Found problems: 963
2016 Saint Petersburg Mathematical Olympiad, 7
A sequence of $N$ consecutive positive integers is called [i]good [/i] if it is possible to choose two of these numbers so that their product is divisible by the sum of the other $N-2$ numbers. For which $N$ do there exist infinitely many [i]good [/i] sequences?
1992 IMO Longlists, 79
Let $ \lfloor x \rfloor$ denote the greatest integer less than or equal to $ x.$ Pick any $ x_1$ in $ [0, 1)$ and define the sequence $ x_1, x_2, x_3, \ldots$ by $ x_{n\plus{}1} \equal{} 0$ if $ x_n \equal{} 0$ and $ x_{n\plus{}1} \equal{} \frac{1}{x_n} \minus{} \left \lfloor \frac{1}{x_n} \right \rfloor$ otherwise. Prove that
\[ x_1 \plus{} x_2 \plus{} \ldots \plus{} x_n < \frac{F_1}{F_2} \plus{} \frac{F_2}{F_3} \plus{} \ldots \plus{} \frac{F_n}{F_{n\plus{}1}},\]
where $ F_1 \equal{} F_2 \equal{} 1$ and $ F_{n\plus{}2} \equal{} F_{n\plus{}1} \plus{} F_n$ for $ n \geq 1.$
2019 Swedish Mathematical Competition, 6
Is there an infinite sequence of positive integers $\{a_n\}_{n = 1}^{\infty}$ which contains each positive integer exactly once and is such that the number $a_n + a_{n + 1} $ is a perfect square for each $n$?
2019 IMO Shortlist, A3
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
\[
\sum_{i \in X} b_{i}=1.
\]
2022 Greece Team Selection Test, 3
Find largest possible constant $M$ such that, for any sequence $a_n$, $n=0,1,2,...$ of real numbers, that satisfies the conditions :
i) $a_0=1$, $a_1=3$
ii) $a_0+a_1+...+a_{n-1} \ge 3 a_n - a_{n+1}$ for any integer $n\ge 1$
to be true that
$$\frac{a_{n+1}}{a_n} >M$$ for any integer $n\ge 0$.
1998 North Macedonia National Olympiad, 5
The sequence $(a_n)$ is defined by $a_1 =\sqrt2$ and $a_{n+1} =\sqrt{2-\sqrt{4-a_n^2}}$.
Let $b_n =2^{n+1}a_n$. Prove that $b_n \le 7$ and $b_n < b_{n+1}$ for all $n$.
2020 Romanian Master of Mathematics Shortlist, C4
A ternary sequence is one whose terms all lie in the set $\{0, 1, 2\}$. Let $w$ be a length $n$ ternary sequence $(a_1,\ldots,a_n)$. Prove that $w$ can be extended leftwards and rightwards to a length $m=6n$ ternary sequence \[(d_1,\ldots,d_m) = (b_1,\ldots,b_p,a_1,\ldots,a_n,c_1,\ldots,c_q), \quad p,q\geqslant 0,\]containing no length $t > 2n$ palindromic subsequence.
(A sequence is called palindromic if it reads the same rightwards and leftwards. A length $t$ subsequence of $(d_1,\ldots,d_m)$ is a sequence of the form $(d_{i_1},\ldots,d_{i_t})$, where $1\leqslant i_1<\cdots<i_t \leqslant m$.)
1973 Swedish Mathematical Competition, 2
The Fibonacci sequence $f_1,f_2,f_3,\dots$ is defined by $f_1=f_2=1$, $f_{n+2}=f_{n+1}+f_n$. Find all $n$ such that $f_n = n^2$.
2010 Germany Team Selection Test, 3
Find all positive integers $n$ such that there exists a sequence of positive integers $a_1$, $a_2$,$\ldots$, $a_n$ satisfying: \[a_{k+1}=\frac{a_k^2+1}{a_{k-1}+1}-1\] for every $k$ with $2\leq k\leq n-1$.
[i]Proposed by North Korea[/i]
1982 IMO Longlists, 17
[b](a)[/b] Find the rearrangement $\{a_1, \dots , a_n\}$ of $\{1, 2, \dots, n\}$ that maximizes
\[a_1a_2 + a_2a_3 + \cdots + a_na_1 = Q.\]
[b](b)[/b] Find the rearrangement that minimizes $Q.$
Mathley 2014-15, 5
Given the sequence $(u_n)_{n=1}^{\infty}$, where $u_1 = 1, u_2 = 2$, and $u_{n + 2} = u_{n + 1} +u_ n+ \frac{(-1)^n-1}{2}$ for any positive integers $n$. Prove that every positive integers can be expressed as the sum of some distinguished numbers of the sequence of numbers $(u_n)_{n=1}^{\infty}$
Nguyen Duy Thai Son, The University of Danang, Da Nang.
2018 IMO Shortlist, A2
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]
2015 APMO, 3
A sequence of real numbers $a_0, a_1, . . .$ is said to be good if the following three conditions hold.
(i) The value of $a_0$ is a positive integer.
(ii) For each non-negative integer $i$ we have $a_{i+1} = 2a_i + 1 $ or $a_{i+1} =\frac{a_i}{a_i + 2} $
(iii) There exists a positive integer $k$ such that $a_k = 2014$.
