Found problems: 1239
1985 Tournament Of Towns, (084) T5
Every member of a given sequence, beginning with the second , is equal to the sum of the preceding one and the sum of its digits . The first member equals $1$ . Is there, among the members of this sequence, a number equal to $123456$ ?
(S. Fomin , Leningrad)
2015 Brazil Team Selection Test, 2
Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.)
[i]Proposed by Hong Kong[/i]
1975 Spain Mathematical Olympiad, 6
Let $\{x_n\}$ and $\{y_n\}$ be two sequences of natural numbers defined as follow:
$x_1 = 1, \,\,\, x_2 = 1, \,\,\, x_{n+2} = x_{n+1} + 2x_n$ for $n = 1, 2, 3, ...$
$y_1 = 1, \,\,\, y_2 = 7, \,\,\, y_{n+2} = 2y_{n+1} + 3y_n$ for $n = 1, 2, 3, ...$
Prove that, except for the case $x_1 = y_1 = 1$, there is no natural value that occurs in the two sequences.
2016 Switzerland - Final Round, 6
Let $a_n$ be a sequence of natural numbers defined by $a_1 = m$ and for $n > 1$. We call apair$ (a_k, a_{\ell })$ [i]interesting [/i] if
(i) $0 < \ell - k < 2016$,
(ii) $a_k$ divides $a_{\ell }$.
Show that there exists a $m$ such that the sequence $a_n$ contains no interesting pair.
2022 Korea National Olympiad, 7
Suppose that the sequence $\{a_n\}$ of positive reals satisfies the following conditions:
[list]
[*]$a_i \leq a_j$ for every positive integers $i <j$.
[*]For any positive integer $k \geq 3$, the following inequality holds:
$$(a_1+a_2)(a_2+a_3)\cdots(a_{k-1}+a_k)(a_k+a_1)\leq (2^k+2022)a_1a_2\cdots a_k$$
[/list]
Prove that $\{a_n\}$ is constant.
2024 VJIMC, 4
Let $(b_n)_{n \ge 0}$ be a sequence of positive integers satisfying $b_n=d\left(\sum_{i=0}^{n-1} b_k\right)$ for all $n \ge 1$. (By $d(m)$ we denote the number of positive divisors of $m$.)
a) Prove that $(b_n)_{n \ge 0}$ is unbounded.
b) Prove that there are infinitely many $n$ such that $b_n>b_{n+1}$.
1966 Swedish Mathematical Competition, 4
Let $f(x) = 1 + \frac{2}{x}$. Put $f_1(x) = f(x)$, $f_2(x) = f(f_1(x))$, $f_3(x) = f(f_2(x))$, $... $. Find the solutions to $x = f_n(x)$ for $n > 0$.
2002 District Olympiad, 1
Determine the sequence of complex numbers $ \left( x_n\right)_{n\ge 1} $ for which $ 1=x_1, $ and for any natural number $ n, $ the following equality is true:
$$ 4\left( x_1x_n+2x_2x_{n-1}+3x_3x_{n-2}+\cdots +nx_nx_1\right) =(1+n)\left( x_1x_2+x_2x_3+\cdots +x_{n-1}x_n +x_nx_{n+1}\right) . $$
1996 IMC, 4
Let $a_{1}=1$, $a_{n}=\frac{1}{n} \sum_{k=1}^{n-1}a_{k}a_{n-k}$ for $n\geq 2$. Show that
i) $\limsup_{n\to \infty} |a_{n}|^{\frac{1}{n}}<2^{-\frac{1}{2}}$;
ii) $\limsup_{n\to \infty} |a_{n}|^{\frac{1}{n}}\geq \frac{2}{3}$
2019 Danube Mathematical Competition, 3
Let be a sequence of $ 51 $ natural numbers whose sum is $ 100. $ Show that for any natural number $ 1\le k<100 $ there are some consecutive numbers from this sequence whose sum is $ k $ or $ 100-k. $
2023 Mongolian Mathematical Olympiad, 3
Let $m$ be a positive integer. We say that a sequence of positive integers written on a circle is [i] good [/i], if the sum of any $m$ consecutive numbers on this circle is a power of $m$.
1. Let $n \geq 2$ be a positive integer. Prove that for any [i] good [/i] sequence with $mn$ numbers, we can remove $m$ numbers such that the remaining $mn-m$ numbers form a [i] good [/i] sequence.
2. Prove that in any [i] good [/i] sequence with $m^2$ numbers, we can always find a number that was repeated at least $m$ times in the sequence.
2024 IMC, 8
Define the sequence $x_1,x_2,\dots$ by the initial terms $x_1=2, x_2=4$, and the recurrence relation
\[x_{n+2}=3x_{n+1}-2x_n+\frac{2^n}{x_n} \quad \text{for} \quad n \ge 1.\]
Prove that $\lim_{n \to \infty} \frac{x_n}{2^n}$ exists and satisfies
\[\frac{1+\sqrt{3}}{2} \le \lim_{n \to \infty} \frac{x_n}{2^n} \le \frac{3}{2}.\]
1977 IMO Longlists, 57
In a finite sequence of real numbers the sum of any seven successive terms is negative and the sum of any eleven successive terms is positive. Determine the maximum number of terms in the sequence.
