Found problems: 963
1967 IMO, 5
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 $c_n = 0.$
2018 China Team Selection Test, 1
Define the polymonial sequence $\left \{ f_n\left ( x \right ) \right \}_{n\ge 1}$ with $f_1\left ( x \right )=1$, $$f_{2n}\left ( x \right )=xf_n\left ( x \right ), \; f_{2n+1}\left ( x \right ) = f_n\left ( x \right )+ f_{n+1} \left ( x \right ), \; n\ge 1.$$ Look for all the rational number $a$ which is a root of certain $f_n\left ( x \right ).$
1980 IMO Shortlist, 14
Let $\{x_n\}$ be a sequence of natural numbers such that \[(a) 1 = x_1 < x_2 < x_3 < \ldots; \quad (b) x_{2n+1} \leq 2n \quad \forall n.\] Prove that, for every natural number $k$, there exist terms $x_r$ and $x_s$ such that $x_r - x_s = k.$
2013 Abels Math Contest (Norwegian MO) Final, 1b
The sequence $a_1, a_2, a_3,...$ is defined so that $a_1 = 1$ and $a_{n+1} =\frac{a_1 + a_2 + ...+ a_n}{n}+1$ for $n \ge 1$. Show that for every positive real number $b$ we can find $a_k$ so that $a_k < bk$.
1984 Dutch Mathematical Olympiad, 3
For $n = 1,2,3,...$. $a_n$ is defined by:
$$a_n =\frac{1 \cdot 4 \cdot 7 \cdot ... (3n-2)}{2 \cdot 5 \cdot 8 \cdot ... (3n-1)}$$
Prove that for every $n$ holds that
$$\frac{1}{\sqrt{3n+1}}\le a_n \le \frac{1}{\sqrt[3]{3n+1}}$$
2010 Saudi Arabia BMO TST, 3
Let $(a_n )_{n \ge o}$ and $(b_n )_{n \ge o}$ be sequences defined by $a_{n+2} = a_{n+1}+ a_n$ , $n = 0 , 1 , . .. $, $a_0 = 1$, $a_1 = 2$, and $b_{n+2} = b_{n+1} + b_n$ , $n = 0 , 1 , . . .$, $b_0 = 2$, $b_1 = 1$. How many integers do the sequences have in common?
2018 Swedish Mathematical Competition, 3
Let m be a positive integer. An $m$-[i]pattern [/i] is a sequence of $m$ symbols of strict inequalities. An $m$-pattern is said to be [i]realized [/i] by a sequence of $m + 1$ real numbers when the numbers meet each of the inequalities in the given order. (For example, the $5$-pattern $ <, <,>, < ,>$ is realized by the sequence of numbers $1, 4, 7, -3, 1, 0$.)
Given $m$, which is the least integer $n$ for which there exists any number sequence $x_1,... , x_n$ such that each $m$-pattern is realized by a subsequence $x_{i_1},... , x_{i_{m + 1}}$ with $1 \le i_1 <... < i_{m + 1} \le n$?
2017 Grand Duchy of Lithuania, 1
The infinite sequence $a_0, a_1, a_2, a_3,... $ is defined by $a_0 = 2$ and
$$a_n =\frac{2a_{n-1} + 1}{a_{n-1} + 2}$$ , $n = 1, 2, 3, ...$ Prove that $1 < a_n < 1 + \frac{1}{3^n}$ for all $n = 1, 2, 3, . .$
Russian TST 2021, P3
Let $\mathcal{S}$ be a set consisting of $n \ge 3$ positive integers, none of which is a sum of two other distinct members of $\mathcal{S}$. Prove that the elements of $\mathcal{S}$ may be ordered as $a_1, a_2, \dots, a_n$ so that $a_i$ does not divide $a_{i - 1} + a_{i + 1}$ for all $i = 2, 3, \dots, n - 1$.
1980 Austrian-Polish Competition, 6
Let $a_1,a_2,a_3,\dots$ be a sequence of real numbers satisfying the inequality \[ |a_{k+m}-a_k-a_m| \leq 1 \quad \text{for all} \ k,m \in \mathbb{Z}_{>0}. \] Show that the following inequality holds for all positive integers $k,m$ \[ \left| \frac{a_k}{k}-\frac{a_m}{m} \right| < \frac{1}{k}+\frac{1}{m}. \]
2010 Korea Junior Math Olympiad, 4
Let there be a sequence $a_n$ such that $a_1 = 2,a_2 = 0, a_3 = 1, a_4 = 0$, and for $n \ge 1, a_{n+4}$ is the remainder when $a_n + 2a_{n+1} + 3a_{n+2} + 4a_{n+3}$ is divided by $9$. Prove that there are no positive integer $k$ such that $$a_k = 0, a_{k+1} = 1, a_{k+2} = 0,a_{k+3} = 2.$$
1999 IMO Shortlist, 3
Prove that there exists two strictly increasing sequences $(a_{n})$ and $(b_{n})$ such that $a_{n}(a_{n}+1)$ divides $b^{2}_{n}+1$ for every natural n.