Find the smallest positive integer $n$ such that there exists a good sequence $a_0, a_1, . . .$ of real numbers with the property that $a_n = 2014$.
[i]Proposed by Wang Wei Hua, Hong Kong[/i]
2023 Indonesia TST, 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 $a^n$.
1984 Poland - Second Round, 6
The sequence $(x_n)$ is defined by formulas
$$
x_1=c,\; x_{n+1} = cx_n + \sqrt{(c^2-1)(x_n^2-1)} \quad\text{ for }\quad n=1,2,\ldots$$
Prove that if $ c $ is a natural number, then all numbers $ x_n $ are natural.
2015 Belarus Team Selection Test, 2
In the sequence of digits $2,0,2,9,3,...$ any digit it equal to the last digit in the decimal representation of the sum of four previous digits. Do the four numbers $2,0,1,5$ in that order occur in the sequence?
Folklore
1997 Akdeniz University MO, 3
$(x_n)$ be a sequence with $x_1=0$,
$$x_{n+1}=5x_n + \sqrt{24x_n^2+1}$$.
Prove that for $k \geq 2$ $x_k$ is a natural number.
2007 Germany Team Selection Test, 1
A sequence of real numbers $ a_{0},\ a_{1},\ a_{2},\dots$ is defined by the formula
\[ a_{i \plus{} 1} \equal{} \left\lfloor a_{i}\right\rfloor\cdot \left\langle a_{i}\right\rangle\qquad\text{for}\quad i\geq 0;
\]here $a_0$ is an arbitrary real number, $\lfloor a_i\rfloor$ denotes the greatest integer not exceeding $a_i$, and $\left\langle a_i\right\rangle=a_i-\lfloor a_i\rfloor$. Prove that $a_i=a_{i+2}$ for $i$ sufficiently large.
[i]Proposed by Harmel Nestra, Estionia[/i]
2017 China National Olympiad, 1
The sequences $\{u_{n}\}$ and $\{v_{n}\}$ are defined by $u_{0} =u_{1} =1$ ,$u_{n}=2u_{n-1}-3u_{n-2}$ $(n\geq2)$ , $v_{0} =a, v_{1} =b , v_{2}=c$ ,$v_{n}=v_{n-1}-3v_{n-2}+27v_{n-3}$ $(n\geq3)$. There exists a positive integer $N$ such that when $n> N$, we have $u_{n}\mid v_{n}$ . Prove that $3a=2b+c$.
2020 European Mathematical Cup, 2
A positive integer $k\geqslant 3$ is called[i] fibby[/i] if there exists a positive integer $n$ and positive integers $d_1 < d_2 < \ldots < d_k$ with the following properties: \\ $\bullet$ $d_{j+2}=d_{j+1}+d_j$ for every $j$ satisfying $1\leqslant j \leqslant k-2$, \\ $\bullet$ $d_1, d_2, \ldots, d_k$ are divisors of $n$, \\ $\bullet$ any other divisor of $n$ is either less than $d_1$ or greater than $d_k$.
Find all fibby numbers. \\ \\ [i]Proposed by Ivan Novak.[/i]
Revenge EL(S)MO 2024, 3
Fix a positive integer $n$. Define sequences $a, b, c \in \mathbb{Q}^{n+1}$ by $(a_0, b_0, c_0) = (0, 0, 1)$ and
\[
a_k = (n-k+1) \cdot c_{k-1}, \quad
b_k = \binom nk - c_k - a_k, \quad \text{and} \quad
c_k = \frac{b_{k-1}}{k}
\]
for each integer $1 \leq k \leq n$.
$ $ $ $ $ $ $ $ $ $ Determine for which $n$ it happens that $a, b, c \in \mathbb{Z}^{n+1}$.
Proposed by [i]Jonathan Du[/i]
2024 Junior Macedonian Mathematical Olympiad, 4
Let $a_1, a_2, ..., a_n$ be a sequence of perfect squares such that $a_{i + 1}$ can be obtained by concatenating a digit to the right of $a_i$. Determine all such sequences that are of maximum length.
[i]Proposed by Ilija Jovčeski[/i]
2014 Balkan MO Shortlist, A6
$\boxed{A6}$The sequence $a_0,a_1,...$ is defined by the initial conditions $a_0=1,a_1=6$ and the recursion $a_{n+1}=4a_n-a_{n-1}+2$ for $n>1.$Prove that $a_{2^k-1}$ has at least three prime factors for every positive integer $k>3.$
2022 Vietnam National Olympiad, 1
Let $a$ be a non-negative real number and a sequence $(u_n)$ defined as:
$u_1=6,u_{n+1} = \frac{2n+a}{n} + \sqrt{\frac{n+a}{n}u_n+4}, \forall n \ge 1$
a) With $a=0$, prove that there exist a finite limit of $(u_n)$ and find that limit
b) With $a \ge 0$, prove that there exist a finite limit of $(u_n)$
2020 Thailand TSTST, 5
Let $\{a_n\}$ be a sequence of positive integers such that $a_{n+1} = a_n^2+1$ for all $n \geq 1$. Prove that there is no positive integer $N$ such that $$\prod_{k=1}^N(a_k^2+a_k+1)$$ is a perfect square.