Found problems: 30
2005 Junior Tuymaada Olympiad, 8
The sequence of natural numbers is based on the following rule: each term, starting with the second, is obtained from the previous addition works of all its various simple divisors (for example, after the number $12$ should be the number $18$, and after the number $125$ , the number $130$).
Prove that any two sequences constructed in this way have a common member.
2021 Lotfi Zadeh Olympiad, 2
Let $a_1, a_2,\cdots , a_n$ and $b_1, b_2,\cdots , b_n$ be (not necessarily distinct) positive integers. We continue the sequences as follows: For every $i>n$, $a_i$ is the smallest positive integer which is not among $b_1, b_2,\cdots , b_{i-1}$, and $b_i$ is the smallest positive integer which is not among $a_1, a_2,\cdots , a_{i-1}$. Prove that there exists $N$ such that for every $i>N$ we have $a_i=b_i$ or for every $i>N$ we have $a_{i+1}=a_i$.
2004 India Regional Mathematical Olympiad, 3
Let $\alpha$ and $\beta$ be the roots of the equation $x^2 + mx -1 = 0$ where $m$ is an odd integer. Let $\lambda _n = \alpha ^n + \beta ^n , n \geq 0$
Prove that
(A) $\lambda _n$ is an integer
(B) gcd ( $\lambda _n , \lambda_{n+1}$) = 1 .
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 $a_1$?
2012 Balkan MO Shortlist, N2
Let the sequences $(a_n)_{n=1}^{\infty}$ and $(b_n)_{n=1}^{\infty}$ satisfy $a_0 = b_0 = 1, a_n = 9a_{n-1} -2b_{n-1}$ and $b_n = 2a_{n-1} + 4b_{n-1}$ for all positive integers $n$. Let $c_n = a_n + b_n$ for all positive integers $n$.
Prove that there do not exist positive integers $k, r, m$ such that $c^2_r = c_kc_m$.
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?
2016 Thailand Mathematical Olympiad, 5
given $p_1,p_2,...$ be a sequence of integer and $p_1=2$,
for positive integer $n$, $p_{n+1}$ is the least prime factor of $np_1^{1!}p_2^{2!}...p_n^{n!}+1 $
prove that all primes appear in the sequence
(Proposed by Beatmania)
2000 Moldova Team Selection Test, 9
The sequence $x_{n}$ is de fined by:
$x_{0}=1, x_{1}=0, x_{2}=1,x_{3}=1, x_{n+3}=\frac{(n^2+n+1)(n+1)}{n}x_{n+2}+(n^2+n+1)x_{n+1}-\frac{n+1}{n}x_{n} (n=1,2,3..)$
Prove that all members of the sequence are perfect squares.
1999 Kazakhstan National Olympiad, 6
In a sequence of natural numbers $ a_1 $, $ a_2 $, $ \dots $, $ a_ {1999} $, $ a_n-a_ {n-1} -a_ {n-2} $ is divisible by $ 100 (3 \leq n \leq 1999) $. It is known that $ a_1 = 19$ and $ a_2 = 99$. Find the remainder of $ a_1 ^ 2 + a_2 ^ 2 + \dots + a_ {1999} ^ 2 $ by $8$.
2020 Canadian Mathematical Olympiad Qualification, 5
We define the following sequences:
• Sequence $A$ has $a_n = n$.
• Sequence $B$ has $b_n = a_n$ when $a_n \not\equiv 0$ (mod 3) and $b_n = 0$ otherwise.
• Sequence $C$ has $c_n =\sum_{i=1}^{n} b_i$
.• Sequence $D$ has $d_n = c_n$ when $c_n \not\equiv 0$ (mod 3) and $d_n = 0$ otherwise.
• Sequence $E$ has $e_n =\sum_{i=1}^{n}d_i$
Prove that the terms of sequence E are exactly the perfect cubes.
2025 Turkey Team Selection Test, 9
Let \(n\) be a positive integer. For every positive integer $1 \leq k \leq n$ the sequence ${\displaystyle {\{ a_{i}+ki\}}_{i=1}^{n }}$ is defined, where $a_1,a_2, \dots ,a_n$ are integers. Among these \(n\) sequences, for at most how many of them does all the elements of the sequence give different remainders when divided by \(n\)?
