Found problems: 1239
2024 TASIMO, 2
Find all positive integers $(r,s)$ such that there is a non-constant sequence $a_n$ os positive integers such that for all $n=1,2,\dots$
\[ a_{n+2}= \left(1+\frac{{a_2}^r}{{a_1}^s} \right ) \left(1+\frac{{a_3}^r}{{a_2}^s} \right ) \dots \left(1+\frac{{a_{n+1}}^r}{{a_n}^s} \right ).\]
Proposed by Navid Safaei, Iran
2021 Dutch IMO TST, 1
The sequence of positive integers $a_0, a_1, a_2, . . .$ is defined by $a_0 = 3$ and $$a_{n+1} - a_n = n(a_n - 1)$$ for all $n \ge 0$. Determine all integers $m \ge 2$ for which $gcd (m, a_n) = 1$ for all $n \ge 0$.
2011 Abels Math Contest (Norwegian MO), 3a
The positive numbers $a_1, a_2,...$ satisfy $a_1 = 1$ and $(m+n)a_{m+n }\le a_m +a_n$ for all positive integers $m$ and $n$. Show that $\frac{1}{a_{200}} > 4 \cdot 10^7$ .
.
1989 Nordic, 3
Let $S$ be the set of all points $t$ in the closed interval $[-1, 1]$ such that for the sequence $x_0, x_1, x_2, ...$ defined by the equations $x_0 = t, x_{n+1} = 2x_n^2-1$, there exists a positive integer $N$ such that $x_n = 1$ for all $n \ge N$. Show that the set $S$ has infinitely many elements.
2022 Thailand TST, 3
Let $a_1,a_2,a_3,\ldots$ be an infinite sequence of positive integers such that $a_{n+2m}$ divides $a_{n}+a_{n+m}$ for all positive integers $n$ and $m.$ Prove that this sequence is eventually periodic, i.e. there exist positive integers $N$ and $d$ such that $a_n=a_{n+d}$ for all $n>N.$
2011 Peru IMO TST, 6
Let $a_1, a_2, \cdots , a_n$ be real numbers, with $n\geq 3,$ such that $a_1 + a_2 +\cdots +a_n = 0$ and $$ 2a_k\leq a_{k-1} + a_{k+1} \ \ \ \text{for} \ \ \ k = 2, 3, \cdots , n-1.$$ Find the least number $\lambda(n),$ such that for all $k\in \{ 1, 2, \cdots, n\} $ it is satisfied that $|a_k|\leq \lambda (n)\cdot \max \{|a_1|, |a_n|\} .$
2023 Germany Team Selection Test, 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 $a^n$.
1983 IMO Longlists, 9
Consider the set of all strictly decreasing sequences of $n$ natural numbers having the property that in each sequence no term divides any other term of the sequence. Let $A = (a_j)$ and $B = (b_j)$ be any two such sequences. We say that $A$ precedes $B$ if for some $k$, $a_k < b_k$ and $a_i = b_i$ for $i < k$. Find the terms of the first sequence of the set under this ordering.
2001 SNSB Admission, 2
Let be a number $ a\in \left[ 1,\infty \right) $ and a function $ f\in\mathcal{C}^2(-a,a) . $ Show that the sequence
$$ \left( \sum_{k=1}^n f\left( \frac{k}{n^2} \right) \right)_{n\ge 1} $$
is convergent, and determine its limit.
2021 Thailand TSTST, 3
A finite sequence of integers $a_0,,a_1,\dots,a_n$ is called [i]quadratic[/i] if for each $i\in\{1,2,\dots n\}$ we have the equality $|a_i-a_{i-1}|=i^2$.
$\text{(i)}$ Prove that for any two integers $b$ and $c$, there exist a positive integer $n$ and a quadratic sequence with $a_0=b$ and $a_n = c$.
$\text{(ii)}$ Find the smallest positive integer $n$ for which there exists a quadratic sequence with $a_0=0$ and $a_n=2021$.
2001 Rioplatense Mathematical Olympiad, Level 3, 3
For every integer $n > 1$, the sequence $\left( {{S}_{n}} \right)$ is defined by ${{S}_{n}}=\left\lfloor {{2}^{n}}\underbrace{\sqrt{2+\sqrt{2+...+\sqrt{2}}}}_{n\ radicals} \right\rfloor $
where $\left\lfloor x \right\rfloor$ denotes the floor function of $x$. Prove that ${{S}_{2001}}=2\,{{S}_{2000}}+1$.
.
2017 IMO, 1
For each integer $a_0 > 1$, define the sequence $a_0, a_1, a_2, \ldots$ for $n \geq 0$ as
$$a_{n+1} =
\begin{cases}
\sqrt{a_n} & \text{if } \sqrt{a_n} \text{ is an integer,} \\
a_n + 3 & \text{otherwise.}
\end{cases}
$$
Determine all values of $a_0$ such that there exists a number $A$ such that $a_n = A$ for infinitely many values of $n$.
