Found problems: 88
2023 Thailand TST, 1
A $\pm 1$-[i]sequence[/i] is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and $$\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.$$
2018 Istmo Centroamericano MO, 3
Determine all sequences of integers $a_1, a_2,. . .,$ such that:
(i) $1 \le a_i \le n$ for all $1 \le i \le n$.
(ii) $| a_i - a_j| = | i - j |$ for any $1 \le i, j \le n$
2015 Rioplatense Mathematical Olympiad, Level 3, 2
Let $a , b , c$ positive integers, coprime. For each whole number $n \ge 1$, we denote by $s ( n )$ the number of elements in the set $\{ a , b , c \}$ that divide $n$. We consider $k_1< k_2< k_3<...$ .the sequence of all positive integers that are divisible by some element of $\{ a , b , c \}$. Finally we define the characteristic sequence of $( a , b , c )$ like the succession $ s ( k_1) , s ( k_2) , s ( k_3) , .... $ .
Prove that if the characteristic sequences of $( a , b , c )$ and $( a', b', c')$ are equal, then $a = a', b = b'$ and $c=c'$
1988 All Soviet Union Mathematical Olympiad, 485
The sequence of integers an is given by $a_0 = 0, a_n = p(a_n-1)$, where $p(x)$ is a polynomial whose coefficients are all positive integers. Show that for any two positive integers $m, k$ with greatest common divisor $d$, the greatest common divisor of $a_m$ and $a_k$ is $a_d$.
2023 Belarus Team Selection Test, 4.1
A $\pm 1$-[i]sequence[/i] is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and $$\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.$$
2000 Mexico National Olympiad, 4
Let $a$ and $b$ be positive integers not divisible by $5$. A sequence of integers is constructed as follows: the first term is $5$, and every consequent term is obtained by multiplying its precedent by $a$ and adding $b$. (For example, if $a = 2$ and $b = 4$, the first three terms are $5,14,32$.) What is the maximum possible number of primes that can occur before encoutering the first composite term?
2019 Bulgaria National Olympiad, 3
Find all real numbers $a,$ which satisfy the following condition:
For every sequence $a_1,a_2,a_3,\ldots$ of pairwise different positive integers, for which the inequality $a_n\leq an$ holds for every positive integer $n,$ there exist infinitely many numbers in the sequence with sum of their digits in base $4038,$ which is not divisible by $2019.$
2001 IMO Shortlist, 3
Let $ a_1 \equal{} 11^{11}, \, a_2 \equal{} 12^{12}, \, a_3 \equal{} 13^{13}$, and $ a_n \equal{} |a_{n \minus{} 1} \minus{} a_{n \minus{} 2}| \plus{} |a_{n \minus{} 2} \minus{} a_{n \minus{} 3}|, n \geq 4.$ Determine $ a_{14^{14}}$.
2018 ELMO Shortlist, 3
Consider infinite sequences $a_1,a_2,\dots$ of positive integers satisfying $a_1=1$ and $$a_n \mid a_k+a_{k+1}+\dots+a_{k+n-1}$$ for all positive integers $k$ and $n.$ For a given positive integer $m,$ find the maximum possible value of $a_{2m}.$
[i]Proposed by Krit Boonsiriseth[/i]
1988 All Soviet Union Mathematical Olympiad, 466
Given a sequence of $19$ positive integers not exceeding $88$ and another sequence of $88$ positive integers not exceeding $19$. Show that we can find two subsequences of consecutive terms, one from each sequence, with the same sum.
2004 Iran MO (3rd Round), 13
Suppose $f$ is a polynomial in $\mathbb{Z}[X]$ and m is integer .Consider the sequence $a_i$ like this $a_1=m$ and $a_{i+1}=f(a_i)$ find all polynomials $f$ and alll integers $m$ that for each $i$:
\[ a_i | a_{i+1}\]
2015 Dutch BxMO/EGMO TST, 2
Given are positive integers $r$ and $k$ and an infi nite sequence of positive integers $a_1 \le a_2 \le ...$ such that $\frac{r}{a_r}= k + 1$. Prove that there is a $t$ satisfying $\frac{t}{a_t}=k$.
1991 All Soviet Union Mathematical Olympiad, 551
A sequence of positive integers is constructed as follows. If the last digit of $a_n$ is greater than $5$, then $a_{n+1}$ is $9a_n$. If the last digit of $a_n$ is $5$ or less and an has more than one digit, then $a_{n+1}$ is obtained from $a_n$ by deleting the last digit. If $a_n$ has only one digit, which is $5$ or less, then the sequence terminates. Can we choose the first member of the sequence so that it does not terminate?
