Found problems: 963
2023 Estonia Team Selection Test, 1
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 $a^n$.
2022 Korea National Olympiad, 3
Suppose that the sequence $\{a_n\}$ of positive integers satisfies the following conditions:
[list]
[*]For an integer $i \geq 2022$, define $a_i$ as the smallest positive integer $x$ such that $x+\sum_{k=i-2021}^{i-1}a_k$ is a perfect square.
[*]There exists infinitely many positive integers $n$ such that $a_n=4\times 2022-3$.
[/list]
Prove that there exists a positive integer $N$ such that $\sum_{k=n}^{n+2021}a_k$ is constant for every integer $n \geq N$.
And determine the value of $\sum_{k=N}^{N+2021}a_k$.
2013 IMO Shortlist, A1
Let $n$ be a positive integer and let $a_1, \ldots, a_{n-1} $ be arbitrary real numbers. Define the sequences $u_0, \ldots, u_n $ and $v_0, \ldots, v_n $ inductively by $u_0 = u_1 = v_0 = v_1 = 1$, and $u_{k+1} = u_k + a_k u_{k-1}$, $v_{k+1} = v_k + a_{n-k} v_{k-1}$ for $k=1, \ldots, n-1.$
Prove that $u_n = v_n.$
2015 Saudi Arabia GMO TST, 2
Find the number of strictly increasing sequences of nonnegative integers with the first term $0$ and the last term $15$, and among any two consecutive terms, exactly one of them is even.
Lê Anh Vinh
2000 Belarus Team Selection Test, 8.2
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.
2021 South East Mathematical Olympiad, 4
Suppose there are $n\geq{5}$ different points arbitrarily arranged on a circle, the labels are $1, 2,\dots $, and $n$, and the permutation is $S$. For a permutation , a “descending chain” refers to several consecutive points on the circle , and its labels is a clockwise descending sequence (the length of sequence is at least $2$), and the descending chain cannot be extended to longer .The point with the largest label in the chain is called the "starting point of descent", and the other points in the chain are called the “non-starting point of descent” . For example: there are two descending chains $5, 2$and $4, 1$ in $5, 2, 4, 1, 3$ arranged in a clockwise direction, and $5$ and $4$ are their starting points of descent respectively, and $2, 1$ is the non-starting point of descent . Consider the following operations: in the first round, find all descending chains in the permutation $S$, delete all non-starting points of descent , and then repeat the first round of operations for the arrangement of the remaining points, until no more descending chains can be found. Let $G(S)$ be the number of all descending chains that permutation $S$ has appeared in the operations, $A(S)$ be the average value of $G(S)$of all possible n-point permutations $S$.
(1) Find $A(5)$.
(2)For $n\ge{6}$ , prove that $\frac{83}{120}n-\frac{1}{2} \le A(S) \le \frac{101}{120}n-\frac{1}{2}.$
1953 Moscow Mathematical Olympiad, 257
Let $x_0 = 10^9$, $x_n = \frac{x^2_{n-1}+2}{2x_{n-1}}$ for $n > 0$. Prove that $0 < x_{36} - \sqrt2 < 10^{-9}$.
2018 Brazil National Olympiad, 5
Consider the sequence in which $a_1 = 1$ and $a_n$ is obtained by juxtaposing the decimal representation of $n$ at the end of the decimal representation of $a_{n-1}$. That is, $a_1 = 1$, $a_2 = 12$, $a_3 = 123$, $\dots$ , $a_9 = 123456789$, $a_{10} = 12345678910$ and so on. Prove that infinitely many numbers of this sequence are multiples of $7$.
2020 Francophone Mathematical Olympiad, 3
Let $(a_i)_{i\in \mathbb{N}}$ be a sequence with $a_1=\frac{3}2$ such that
$$a_{n+1}=1+\frac{n}{a_n}$$
Find $n$ such that $2020\le a_n <2021$
2016 China Team Selection Test, 4
Let $c,d \geq 2$ be naturals. Let $\{a_n\}$ be the sequence satisfying $a_1 = c, a_{n+1} = a_n^d + c$ for $n = 1,2,\cdots$.
Prove that for any $n \geq 2$, there exists a prime number $p$ such that $p|a_n$ and $p \not | a_i$ for $i = 1,2,\cdots n-1$.
