Found problems: 1239
2001 BAMO, 5
For each positive integer $n$, let $a_n$ be the number of permutations $\tau$ of $\{1, 2, ... , n\}$ such that $\tau (\tau (\tau (x))) = x$ for $x = 1, 2, ..., n$. The first few values are $a_1 = 1, a_2 = 1, a_3 = 3, a_4 = 9$.
Prove that $3^{334}$ divides $a_{2001}$.
(A permutation of $\{1, 2, ... , n\}$ is a rearrangement of the numbers $\{1, 2, ... , n\}$ or equivalently, a one-to-one and
onto function from $\{1, 2, ... , n\}$ to $\{1, 2, ... , n\}$. For example, one permutation of $\{1, 2, 3\}$ is the rearrangement $\{2, 1, 3\}$, which is equivalent to the function $\sigma : \{1, 2, 3\} \to \{1, 2, 3\}$ defined by $\sigma (1) = 2, \sigma (2) = 1, \sigma (3) = 3$.)
2025 Philippine MO, P3
Let $d$ be a positive integer. Define the sequence $a_1, a_2, a_3, \dots$ such that \[\begin{cases} a_1 = 1 \\ a_{n+1} = n\left\lfloor\frac{a_n}{n}\right\rfloor + d, \quad n \ge 1.\end{cases}\] Prove that there exists a positive integer $M$ such that $a_M, a_{M+1}, a_{M+2}, \dots$ is an arithmetic sequence.
1999 Romania National Olympiad, 2
Let $k$ be a positive integer, let $z_1,z_2, \ldots, z_k \in \mathbb{C}$ be distinct and let $u_1,u_2,\ldots,u_k \in \mathbb{C}$ be such that the set $\big\{a_n=u_1z_1^n+u_2z_2^n+\ldots+u_kz_k^n : n \in \mathbb{Z}_{>0} \big\}$ is finite. Prove that there exists a positive integer $p$ such that $a_n=a_{n+p},$ for any positive integer $n.$
2020 Brazil Undergrad MO, Problem 2
For a positive integer $a$, define $F_1 ^{(a)}=1$, $F_2 ^{(a)}=a$ and for $n>2$, $F_n ^{(a)}=F_{n-1} ^{(a)}+F_{n-2} ^{(a)}$. A positive integer is fibonatic when it is equal to $F_n ^{(a)}$ for a positive integer $a$ and $n>3$. Prove that there are infintely many not fibonatic integers.
2022 Greece Team Selection Test, 3
Find largest possible constant $M$ such that, for any sequence $a_n$, $n=0,1,2,...$ of real numbers, that satisfies the conditions :
i) $a_0=1$, $a_1=3$
ii) $a_0+a_1+...+a_{n-1} \ge 3 a_n - a_{n+1}$ for any integer $n\ge 1$
to be true that
$$\frac{a_{n+1}}{a_n} >M$$ for any integer $n\ge 0$.
1972 Putnam, B1
Let $\sum_{n=0}^{\infty} \frac{x^n (x-1)^{2n}}{n!}=\sum_{n=0}^{\infty} a_{n}x^{n}$. Show that no three consecutive $a_n$ can be equal to $0$.
2013 Nordic, 3
Define a sequence ${(n_k)_{k\ge 0}}$ by ${n_{0 }= n_{1} = 1}$, and ${n_{2k} = n_k + n_{k-1} }$ and ${n_{2k+1} = n_k}$ for ${k \ge 1}$. Let further ${q_k = n_k }$ / ${ n_{k-1} }$ for each ${k \ge 1}$. Show that every positive rational number is present exactly once in the sequence ${(q_k)_{k\ge 1}}$
2017 District Olympiad, 1
Let $ \left( a_n \right)_{n\ge 1} $ be a sequence of real numbers such that $ a_1>2 $ and $ a_{n+1} =a_1+\frac{2}{a_n} , $ for all natural numbers $ n. $
[b]a)[/b] Show that $ a_{2n-1} +a_{2n} >4 , $ for all natural numbers $ n, $ and $ \lim_{n\to\infty} a_n =2. $
[b]b)[/b] Find the biggest real number $ a $ for which the following inequality is true:
$$ \sqrt{x^2+a_1^2} +\sqrt{x^2+a_2^2} +\sqrt{x^2+a_3^2} +\cdots +\sqrt{x^2+a_n^2} > n\sqrt{x^2+a^2}, \quad\forall x\in\mathbb{R} ,\quad\forall n\in\mathbb{N} . $$
2019 Jozsef Wildt International Math Competition, W. 9
Let $\alpha > 0$ be a real number. Compute the limit of the sequence $\{x_n\}_{n\geq 1}$ defined by $$x_n=\begin{cases} \sum \limits_{k=1}^n \sinh \left(\frac{k}{n^2}\right),& \text{when}\ n>\frac{1}{\alpha}\\ 0,& \text{when}\ n\leq \frac{1}{\alpha}\end{cases}$$
2023 JBMO Shortlist, A7
Let $a_1,a_2,a_3,\ldots,a_{250}$ be real numbers such that $a_1=2$ and
$$a_{n+1}=a_n+\frac{1}{a_n^2}$$
for every $n=1,2, \ldots, 249$. Let $x$ be the greatest integer which is less than
$$\frac{1}{a_1}+\frac{1}{a_2}+\ldots+\frac{1}{a_{250}}$$
How many digits does $x$ have?
