Found problems: 1239
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.
2003 Olympic Revenge, 2
Let $x_n$ the sequence defined by any nonnegatine integer $x_0$ and $x_{n+1}=1+\prod_{0 \leq i \leq n}{x_i}$
Show that there exists prime $p$ such that $p\not|x_n$ for any $n$.
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]
2014 India PRMO, 2
The first term of a sequence is $2014$. Each succeeding term is the sum of the cubes of the digits of the previous term. What is the $2014$ th term of the sequence?
2017 China National Olympiad, 1
The sequences $\{u_{n}\}$ and $\{v_{n}\}$ are defined by $u_{0} =u_{1} =1$ ,$u_{n}=2u_{n-1}-3u_{n-2}$ $(n\geq2)$ , $v_{0} =a, v_{1} =b , v_{2}=c$ ,$v_{n}=v_{n-1}-3v_{n-2}+27v_{n-3}$ $(n\geq3)$. There exists a positive integer $N$ such that when $n> N$, we have $u_{n}\mid v_{n}$ . Prove that $3a=2b+c$.
2007 Denmark MO - Mohr Contest, 5
The sequence of numbers $a_0,a_1,a_2,...$ is determined by $a_0 = 0$, and
$$a_n= \begin{cases} 1+a_{n-1} \,\,\, when\,\,\, n \,\,\, is \,\,\, positive \,\,\, and \,\,\, odd \\
3a_{n/2} \,\,\,when \,\,\,n \,\,\,is \,\,\,positive \,\,\,and \,\,\,even\end{cases}$$
How many of these numbers are less than $2007$ ?
2019 Taiwan APMO Preliminary Test, P7
Let positive integer $k$ satisfies $1<k<100$. For the permutation of $1,2,...,100$ be $a_1,a_2,...,a_{100}$, take the minimum $m>k$ such that $a_m$ is at least less than $(k-1)$ numbers of $a_1,a_2,...,a_k$. We know that the number of sequences satisfies $a_m=1$ is $\frac{100!}{4}$. Find the all possible values of $k$.
1977 IMO Shortlist, 13
Let $B$ be a set of $k$ sequences each having $n$ terms equal to $1$ or $-1$. The product of two such sequences $(a_1, a_2, \ldots , a_n)$ and $(b_1, b_2, \ldots , b_n)$ is defined as $(a_1b_1, a_2b_2, \ldots , a_nb_n)$. Prove that there exists a sequence $(c_1, c_2, \ldots , c_n)$ such that the intersection of $B$ and the set containing all sequences from $B$ multiplied by $(c_1, c_2, \ldots , c_n)$ contains at most $\frac{k^2}{2^n}$ sequences.
2023 Indonesia TST, 2
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 $a^n$.
2016 239 Open Mathematical Olympiad, 4
The sequences of natural numbers $p_n$ and $q_n$ are given such that
$$p_1 = 1,\ q_1 = 1,\ p_{n + 1} = 2q_n^2-p_n^2,\ q_{n + 1} = 2q_n^2+p_n^2 $$
Prove that $p_n$ and $q_m$ are coprime for any m and n.
2021 USA TSTST, 2
Let $a_1<a_2<a_3<a_4<\cdots$ be an infinite sequence of real numbers in the interval $(0,1)$. Show that there exists a number that occurs exactly once in the sequence
\[ \frac{a_1}{1},\frac{a_2}{2},\frac{a_3}{3},\frac{a_4}{4},\ldots.\]
[i]Merlijn Staps[/i]
2001 Regional Competition For Advanced Students, 4
Let $A_o =\{1, 2\}$ and for $n> 0, A_n$ results from $A_{n-1}$ by adding the natural numbers to $A_{n-1}$ which can be represented as the sum of two different numbers from $A_{n-1}$. Let $a_n = |A_n |$ be the number of numbers in $A_n$. Determine $a_n$ as a function of $n$.
1998 Brazil Team Selection Test, Problem 5
Consider $k$ positive integers $a_1,a_2,\ldots,a_k$ satisfying $1\le a_1<a_2<\ldots<a_k\le n$ and $\operatorname{lcm}(a_i,a_j)\le n$ for any $i,j$. Prove that
$$k\le2\lfloor\sqrt n\rfloor.$$
2023 AMC 10, 6
Let $L_1=1$, $L_2=3$, and $L_{n+2}=L_{n+1}+L_n$ for $n\geq1$. How many terms in the sequence $L_1, L_2, L_3, \dots, L_{2023}$ are even?
$\textbf{(A) }673\qquad\textbf{(B) }1011\qquad\textbf{(C) }675\qquad\textbf{(D) }1010\qquad\textbf{(E) }674$
2024 Mexican University Math Olympiad, 5
Consider two finite sequences of real numbers \( a_1, a_2, \dots, a_n \) and \( b_1, b_2, \dots, b_n \). Let \( \alpha(x) = \#\{i | a_i = x \} \) and \( \beta(x) = \#\{i | b_i = -x \} \). Prove that there exists a permutation \( \sigma \in S_n \) (the symmetric group of \( n \) elements) such that \( a_{\sigma(i)} + b_i \neq 0 \) for all \( i = 1, \dots, n \) if and only if \( \alpha(x) + \beta(x) \leq n \) for all \( x \in \mathbb{R} \).
1994 All-Russian Olympiad, 5
Let $a_1$ be a natural number not divisible by $5$. The sequence $a_1,a_2,a_3, . . .$ is defined by $a_{n+1} =a_n+b_n$, where $b_n$ is the last digit of $a_n$. Prove that the sequence contains infinitely many powers of two.
(N. Agakhanov)
1980 All Soviet Union Mathematical Olympiad, 297
Let us denote with $P(n)$ the product of all the digits of $n$. Consider the sequence $$n_{k+1} = n_k + P(n_k)$$ Can it be unbounded for some $n_1$?
2015 IMO Shortlist, N1
Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \cdots$ defined by \[ a_0 = M + \frac{1}{2} \qquad \textrm{and} \qquad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots \] contains at least one integer term.
1967 IMO Longlists, 57
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 $c_n = 0.$
1990 Czech and Slovak Olympiad III A, 1
Let $(a_n)_{n\ge1}$ be a sequence given by
\begin{align*}
a_1 &= 1, \\
a_{2^k+j} &= -a_j\text{ for any } k\ge0,1\le j\le 2^k.
\end{align*}
Show that the sequence is not periodic.
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.)
1989 IMO Shortlist, 30
Prove that for each positive integer $ n$ there exist $ n$ consecutive positive integers none of which is an integral power of a prime number.
2012 Junior Balkan Team Selection Tests - Romania, 1
Let $a_1, a_2, ..., a_n$ be real numbers such that $a_1 = a_n = a$ and $a_{k+1} \le \frac{a_k + a_{k+2}}{2} $, for all $k = 1, 2, ..., n - 2$. Prove that $a_k \le a,$ for all $k = 1, 2, ..., n.$
2014 Federal Competition For Advanced Students, 3
Let $a_n$ be a sequence de fined by some $a_0$ and the recursion $a_{n+1} = a_n + 2 \cdot 3^n$ for $n \ge 0$.
Determine all rational values of $a_0$ such that $a^j_k / a^k_j$ is an integer for all integers $j$ and $k$ with $0 < j < k$.
2023 Estonia Team Selection Test, 5
Let $(a_n)_{n\geq 1}$ be a sequence of positive real numbers with the property that
$$(a_{n+1})^2 + a_na_{n+2} \leq a_n + a_{n+2}$$
for all positive integers $n$. Show that $a_{2022}\leq 1$.