequence of real numbers $a_1, a_2, ... , a_n$. Prove that it is possible to choose some of the numbers providing $3$ conditions:
a) not a triple of successive members is chosen,
b) at least one of every triple of successive members is chosen,
c) the absolute value of chosen numbers sum is not less that one sixth part of the initial numbers' absolute values sum.
2001 Miklós Schweitzer, 2
Let $\alpha \leq -2$ be an integer. Prove that for every pair $(\beta_0, \beta_1)$ of integers there exists a uniquely determined sequence $0\leq q_0, \ldots, q_k<\alpha ^ 2 - \alpha$ of integers, such that $q_k\neq 0$ if $(\beta_0, \beta 1)\neq (0,0)$ and
$$\beta_i=\sum_{j=0}^k q_j(\alpha - i)^j,\text{ for }i=0,1$$
2019 Dutch BxMO TST, 4
Do there exist a positive integer $k$ and a non-constant sequence $a_1, a_2, a_3, ...$ of positive integers such that $a_n = gcd(a_{n+k}, a_{n+k+1})$ for all positive integers $n$?
1998 Bundeswettbewerb Mathematik, 2
Prove that there exists an infinite sequence of perfect squares with the following properties:
(i) The arithmetic mean of any two consecutive terms is a perfect square,
(ii) Every two consecutive terms are coprime,
(iii) The sequence is strictly increasing.
2005 Estonia National Olympiad, 4
A sequence of natural numbers $a_1, a_2, a_3,..$ is called [i]periodic modulo[/i] $n$ if there exists a positive integer $k$ such that, for any positive integer $i$, the terms $a_i$ and $a_{i+k}$ are equal modulo $n$. Does there exist a strictly increasing sequence of natural numbers that
a) is not periodic modulo finitely many positive integers and is periodic modulo all the other positive integers?
b) is not periodic modulo infinitely many positive integers and is periodic modulo infinitely many positive integers?
1997 VJIMC, Problem 2
Let $\alpha\in(0,1]$ be a given real number and let a real sequence $\{a_n\}^\infty_{n=1}$ satisfy the inequality
$$a_{n+1}\le\alpha a_n+(1-\alpha)a_{n-1}\qquad\text{for }n=2,3,\ldots$$Prove that if $\{a_n\}$ is bounded, then it must be convergent.
2004 Alexandru Myller, 1
[b]a)[/b] Let $ \left( x_n \right)_{n\ge 1} $ be a sequence of real numbers having the property that $ \left| x_{n+1} -x_n \right|\leqslant 1/2^n, $ for any $ n\geqslant 1. $
Show that $ \left( x_n \right)_{n\ge 1} $ is convergent.
[b]b)[/b] Create a sequence $ \left( y_n \right)_{n\ge 1} $ of real numbers that has the following properties:
$ \text{(i) } \lim_{n\to\infty } \left( y_{n+1} -y_n \right) = 0 $
$ \text{(ii) } $ is bounded
$ \text{(iii) } $ is divergent
[i]Eugen Popa[/i]
Mathematical Minds 2024, P6
Consider the sequence $a_1, a_2, \dots$ of positive integers such that $a_1=2$ and $a_{n+1}=a_n^4+a_n^3-3a_n^2-a_n+2$, for all $n\geqslant 1$. Prove that there exist infinitely many prime numbers that don't divide any term of the sequence.
[i]Proposed by Pavel Ciurea[/i]
2013 Junior Balkan Team Selection Tests - Romania, 4
For any sequence ($a_1,a_2,...,a_{2013}$) of integers, we call a triple ($i,j, k$) satisfying $1 \le i < j < k \le 2013$ to be [i]progressive [/i] if $a_k-a_j = a_j -a_i = 1$. Determine the maximum number of progressive triples that a sequence of $2013$ integers could have.
2017 Azerbaijan Junior National Olympiad, P5
A student firstly wrote $x=3$ on the board. For each procces, the stutent deletes the number x and replaces it with either $(2x+4)$ or $(3x+8)$ or $(x^2+5x)$. Is this possible to make the number $(20^{17}+2016)$ on the board? \\
(Explain your answer) \\
[hide=Note]This type of the question is well known but I am going to make a collection so, :blush: [/hide]
2010 Ukraine Team Selection Test, 8
Consider an infinite sequence of positive integers in which each positive integer occurs exactly once. Let $\{a_n\}, n\ge 1$ be such a sequence. We call it [i]consistent [/i] if, for an arbitrary natural $k$ and every natural $n ,m$ such that $a_n <a_m$, the inequality $a_{kn} <a _{km}$ also holds. For example, the sequence $a_n = n$ is consistent .
a) Prove that there are consistent sequences other than $a_n = n$.
b) Are there consistent sequences for which $a_n \ne n, n\ge 2$ ?
c) Are there consistent sequences for which $a n \ne n, n\ge 1$ ?
1988 IMO Longlists, 80
The sequence $ \{a_n\}$ of integers is defined by
\[ a_1 \equal{} 2, a_2 \equal{} 7
\]
and
\[ \minus{} \frac {1}{2} < a_{n \plus{} 1} \minus{} \frac {a^2_n}{a_{n \minus{} 1}} \leq \frac {}{}, n \geq 2.
\]
Prove that $ a_n$ is odd for all $ n > 1.$
1967 IMO, 5
Let $a_1,\ldots,a_8$ be reals, not all equal to zero. Let
\[ c_n = \sum^8_{k=1} a^n_k\]
for $n=1,2,3,\ldots$. Given that among the numbers of the sequence $(c_n)$, there are infinitely many equal to zero, determine all the values of $n$ for which $c_n = 0.$