Found problems: 1239
2014 Taiwan TST Round 2, 2
Let $r$ be a positive integer, and let $a_0 , a_1 , \cdots $ be an infinite sequence of real numbers. Assume that for all nonnegative integers $m$ and $s$ there exists a positive integer $n \in [m+1, m+r]$ such that
\[ a_m + a_{m+1} +\cdots +a_{m+s} = a_n + a_{n+1} +\cdots +a_{n+s} \]
Prove that the sequence is periodic, i.e. there exists some $p \ge 1 $ such that $a_{n+p} =a_n $ for all $n \ge 0$.
2024 Middle European Mathematical Olympiad, 4
A finite sequence $x_1,\dots,x_r$ of positive integers is a [i]palindrome[/i] if $x_i=x_{r+1-i}$ for all integers
$1 \le i \le r$.
Let $a_1,a_2,\dots$ be an infinite sequence of positive integers. For a positive integer $j \ge 2$, denote by
$a[j]$ the finite subsequence $a_1,a_2,\dots,a_{j-1}$. Suppose that there exists a strictly increasing infinite
sequence $b_1,b_2,\dots$ of positive integers such that for every positive integer $n$, the subsequence
$a[b_n]$ is a palindrome and $b_{n+2} \le b_{n+1}+b_n$. Prove that there exists a positive integer $T$ such
that $a_i=a_{i+T}$ for every positive integer $i$.
2012 District Olympiad, 4
A sequence $ \left( a_n \right)_{n\ge 1} $ has the property that it´s nondecreasing, nonconstant and, for every natural $ n, a_n\big| n^2. $ Show that at least one of the following affirmations are true.
$ \text{(i)} $ There exists an index $ n_1 $ such that $ a_n=n, $ for all $ n\ge n_1. $
$ \text{(ii)} $ There exists an index $ n_2 $ such that $ a_n=n^2, $ for all $ n\ge n_2. $
2010 VJIMC, Problem 2
Prove or disprove that if a real sequence $(a_n)$ satisfies $a_{n+1}-a_n\to0$ and $a_{2n}-2a_n\to0$ as $n\to\infty$, then $a_n\to0$.
1973 IMO, 3
Let $a_1, \ldots, a_n$ be $n$ positive numbers and $0 < q < 1.$ Determine $n$ positive numbers $b_1, \ldots, b_n$ so that:
[i]a.)[/i] $ a_{k} < b_{k}$ for all $k = 1, \ldots, n,$
[i]b.)[/i] $q < \frac{b_{k+1}}{b_{k}} < \frac{1}{q}$ for all $k = 1, \ldots, n-1,$
[i]c.)[/i] $\sum \limits^n_{k=1} b_k < \frac{1+q}{1-q} \cdot \sum \limits^n_{k=1} a_k.$
2014 German National Olympiad, 2
For a positive integer $n$, let $y_n$ be the number of $n$-digit positive integers containing only the digits $2,3,5, 7$ and which do not have a $5$ directly to the right of a $2.$ If $r\geq 1$ and $m\geq 2$ are integers, prove that $y_{m-1}$ divides $y_{rm-1}.$
2023 Simon Marais Mathematical Competition, A4
Let $x_0, x_1, x_2 \dots$ be a sequence of positive real numbers such that for all $n \geq 0$, $$x_{n+1} = \dfrac{(n^2+1)x_n^2}{x_n^3+n^2}$$ For which values of $x_0$ is this sequence bounded?
Gheorghe Țițeica 2025, P3
Let $(a_n)_{n\geq 0}$ be a sequence defined by $a_0\geq 0$ and the recurrence relation $$a_{n+1}=\frac{a_n^2-1}{n+1},$$ for all $n\geq 0$. Prove that here exists a real number $a> 0$ such that:
[list]
[*] if $a_0\geq a,$ $\lim_{n\rightarrow\infty}a_n = \infty$;
[*] if $a_0\in [0,a),$ $\lim_{n\rightarrow\infty}a_n = 0$.
2012 District Olympiad, 3
Let be a sequence of natural numbers $ \left( a_n \right)_{n\ge 1} $ such that $ a_n\le n $ for all natural numbers $ n, $ and
$$ \sum_{j=1}^{k-1} \cos \frac{\pi a_j}{k} =0, $$
for all natural $ k\ge 2. $
[b]a)[/b] Find $ a_2. $
[b]b)[/b] Determine this sequence.
2009 Kyiv Mathematical Festival, 5
The sequence of positive integers $\{a_n, n\ge 1\}$ is such that $a_n\le a_{n+1}\le a_n+5$ and $a_n$ is divisible by $n$ for all $n \ge 1$. What are the possible values of $a_1$?
1969 IMO Longlists, 28
$(GBR 5)$ Let us define $u_0 = 0, u_1 = 1$ and for $n\ge 0, u_{n+2} = au_{n+1}+bu_n, a$ and $b$ being positive integers. Express $u_n$ as a polynomial in $a$ and $b.$ Prove the result. Given that $b$ is prime, prove that $b$ divides $a(u_b -1).$
2020 Baltic Way, 1
Let $a_0>0$ be a real number, and let
$$a_n=\frac{a_{n-1}}{\sqrt{1+2020\cdot a_{n-1}^2}}, \quad \textrm{for } n=1,2,\ldots ,2020.$$
Show that $a_{2020}<\frac1{2020}$.