1947 Putnam, A5
Let $a_1 , b_1 , c_1$ be positive real numbers whose sum is $1,$ and for $n=1, 2, \ldots$ we define
$$a_{n+1}= a_{n}^{2} +2 b_n c_n, \;\;\;b_{n+1}= b_{n}^{2} +2 a_n c_n, \;\;\; c_{n+1}= c_{n}^{2} +2 a_n b_n.$$
Show that $a_n , b_n ,c_n$ approach limits as $n\to \infty$ and find those limits.
2021 Science ON all problems, 1
Consider the sequence $(a_n)_{n\ge 1}$ such that $a_1=1$ and $a_{n+1}=\sqrt{a_n+n^2}$, $\forall n\ge 1$.
$\textbf{(a)}$ Prove that there is exactly one rational number among the numbers $a_1,a_2,a_3,\dots$.
$\textbf{(b)}$ Consider the sequence $(S_n)_{n\ge 1}$ such that
$$S_n=\sum_{i=1}^n\frac{4}{\left (\left \lfloor a_{i+1}^2\right \rfloor-\left \lfloor a_i^2\right \rfloor\right)\left(\left \lfloor a_{i+2}^2\right \rfloor-\left \lfloor a_{i+1}^2\right \rfloor\right)}.$$
Prove that there exists an integer $N$ such that $S_n>0.9$, $\forall n>N$.
[i] (Stefan Obadă)[/i]
1981 IMO Shortlist, 9
A sequence $(a_n)$ is defined by means of the recursion
\[a_1 = 1, a_{n+1} = \frac{1 + 4a_n +\sqrt{1+ 24a_n}}{16}.\]
Find an explicit formula for $a_n.$
2019 Saudi Arabia Pre-TST + Training Tests, 3.3
Define sequence of positive integers $(a_n)$ as $a_1 = a$ and $a_{n+1} = a^2_n + 1$ for $n \ge 1$. Prove that there is no index $n$ for which $$\prod_{k=1}^{n} \left(a^2_k + a_k + 1\right)$$ is a perfect square.
2006 Thailand Mathematical Olympiad, 12
Let $a_n = 2^{3n-1} + 3^{6n-2} + 5^{6n-3}$. Compute gcd$(a_1, a_2, ... , a_{25})$
2021 Belarusian National Olympiad, 10.1
An arbitrary positive number $a$ is given. A sequence ${a_n}$ is defined by equalities $a_1=\frac{a}{a+1}$ and $a_{n+1}=\frac{aa_n}{a^2+a_n-aa_n}$ for all $n \geq 1$
Find the minimal constant $C$ such that inequality $$a_1+a_1a_2+\ldots+a_1\ldots a_m<C$$ holds for all positive integers $m$ regardless of $a$
2025 AIME, 13
Let the sequence of rationals $x_1,x_2,\dots$ be defined such that $x_1=\frac{25}{11}$ and
\[x_{k+1}=\frac{1}{3}\left(x_k+\frac{1}{x_k}-1\right).\]
$x_{2025}$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find the remainder when $m+n$ is divided by $1000$.
1987 Greece National Olympiad, 3
There is no sequence $x_n$ strictly increasing with terms natural numbers such that : $$ x_n+x_{k}=x_{nk}, \ \ for \, any \,\,\, n, k \in \mathbb{N}^*$$
2004 Thailand Mathematical Olympiad, 7
Let f be a function such that $f(0) = 0, f(1) = 1$, and $f(n) = 2f(n-1)- f(n- 2) + (-1)^n(2n - 4)$ for all integers $n \ge 2$. Find f(n) in terms of $n$.
2019 Serbia National MO, 6
Sequences $(a_n)_{n=0}^{\infty}$ and $(b_n)_{n=0}^{\infty}$ are defined with recurrent relations :
$$a_0=0 , \;\;\; a_1=1, \;\;\;\; a_{n+1}=\frac{2018}{n} a_n+ a_{n-1}\;\;\; \text {for }\;\;\; n\geq 1$$ and
$$b_0=0 , \;\;\; b_1=1, \;\;\;\; b_{n+1}=\frac{2020}{n} b_n+ b_{n-1}\;\;\; \text {for }\;\;\; n\geq 1$$
Prove that :$$\frac{a_{1010}}{1010}=\frac{b_{1009}}{1009}$$
1999 Singapore Senior Math Olympiad, 3
Let $\{a_1,a_2,...,a_{100}\}$ be a sequence of $100$ distinct real numbers. Show that there exists either an increasing subsequence
$a_{i_1}<a_{i_2}<...<a_{i_{10}}$ $(i_1<i_2<...<i_{10})$ of $10$ numbers, or a decreasing subsequence
$ a_{j_1}>a_{j_2}>...>a_{j_{12}}$ $(j_1<j_2<...<j_{12})$ of $12$ numbers, or both.
2021 Romanian Master of Mathematics Shortlist, A4
Let $f: \mathbb{R} \to \mathbb{R}$ be a non-decreasing function such that $f(y) - f(x) < y - x$ for all real numbers
$x$ and $y > x$. The sequence $u_1,u_2,\ldots$ of real numbers is such that $u_{n+2} = f(u_{n+1}) - f(u_n)$ for all $n\geq 1$. Prove that for any $\varepsilon > 0$ there exists a positive integer $N$ such that $|u_n| < \varepsilon$ for all $n\geq N$.