[i]Proposed by Stephan Wagner, South Africa[/i]
2023 Brazil National Olympiad, 1
Show an infinite sequence $a_1, a_2, \ldots$ of integers with both of the following properties:
• $a_i \neq 0$ for every positive integer $i$, that is, no term in the sequence is equal to zero;
• for all positive integer $n$, $a_n + a_{2n} + \ldots + a_{2023n} = 0$.
2018 Pan African, 3
For any positive integer $x$, we set
$$
g(x) = \text{ largest odd divisor of } x,
$$
$$
f(x) = \begin{cases}
\frac{x}{2} + \frac{x}{g(x)} & \text{ if } x \text{ is even;} \\
2^{\frac{x+1}{2}} & \text{ if } x \text{ is odd.}
\end{cases}
$$
Consider the sequence $(x_n)_{n \in \mathbb{N}}$ defined by $x_1 = 1$, $x_{n + 1} = f(x_n)$. Show that the integer $2018$ appears in this sequence, determine the least integer $n$ such that $x_n = 2018$, and determine whether $n$ is unique or not.
2012 District Olympiad, 1
Consider the sequence $ \left( x_n \right)_{n\ge 1} $ having $ x_1>1 $ and satisfying the equation
$$ x_1+x_2+\cdots +x_{n+1} =x_1x_2\cdots x_{n+1} ,\quad\forall n\in\mathbb{N} . $$
Show that this sequence is convergent and find its limit.
2020 Paraguay Mathematical Olympiad, 5
The general term of a sequence of numbers is defined as $a_n =\frac{1}{n^2 - n}$, for every integer $n \ge 3$.
That is, $a_3 =\frac16$, $a_4 =\frac{1}{12}$, $a_5 =\frac{1}{20}$, and so on.
Find a general expression for the sum $S_n$, which is the sum of all terms from $a_3$ until $a_n$.
2020 Jozsef Wildt International Math Competition, W38
Let $(a_n)_{n\in\mathbb N}$ be a sequence, given by the recurrence:
$$ma_{n+1}+(m-2)a_n-a_{n-1}=0$$
where $m\in\mathbb R$ is a parameter and the first two terms of $a_n$ are fixed known real numbers. Find $m\in\mathbb R$, so that
$$\lim_{n\to\infty}a_n=0$$
[i]Proposed by Laurențiu Modan[/i]
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 $a_{2022}\leq 1$.
2016 Belarus Team Selection Test, 1
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.
VMEO II 2005, 3
Given positive integers $a_1$, $a_2$, $...$, $a_m$ ($m \ge 1$). Consider the sequence $\{u_n\}_{n=1}^{\infty}$, with $$u_n = a_1^n + a_2^n + ... + a_m^n.$$ We know that this sequence has a finite number of prime divisors. Prove that $a_1 = a_2 = ...= a_m$.
2018 Serbia National Math Olympiad, 2
Let $n>1$ be an integer. Call a number beautiful if its square leaves an odd remainder upon divison by $n$. Prove that the number of consecutive beautiful numbers is less or equal to $1+\lfloor \sqrt{3n} \rfloor$.
1962 Dutch Mathematical Olympiad, 2
The $n^{th}$ term of a sequence is $t_n$. For $n \ge 1$, $t_n$ is given by the relation:
$$t_n= n^3+\frac12 n^2+ \frac13 n + \frac14$$
The $n^{th}$ term of a second sequence $T_n$, where $T_n$ represents the smallest integer greater than $t_n$. Calculate: $$(T_1+T_2+...+T_{1014}) -(t_1+t_2+...+t_{1014}) $$
2015 Singapore MO Open, 2
A boy lives in a small island in which there are three roads at every junction. He starts
from his home and walks along the roads. At each junction he would choose to turn
to the road on his right or left alternatively, i.e., his choices would be . . ., left, right,
left,... Prove that he will eventually return to his home.
1988 IMO Shortlist, 24
Let $ \{a_k\}^{\infty}_1$ be a sequence of non-negative real numbers such that:
\[ a_k \minus{} 2 a_{k \plus{} 1} \plus{} a_{k \plus{} 2} \geq 0
\]
and $ \sum^k_{j \equal{} 1} a_j \leq 1$ for all $ k \equal{} 1,2, \ldots$. Prove that:
\[ 0 \leq a_{k} \minus{} a_{k \plus{} 1} < \frac {2}{k^2}
\]
for all $ k \equal{} 1,2, \ldots$.
2025 Bulgarian Winter Tournament, 12.3
Determine all functions $f: \mathbb{Z}_{\geq 2025} \to \mathbb{Z}_{>0}$ such that $mn+1$ divides $f(m)f(n) + 1$ for any integers $m,n \geq 2025$ and there exists a polynomial $P$ with integer coefficients, such that $f(n) \leq P(n)$ for all $n\geq 2025$.