Found problems: 963
2005 Korea Junior Math Olympiad, 3
For a positive integer $K$, de fine a sequence, $\{a_n\}$, as following: $a_1 = K$ and
$a_{n+1} =a_n -1$ if $a_n$ is even
$a_{n+1} =\frac{a_n - 1}{2}$ if $a_n$ is odd , for all $n \ge 1$.
Find the smallest value of $K$, which makes $a_{2005}$ the first term equal to $0$.
1997 Switzerland Team Selection Test, 1
1. A finite sequence of integers $a_0,a_1,...,a_n$ is called quadratic if $|a_k -a_{k-1}| = k^2$
for $n\geq k\geq1$.
(a) Prove that for any two integers $b$ and $c$, there exist a natural number $n$ and a quadratic sequence
with $a_0 = b$ and $a_n =c$.
(b) Find the smallest natural number $n$ for which there exists a quadratic sequence
with $a_0 = 0$ and $a_n = 1997$
1994 Tournament Of Towns, (428) 5
The periods of two periodic sequences are $7$ and $13$. What is the maximal length of initial sections of the two sequences which can coincide? (The period $p$ of a sequence $a_1$,$a_2$, $...$ is the minimal $p$ such that $a_n = a_{n+p}$ for all $n$.)
(AY Belov)
2021 Thailand Mathematical Olympiad, 2
Determine all sequences $a_1,a_2,a_3,\dots$ of positive integers that satisfy the equation
$$(n^2+1)a_{n+1} - a_n = n^3+n^2+1$$
for all positive integers $n$.
2020 Vietnam National Olympiad, 3
Let a sequence $(a_n)$ satisfy: $a_1=5,a_2=13$ and $a_{n+1}=5a_n-6a_{n-1},\forall n\ge2$
a) Prove that $(a_n, a_{n+1})=1,\forall n\ge1$
b) Prove that: $2^{k+1}|p-1\forall k\in\mathbb{N}$, if p is a prime factor of $a_{2^k}$
2021 Balkan MO Shortlist, N2
Denote by $l(n)$ the largest prime divisor of $n$. Let $a_{n+1} = a_n + l(a_n)$ be a recursively
defined sequence of integers with $a_1 = 2$. Determine all natural numbers $m$ such that there
exists some $i \in \mathbb{N}$ with $a_i = m^2$.
[i]Proposed by Nikola Velov, North Macedonia[/i]
KoMaL A Problems 2017/2018, A. 712
We say that a strictly increasing positive real sequence $a_1,a_2,\cdots $ is an [i]elf sequence[/i] if for any $c>0$ we can find an $N$ such that $a_n<cn$ for $n=N,N+1,\cdots$. Furthermore, we say that $a_n$ is a [i]hat[/i] if $a_{n-i}+a_{n+i}<2a_n$ for $\displaystyle 1\le i\le n-1$. Is it true that every elf sequence has infinitely many hats?
2021 Macedonian Mathematical Olympiad, Problem 1
Let $(a_n)^{+\infty}_{n=1}$ be a sequence defined recursively as follows: $a_1=1$ and $$a_{n+1}=1 + \sum\limits_{k=1}^{n}ka_k$$
For every $n > 1$, prove that $\sqrt[n]{a_n} < \frac {n+1}{2}$.
2021 Saudi Arabia Training Tests, 32
Let $N$ be a positive integer. Consider the sequence $a_1, a_2, ..., a_N$ of positive integers, none of which is a multiple of $2^{N+1}$. For $n \ge N +1$, the number $a_n$ is defined as follows: choose $k$ to be the number among $1, 2, ..., n - 1$ for which the remainder obtained when $a_k$ is divided by $2^n$ is the smallest, and define $a_n = 2a_k$ (if there are more than one such $k$, choose the largest such $k$). Prove that there exist $M$ for which $a_n = a_M$ holds for every $n \ge M$.
2015 EGMO, 4
Determine whether there exists an infinite sequence $a_1, a_2, a_3, \dots$ of positive integers
which satisfies the equality \[a_{n+2}=a_{n+1}+\sqrt{a_{n+1}+a_{n}} \] for every positive integer $n$.
2021 Middle European Mathematical Olympiad, 1
Determine all real numbers A such that every sequence of non-zero real numbers $x_1, x_2, \ldots$ satisfying
\[ x_{n+1}=A-\frac{1}{x_n} \]
for every integer $n \ge 1$, has only finitely many negative terms.
2007 IMO Shortlist, 5
Let $ c > 2,$ and let $ a(1), a(2), \ldots$ be a sequence of nonnegative real numbers such that
\[ a(m \plus{} n) \leq 2 \cdot a(m) \plus{} 2 \cdot a(n) \text{ for all } m,n \geq 1,
\]
and $ a\left(2^k \right) \leq \frac {1}{(k \plus{} 1)^c} \text{ for all } k \geq 0.$ Prove that the sequence $ a(n)$ is bounded.
[i]Author: Vjekoslav Kovač, Croatia[/i]
1996 Tuymaada Olympiad, 6
Given the sequence
$f_1(a)=sin(0,5\pi a)$
$f_2(a)=sin(0,5\pi (sin(0,5\pi a)))$
$...$
$f_n(a)=sin(0,5\pi (sin(...(sin(0,5\pi a))...)))$ , where $a$ is any real number.