[i]Proposed by Miroslav Marinov, Bulgaria[/i]
1976 IMO Shortlist, 2
Let $a_0, a_1, \ldots, a_n, a_{n+1}$ be a sequence of real numbers satisfying the following conditions:
\[a_0 = a_{n+1 }= 0,\]\[ |a_{k-1} - 2a_k + a_{k+1}| \leq 1 \quad (k = 1, 2,\ldots , n).\]
Prove that $|a_k| \leq \frac{k(n+1-k)}{2} \quad (k = 0, 1,\ldots ,n + 1).$
2017 IFYM, Sozopol, 6
The sequence $a_1,a_2…$ , is defined by the equations $a_1=1$ and $a_n=n.a_{[n/2]}$ for $n>1$. Prove that $a_n>n^2$ for $n>11$.
2017 Bundeswettbewerb Mathematik, 4
The sequence $a_0,a_1,a_2,\dots$ is recursively defined by \[ a_0 = 1 \quad \text{and} \quad a_n = a_{n-1} \cdot \left(4-\frac{2}{n} \right) \quad \text{for } n \geq 1. \] Prove for each integer $n \geq 1$:
(a) The number $a_n$ is a positive integer.
(b) Each prime $p$ with $n < p \leq 2n$ is a divisor of $a_n$.
(c) If $n$ is a prime, then $a_n-2$ is divisible by $n$.
1983 IMO Shortlist, 19
Let $(F_n)_{n\geq 1} $ be the Fibonacci sequence $F_1 = F_2 = 1, F_{n+2} = F_{n+1} + F_n (n \geq 1),$ and $P(x)$ the polynomial of degree $990$ satisfying
\[ P(k) = F_k, \qquad \text{ for } k = 992, . . . , 1982.\]
Prove that $P(1983) = F_{1983} - 1.$
1996 VJIMC, Problem 2
Let $\{a_n\}^\infty_{n=0}$ be the sequence of integers such that $a_0=1$, $a_1=1$, $a_{n+2}=2a_{n+1}-2a_n$. Decide whether
$$a_n=\sum_{k=0}^{\left\lfloor\frac n2\right\rfloor}\binom n{2k}(-1)^k.$$
2018 Vietnam National Olympiad, 1
The sequence $(x_n)$ is defined as follows:
$$x_1=2,\, x_{n+1}=\sqrt{x_n+8}-\sqrt{x_n+3}$$
for all $n\geq 1$.
a. Prove that $(x_n)$ has a finite limit and find that limit.
b. For every $n\geq 1$, prove that
$$n\leq x_1+x_2+\dots +x_n\leq n+1.$$
1998 North Macedonia National Olympiad, 5
The sequence $(a_n)$ is defined by $a_1 =\sqrt2$ and $a_{n+1} =\sqrt{2-\sqrt{4-a_n^2}}$.
Let $b_n =2^{n+1}a_n$. Prove that $b_n \le 7$ and $b_n < b_{n+1}$ for all $n$.
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)
2023 Thailand TST, 1
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 $a_{2022}\leq 1$.
2006 IMO Shortlist, 2
The sequence of real numbers $a_0,a_1,a_2,\ldots$ is defined recursively by \[a_0=-1,\qquad\sum_{k=0}^n\dfrac{a_{n-k}}{k+1}=0\quad\text{for}\quad n\geq 1.\]Show that $ a_{n} > 0$ for all $ n\geq 1$.
[i]Proposed by Mariusz Skalba, Poland[/i]
2019-IMOC, N5
Initially, Alice is given a positive integer $a_0$. At time $i$, Alice has two choices,
$$\begin{cases}a_i\mapsto\frac1{a_{i-1}}\\a_i\mapsto2a_{i-1}+1\end{cases}$$
Note that it is dangerous to perform the first operation, so Alice cannot choose this operation in two consecutive turns. However, if $x>8763$, then Alice could only perform the first operation. Determine all $a_0$ so that $\{i\in\mathbb N\mid a_i\in\mathbb N\}$ is an infinite set.
2018 Saudi Arabia GMO TST, 1
Let $\{x_n\}$ be a sequence defined by $x_1 = 2$ and $x_{n+1} = x_n^2 - x_n + 1$ for $n \ge 1$. Prove that
$$1 -\frac{1}{2^{2^{n-1}}} < \frac{1}{x_1}+\frac{1}{x_2}+ ... +\frac{1}{x_n}< 1 -\frac{1}{2^{2^n}}$$
for all $n$
1978 Swedish Mathematical Competition, 4
$b_0, b_1, b_2, \dots$ is a sequence of positive reals such that the sequence $b_0,c b_1, c^2b_2,c^3b_3,\dots$ is convex for all $c > 0$. (A sequence is convex if each term is at most the arithmetic mean of its two neighbors.) Show that $\ln b_0, \ln b_1, \ln b_2, \dots$ is convex.
1996 Czech And Slovak Olympiad IIIA, 1
A sequence $(G_n)_{n=0}^{\infty}$ satisfies $G(0) = 0$ and $G(n) = n-G(G(n-1))$ for each $n \in N$. Show that
(a) $G(k) \ge G(k -1)$ for every $k \in N$;
(b) there is no integer $k$ for which $G(k -1) = G(k) = G(k +1)$.
2015 District Olympiad, 3
Consider the following sequence of sets: $ \{ 1,2\} ,\{ 3,4,5\}, \{ 6,7,8,9\} ,... $
[b]a)[/b] Find the samllest element of the $ 100\text{-th} $ term.
[b]b)[/b] Is $ 2015 $ the largest element of one of these sets?