2017 Federal Competition For Advanced Students, P2, 3
Let $(a_n)_{n\ge 0}$ be the sequence of rational numbers with $a_0 = 2016$ and $a_{n+1} = a_n + \frac{2}{a_n}$ for all $n \ge 0$.
Show that the sequence does not contain a square of a rational number.
Proposed by Theresia Eisenkölbl
2014 Contests, 3
The sequence $(a_n)$ is defined with the recursion $a_{n + 1} = 5a^6_n + 3a^3_{n-1} + a^2_{n-2}$ for $n\ge 2$ and the set of initial values $\{a_0, a_1, a_2\} = \{2013, 2014, 2015\}$. (That is, the initial values are these three numbers in any order.)
Show that the sequence contains no sixth power of a natural number.
2022 Saudi Arabia IMO TST, 1
Let $(a_n)$ be the integer sequence which is defined by $a_1= 1$ and
$$ a_{n+1}=a_n^2 + n \cdot a_n \,\, , \,\, \forall n \ge 1.$$
Let $S$ be the set of all primes $p$ such that there exists an index $i$ such that $p|a_i$.
Prove that the set $S$ is an infinite set and it is not equal to the set of all primes.
2005 Miklós Schweitzer, 2
Let $(a_{n})_{n \ge 1}$ be a sequence of integers satisfying the inequality \[ 0\le a_{n-1}+\frac{1-\sqrt{5}}{2}a_{n}+a_{n+1} <1 \] for all $n \ge 2$. Prove that the sequence $(a_{n})$ is periodic.
Any Hints or Sols for this hard problem?? :help:
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)
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).
2015 Bulgaria National Olympiad, 3
The sequence $a_1, a_2,...$ is de fined by the equalities $a_1 = 2, a_2 = 12$ and $a_{n+1} = 6a_n-a_{n-1}$ for every positive integer $n \ge 2$. Prove that no member of this sequence is equal to a perfect power (greater than one) of a positive integer.
1999 Bundeswettbewerb Mathematik, 2
The sequences $(a_n)$ and $(b_n)$ are defined by $a_1 = b_1 = 1$ and $a_{n+1} = a_n +b_n, b_{n+1} = a_nb_n$ for $n = 1,2,...$ Show that every two distinct terms of the sequence $(a_n)$ are coprime
2007 Korea Junior Math Olympiad, 1
A sequence $a_1,a_2,...,a_{2007}$ where $a_i \in\{2,3\}$ for $i = 1,2,...,2007$ and an integer sequence $x_1,x_2,...,x_{2007}$ satis fies the following: $a_ix_i + x_{i+2 }\equiv 0$ ($mod 5$) , where the indices are taken modulo $2007$. Prove that $x_1,x_2,...,x_{2007}$ are all multiples of $5$.
1997 Belarusian National Olympiad, 2
A sequence $(a_n)_{-\infty}^{-\infty}$ of zeros and ones is given. It is known that $a_n = 0$ if and only if $a_{n-6} + a_{n-5} +...+ a_{n-1}$ is a multiple of $3$, and not all terms of the sequence are zero. Determine the maximum possible number of zeros among $a_0,a_1,...,a_{97}$.
2018 Peru IMO TST, 10
For each positive integer $m> 1$, let $P (m)$ be the product of all prime numbers that divide $m$.
Define the sequence $a_1, a_2, a_3,...$ as followed:
$a_1> 1$ is an arbitrary positive integer,
$a_{n + 1} = a_n + P (a_n)$ for each positive integer $n$.
Prove that there exist positive integers $j$ and $k$ such that $a_j$ is the product of the first $k$ prime numbers.
2016 Saint Petersburg Mathematical Olympiad, 1
In the sequence of integers $(a_n)$, the sum $a_m + a_n$ is divided by $m + n$ with any different $m$ and $n$. Prove that $a_n$ is a multiple of $n$ for any $n$.
2017 Iran MO (3rd round), 2
Consider a sequence $\{a_i\}^\infty_{i\ge1}$ of positive integers. For all positvie integers $n$ prove that there exists infinitely many positive integers $k$ such that there is no pair $(m,t)$ of positive integers where $m>n$ and
$$kn+a_n=tm(m+1)+a_m$$
2014 Belarus Team Selection Test, 2
Find all sequences $(a_n)$ of positive integers satisfying the equality $a_n=a_{a_{n-1}}+a_{a_{n+1}}$
a) for all $n\ge 2$
b) for all $n \ge 3$
(I. Gorodnin)