What limit aspire the members of this sequence as $n \to \infty$?
1965 Swedish Mathematical Competition, 4
Find constants $A > B$ such that $\frac{f\left( \frac{1}{1+2x}\right) }{f(x)}$ is independent of $x$, where $f(x) = \frac{1 + Ax}{1 + Bx}$ for all real $x \ne - \frac{1}{B}$. Put $a_0 = 1$, $a_{n+1} = \frac{1}{1 + 2a_n}$. Find an expression for an by considering $f(a_0), f(a_1), ...$.
2010 Belarus Team Selection Test, 1.4
$x_1=\frac{1}{2}$ and $x_{k+1}=\frac{x_k}{x_1^2+...+x_k^2}$
Prove that $\sqrt{x_k^4+4\frac{x_{k-1}}{x_{k+1}}}$ is rational
Russian TST 2020, P1
Let $n \geqslant 3$ be a positive integer and let $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ be a strictly increasing sequence of $n$ positive real numbers with sum equal to 2. Let $X$ be a subset of $\{1,2, \ldots, n\}$ such that the value of
\[
\left|1-\sum_{i \in X} a_{i}\right|
\]
is minimised. Prove that there exists a strictly increasing sequence of $n$ positive real numbers $\left(b_{1}, b_{2}, \ldots, b_{n}\right)$ with sum equal to 2 such that
\[
\sum_{i \in X} b_{i}=1.
\]
2016 NZMOC Camp Selection Problems, 8
Two positive integers $r$ and $k$ are given as is an infinite sequence of positive integers $a_1 \le a_2 \le a_3 \le ..$ such that $\frac{r}{a_r}= k + 1$. Prove that there is a positive integer $t$ such that $\frac{t}{a_t}= k$.
KoMaL A Problems 2023/2024, A. 872
For every positive integer $k$ let $a_{k,1},a_{k,2},\ldots$ be a sequence of positive integers. For every positive integer $k$ let sequence $\{a_{k+1,i}\}$ be the difference sequence of $\{a_{k,i}\}$, i.e. for all positive integers $k$ and $i$ the following holds: $a_{k,i+1}-a_{k,i}=a_{k+1,i}$. Is it possible that every positive integer appears exactly once among numbers $a_{k,i}$?
[i]Proposed by Dávid Matolcsi, Berkeley[/i]
1970 Putnam, A4
Given a sequence $(x_n )$ such that $\lim_{n\to \infty} x_n - x_{n-2}=0,$ prove that
$$\lim_{n\to \infty} \frac{x_n -x_{n-1}}{n}=0.$$
1967 IMO Shortlist, 6
Given a segment $AB$ of the length 1, define the set $M$ of points in the
following way: it contains two points $A,B,$ and also all points obtained from $A,B$ by iterating the following rule: With every pair of points $X,Y$ the set $M$ contains also the point $Z$ of the segment $XY$ for which $YZ = 3XZ.$
2014 BAMO, 4
Let $F_1, F_2, F_3 \cdots$ be the Fibonacci sequence, the sequence of positive integers satisfying $$F_1 =F_2=1$$ and $$F_{n+2} = F_{n+1} + F_n$$ for all $n \ge 1$.
Does there exist an $n \ge 1$ such that $F_n$ is divisible by $2014$? Prove your answer.
2011 Singapore Junior Math Olympiad, 4
Any positive integer $n$ can be written in the form $n = 2^aq$, where $a \ge 0$ and $q$ is odd. We call $q$ the [i]odd part[/i] of $n$. Define the sequence $a_0,a_1,...$ as follows: $a_0 = 2^{2011}-1$ and for $m > 0, a_{m+i}$ is the odd part of $3a_m + 1$. Find $a_{2011}$.
2021 Estonia Team Selection Test, 3
For any odd prime $p$ and any integer $n,$ let $d_p (n) \in \{ 0,1, \dots, p-1 \}$ denote the remainder when $n$ is divided by $p.$ We say that $(a_0, a_1, a_2, \dots)$ is a [i]p-sequence[/i], if $a_0$ is a positive integer coprime to $p,$ and $a_{n+1} =a_n + d_p (a_n)$ for $n \geqslant 0.$
(a) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $(a_0, a_1, a_2, \dots)$ and $(b_0, b_1, b_2, \dots)$ such that $a_n >b_n$ for infinitely many $n,$ and $b_n > a_n$ for infinitely many $n?$
(b) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $(a_0, a_1, a_2, \dots)$ and $(b_0, b_1, b_2, \dots)$ such that $a_0 <b_0,$ but $a_n >b_n$ for all $n \geqslant 1?$
[I]United Kingdom[/i]
2011 Abels Math Contest (Norwegian MO), 1
Let $n$ be the number that is produced by concatenating the numbers $1, 2,... , 4022$,
that is, $n = 1234567891011...40214022$.
a. Show that $n$ is divisible by $3$.
b. Let $a_1 = n^{2011}$, and let $a_i$ be the sum of the digits of $a_{i-1}$ for $i > 1$. Find $a_4$
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 $n \ge 0$.