2022 Iran MO (3rd Round), 4
$a_1,a_2,\ldots$ is a sequence of [u]nonzero integer[/u] numbers that for all $n\in\mathbb{N}$, if $a_n=2^\alpha k$ such that $k$ is an odd integer and $\alpha$ is a nonnegative integer then: $a_{n+1}=2^\alpha-k$. Prove that if this sequence is periodic, then for all $n\in\mathbb{N}$ we have: $a_{n+2}=a_n$. (The sequence $a_1,a_2,\ldots$ is periodic iff there exists natural number $d$ that for all $n\in\mathbb{N}$ we have: $a_{n+d}=a_n$)
2018 ELMO Problems, 2
Consider infinite sequences $a_1,a_2,\dots$ of positive integers satisfying $a_1=1$ and $$a_n \mid a_k+a_{k+1}+\dots+a_{k+n-1}$$ for all positive integers $k$ and $n.$ For a given positive integer $m,$ find the maximum possible value of $a_{2m}.$
[i]Proposed by Krit Boonsiriseth[/i]
1976 Vietnam National Olympiad, 1
Find all integer solutions to $m^{m+n} = n^{12}, n^{m+n} = m^3$.
1983 Tournament Of Towns, (047) 4
$a_1,a_2,a_3,...$ is a monotone increasing sequence of natural numbers. It is known that for any $k, a_{a_k} = 3k$.
a) Find $a_{100}$.
b) Find $a_{1983}$.
(A Andjans, Riga)
PS. (a) for Juniors, (b) for Seniors
2013 Spain Mathematical Olympiad, 5
Study if it there exist an strictly increasing sequence of integers $0=a_0<a_1<a_2<...$ satisfying the following conditions
$i)$ Any natural number can be written as the sum of two terms of the sequence (not necessarily distinct).
$ii)$For any positive integer $n$ we have $a_n > \frac{n^2}{16}$
2019 Vietnam National Olympiad, Day 1
Let $({{x}_{n}})$ be an integer sequence such that $0\le {{x}_{0}}<{{x}_{1}}\le 100$ and
$${{x}_{n+2}}=7{{x}_{n+1}}-{{x}_{n}}+280,\text{ }\forall n\ge 0.$$
a) Prove that if ${{x}_{0}}=2,{{x}_{1}}=3$ then for each positive integer $n,$ the sum of divisors of the following number is divisible by $24$ $${{x}_{n}}{{x}_{n+1}}+{{x}_{n+1}}{{x}_{n+2}}+{{x}_{n+2}}{{x}_{n+3}}+2018.$$
b) Find all pairs of numbers $({{x}_{0}},{{x}_{1}})$ such that ${{x}_{n}}{{x}_{n+1}}+2019$ is a perfect square for infinitely many nonnegative integer numbers $n.$
2012 BAMO, 3
Let $x_1,x_2,...,x_k$ be a sequence of integers. A rearrangement of this sequence (the numbers in the sequence listed in some other order) is called a [b]scramble[/b] if no number in the new sequence is equal to the number originally in its location. For example, if the original sequence is $1,3,3,5$ then $3,5,1,3$ is a scramble, but $3,3,1,5$ is not.
A rearrangement is called a [b]two-two[/b] if exactly two of the numbers in the new sequence are each exactly two more than the numbers that originally occupied those locations. For example, $3,5,1,3$ is a two-two of the sequence $1,3,3,5$ (the first two values $3$ and $5$ of the new sequence are exactly two more than their original values $1$ and $3$).
Let $n\geq 2$. Prove that the number of scrambles of $1,1,2,3,...,n-1,n$ is equal to the number of two-twos of $1,2,3,...,n,n+1$.
(Notice that both sequences have $n+1$ numbers, but the first one contains two 1s.)
1996 IMO Shortlist, 5
Let $ p,q,n$ be three positive integers with $ p \plus{} q < n$. Let $ (x_{0},x_{1},\cdots ,x_{n})$ be an $ (n \plus{} 1)$-tuple of integers satisfying the following conditions :
(a) $ x_{0} \equal{} x_{n} \equal{} 0$, and
(b) For each $ i$ with $ 1\leq i\leq n$, either $ x_{i} \minus{} x_{i \minus{} 1} \equal{} p$ or $ x_{i} \minus{} x_{i \minus{} 1} \equal{} \minus{} q$.
Show that there exist indices $ i < j$ with $ (i,j)\neq (0,n)$, such that $ x_{i} \equal{} x_{j}$.
2016 Saint Petersburg Mathematical Olympiad, 7
A sequence of $N$ consecutive positive integers is called [i]good [/i] if it is possible to choose two of these numbers so that their product is divisible by the sum of the other $N-2$ numbers. For which $N$ do there exist infinitely many [i]good [/i] sequences?
2023 Thailand Online MO, 9
Find all sequences of positive integers $a_1,a_2,\dots$ such that $$(n^2+1)a_n = n(a_{n^2}+1)$$ for all positive integers $n$.
1994 IMO Shortlist, 6
Define the sequence $ a_1, a_2, a_3, ...$ as follows. $ a_1$ and $ a_2$ are coprime positive integers and $ a_{n \plus{} 2} \equal{} a_{n \plus{} 1}a_n \plus{} 1$. Show that for every $ m > 1$ there is an $ n > m$ such that $ a_m^m$ divides $ a_n^n$. Is it true that $ a_1$ must divide $ a_n^n$ for some $ n > 1$?
2023 Estonia Team Selection Test, 4
A $\pm 1$-[i]sequence[/i] is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and $$\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.$$