2010 IMO Shortlist, 4
A sequence $x_1, x_2, \ldots$ is defined by $x_1 = 1$ and $x_{2k}=-x_k, x_{2k-1} = (-1)^{k+1}x_k$ for all $k \geq 1.$ Prove that $\forall n \geq 1$ $x_1 + x_2 + \ldots + x_n \geq 0.$
[i]Proposed by Gerhard Wöginger, Austria[/i]
1978 Germany Team Selection Test, 3
Let $n$ be an integer greater than $1$. Define
\[x_1 = n, y_1 = 1, x_{i+1} =\left[ \frac{x_i+y_i}{2}\right] , y_{i+1} = \left[ \frac{n}{x_{i+1}}\right], \qquad \text{for }i = 1, 2, \ldots\ ,\]
where $[z]$ denotes the largest integer less than or equal to $z$. Prove that
\[ \min \{x_1, x_2, \ldots, x_n \} =[ \sqrt n ]\]
1984 All Soviet Union Mathematical Olympiad, 389
Given a sequence $\{x_n\}$, $$x_1 = x_2 = 1, x_{n+2} = x^2_{n+1} - \frac{x_n}{2}$$ Prove that the sequence has limit and find it.
2000 German National Olympiad, 6
A sequence ($a_n$) satisfies the following conditions:
(i) For each $m \in N$ it holds that $a_{2^m} = 1/m$.
(ii) For each natural $n \ge 2$ it holds that $a_{2n-1}a_{2n} = a_n$.
(iii) For all integers $m,n$ with $2m > n \ge 1$ it holds that $a_{2n}a_{2n+1} = a_{2^m+n}$.
Determine $a_{2000}$. You may assume that such a sequence exists.
2025 China National Olympiad, 1
Let $\alpha > 1$ be an irrational number and $L$ be a integer such that $L > \frac{\alpha^2}{\alpha - 1}$. A sequence $x_1, x_2, \cdots$ satisfies that $x_1 > L$ and for all positive integers $n$, \[ x_{n+1} = \begin{cases} \left \lfloor \alpha x_n \right \rfloor & \textup{if} \; x_n \leqslant L \\\left \lfloor \frac{x_n}{\alpha} \right \rfloor & \textup{if} \; x_n > L \end{cases}. \]
Prove that
(i) $\left\{x_n\right\}$ is eventually periodic.
(ii) The eventual fundamental period of $\left\{x_n\right\}$ is an odd integer which doesn't depend on the choice of $x_1$.
2020 Bulgaria National Olympiad, P2
Let $b_1$, $\dots$ , $b_n$ be nonnegative integers with sum $2$ and $a_0$, $a_1$, $\dots$ , $a_n$ be real numbers such that $a_0=a_n=0$ and $|a_i-a_{i-1}|\leq b_i$ for each $i=1$, $\dots$ , $n$. Prove that
$$\sum_{i=1}^n(a_i+a_{i-1})b_i\leq 2$$
[hide]I believe that the original problem was for nonnegative real numbers and it was a typo on the version of the exam paper we had but I'm not sure the inequality would hold[/hide]
1993 Austrian-Polish Competition, 7
The sequence $(a_n)$ is defined by $a_0 = 0$ and $a_{n+1} = [\sqrt[3]{a_n +n}]^3$ for $n \ge 0$.
(a) Find $a_n$ in terms of $n$.
(b) Find all $n$ for which $a_n = n$.
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.
2021 Thailand TST, 2
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]
2021 Brazil Team Selection Test, 2
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]
2014 IFYM, Sozopol, 2
We define the following sequence: $a_0=a_1=1$, $a_{n+1}=14a_n-a_{n-1}$. Prove that
$2a_n-1$ is a perfect square.
2009 China Northern MO, 1
Sequence {$x_n$} satisfies: $x_1=1$ , ${x_n=\sqrt{x_{n-1}^2+x_{n-1}}+x_{n-1}}$ ( ${n>=2}$ )
Find the general term of {$x_n$}
2012 Estonia Team Selection Test, 2
For a given positive integer $n$ one has to choose positive integers $a_0, a_1,...$ so that the following conditions hold:
(1) $a_i = a_{i+n}$ for any $i$,
(2) $a_i$ is not divisible by $n$ for any $i$,
(3) $a_{i+a_i}$ is divisible by $a_i$ for any $i$.
For which positive integers $n > 1$ is this possible only if the numbers $a_0, a_1, ...$ are all equal?