1997 IMO Shortlist, 26
For every integer $ n \geq 2$ determine the minimum value that the sum $ \sum^n_{i\equal{}0} a_i$ can take for nonnegative numbers $ a_0, a_1, \ldots, a_n$ satisfying the condition $ a_0 \equal{} 1,$ $ a_i \leq a_{i\plus{}1} \plus{} a_{i\plus{}2}$ for $ i \equal{} 0, \ldots, n \minus{} 2.$
1989 IMO Shortlist, 16
The set $ \{a_0, a_1, \ldots, a_n\}$ of real numbers satisfies the following conditions:
[b](i)[/b] $ a_0 \equal{} a_n \equal{} 0,$
[b](ii)[/b] for $ 1 \leq k \leq n \minus{} 1,$ \[ a_k \equal{} c \plus{} \sum^{n\minus{}1}_{i\equal{}k} a_{i\minus{}k} \cdot \left(a_i \plus{} a_{i\plus{}1} \right)\]
Prove that $ c \leq \frac{1}{4n}.$
2010 Bundeswettbewerb Mathematik, 2
The sequence of numbers $a_1, a_2, a_3, ...$ is defined recursively by $a_1 = 1, a_{n + 1} = \lfloor \sqrt{a_1+a_2+...+a_n} \rfloor $ for $n \ge 1$. Find all numbers that appear more than twice at this sequence.
2020 Jozsef Wildt International Math Competition, W26
Let $P_n$ denote the $n$-th Pell number defined by $P_{n+1}=2P_n+P_{n-1}$, $P_0=0$, $P_1=1$. Furthermore, let $T_n$ denote the $n$-th triangular number, that is
$T_n=\binom{n+1}2$. Show that
$$\sum_{n=0}^\infty4T_n\cdot\frac{P_n}{3^{n+2}}=P_3+P_4$$
[i]Proposed by Ángel Plaza[/i]
1980 IMO, 2
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 $x_r - x_s = k.$
2005 Slovenia Team Selection Test, 4
Find the number of sequences of $2005$ terms with the following properties:
(i) No three consecutive terms of the sequence are equal,
(ii) Every term equals either $1$ or $-1$,
(iii) The sum of all terms of the sequence is at least $666$.
2004 Germany Team Selection Test, 1
Consider pairs of the sequences of positive real numbers \[a_1\geq a_2\geq a_3\geq\cdots,\qquad b_1\geq b_2\geq b_3\geq\cdots\] and the sums \[A_n = a_1 + \cdots + a_n,\quad B_n = b_1 + \cdots + b_n;\qquad n = 1,2,\ldots.\] For any pair define $c_n = \min\{a_i,b_i\}$ and $C_n = c_1 + \cdots + c_n$, $n=1,2,\ldots$.
(1) Does there exist a pair $(a_i)_{i\geq 1}$, $(b_i)_{i\geq 1}$ such that the sequences $(A_n)_{n\geq 1}$ and $(B_n)_{n\geq 1}$ are unbounded while the sequence $(C_n)_{n\geq 1}$ is bounded?
(2) Does the answer to question (1) change by assuming additionally that $b_i = 1/i$, $i=1,2,\ldots$?
Justify your answer.
2018 IMO, 2
Find all integers $n \geq 3$ for which there exist real numbers $a_1, a_2, \dots a_{n + 2}$ satisfying $a_{n + 1} = a_1$, $a_{n + 2} = a_2$ and
$$a_ia_{i + 1} + 1 = a_{i + 2},$$
for $i = 1, 2, \dots, n$.
[i]Proposed by Patrik Bak, Slovakia[/i]
1989 IMO Longlists, 55
The set $ \{a_0, a_1, \ldots, a_n\}$ of real numbers satisfies the following conditions:
[b](i)[/b] $ a_0 \equal{} a_n \equal{} 0,$
[b](ii)[/b] for $ 1 \leq k \leq n \minus{} 1,$ \[ a_k \equal{} c \plus{} \sum^{n\minus{}1}_{i\equal{}k} a_{i\minus{}k} \cdot \left(a_i \plus{} a_{i\plus{}1} \right)\]
Prove that $ c \leq \frac{1}{4n}.$
1975 All Soviet Union Mathematical Olympiad, 217
Given a polynomial $P(x)$ with
a) natural coefficients;
b) integer coefficients;
Let us denote with $a_n$ the sum of the digits of $P(n)$ value. Prove that there is a number encountered in the sequence $a_1, a_2, ... , a_n, ...$ infinite times.
2021 Romania Team Selection Test, 3
Let $\alpha$ be a real number in the interval $(0,1).$ Prove that there exists a sequence $(\varepsilon_n)_{n\geq 1}$ where each term is either $0$ or $1$ such that the sequence $(s_n)_{n\geq 1}$ \[s_n=\frac{\varepsilon_1}{n(n+1)}+\frac{\varepsilon_2}{(n+1)(n+2)}+...+\frac{\varepsilon_n}{(2n-1)2n}\]verifies the inequality \[0\leq \alpha-2ns_n\leq\frac{2}{n+1}\] for any $n\geq 2.$
2012 Bogdan Stan, 2
Let be a bounded sequence of positive real numbers $ \left( x_n \right)_{n\ge 1} $ satisfying the recurrence:
$$ x_{n+3} =\sqrt[3]{3x_n-2} . $$
Prove that $ \left( x_n \right)_{n\ge 1} $ is convergent.
[i]Cristinel Mortici[/i]
2014 Contests, 2
Consider the following sequence
$$(a_n)_{n=1}^{\infty}=(1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,1,\dots)$$
Find all pairs $(\alpha, \beta)$ of positive real numbers such that $\lim_{n\to \infty}\frac{\displaystyle\sum_{k=1}^n a_k}{n^{\alpha}}=\beta$.
(Proposed by Tomas Barta, Charles University, Prague)