2024 Czech and Slovak Olympiad III A, 5
Let $(a_k)^{\infty}_{k=0}$ be a sequence of real numbers such that if $k$ is a non-negative integer, then
$$a_{k+1} = 3a_k - \lfloor 2a_k \rfloor - \lfloor a_k \rfloor.$$
Definitely all positive integers $n$ such that if $a_0 = 1/n$, then this sequence is constant after a certain term.
1980 IMO Longlists, 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 $x_r - x_s = k.$
1996 Singapore Team Selection Test, 3
Let $S = \{0, 1, 2, .., 1994\}$. Let $a$ and $b$ be two positive numbers in $S$ which are relatively prime. Prove that the elements of $S$ can be arranged into a sequence $s_1, s_2, s_3,... , s_{1995}$ such that $s_{i+1} - s_i \equiv \pm a$ or $\pm b$ (mod $1995$) for $i = 1, 2, ... , 1994$
2023 Francophone Mathematical Olympiad, 1
Let $u_0, u_1, u_2, \ldots$ be integers such that $u_0 = 100$; $u_{k+2} \geqslant 2 + u_k$ for all $k \geqslant 0$; and $u_{\ell+5} \leqslant 5 + u_\ell$ for all $\ell \geqslant 0$. Find all possible values for the integer $u_{2023}$.
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 $x_r - x_s = k.$
1994 Austrian-Polish Competition, 2
The sequences $(a_n)$ and (c_n) are given by $a_0 =\frac12$, $c_0=4$ , and for $n \ge 0$ , $a_{n+1}=\frac{2a_n}{1+a_n^2}$, $c_{n+1}=c_n^2-2c_n+2$
Prove that for all $n\ge 1$, $a_n=\frac{2c_0c_1...c_{n-1}}{c_n}$
1978 IMO Longlists, 24
Let $0<f(1)<f(2)<f(3)<\ldots$ a sequence with all its terms positive$.$ The $n-th$ positive integer which doesn't belong to the sequence is $f(f(n))+1.$ Find $f(240).$
2010 IMO Shortlist, 7
Let $P_1, \ldots , P_s$ be arithmetic progressions of integers, the following conditions being satisfied:
[b](i)[/b] each integer belongs to at least one of them;
[b](ii)[/b] each progression contains a number which does not belong to other progressions.
Denote by $n$ the least common multiple of the ratios of these progressions; let $n=p_1^{\alpha_1} \cdots p_k^{\alpha_k}$ its prime factorization.
Prove that \[s \geq 1 + \sum^k_{i=1} \alpha_i (p_i - 1).\]
[i]Proposed by Dierk Schleicher, Germany[/i]
2025 Philippine MO, P3
Let $d$ be a positive integer. Define the sequence $a_1, a_2, a_3, \dots$ such that \[\begin{cases} a_1 = 1 \\ a_{n+1} = n\left\lfloor\frac{a_n}{n}\right\rfloor + d, \quad n \ge 1.\end{cases}\] Prove that there exists a positive integer $M$ such that $a_M, a_{M+1}, a_{M+2}, \dots$ is an arithmetic sequence.
2019 Teodor Topan, 3
Let be a natural number $ m\ge 2. $
[b]a)[/b] Let be $ m $ pairwise distinct rational numbers. Prove that there is an ordering of these numbers such that these are terms of an arithmetic progression.
[b]b)[/b] Given that for any $ m $ pairwise distinct real numbers there is an ordering of them such that they are terms of an arithmetic sequence, determine the number $ m. $
[i]Bogdan Blaga[/i]
2008 Singapore MO Open, 3
let n,m be positive integers st $m>n\geq 5$ with m depending on n.
consider the sequence $a_1,a_2,...a_m$ where
$a_i=i$ for $i=1,...,n$
$a_{n+j}=a_{3j}+a_{3j-1}+a_{3j-2}$ for $j=1,..,m-n$
with $m-3(m-n)=$1 or 2, ie $a_m=a_{m-k}+a_{m-k-1}+a_{m-k-2}$ where k=1 or 2
(Thus if $n=5$, the sequence is 1,2,3,4,5,6,15
and if $n=8$, the sequence is 1,2,3,4,5,6,7,8,6,15,21)
Find $S=a_1+...+a_m$ if (i) $n=2007$ (ii) $n=2008$
1983 IMO Longlists, 52
Let $(F_n)_{n\geq 1} $ be the Fibonacci sequence $F_1 = F_2 = 1, F_{n+2} = F_{n+1} + F_n (n \geq 1),$ and $P(x)$ the polynomial of degree $990$ satisfying
\[ P(k) = F_k, \qquad \text{ for } k = 992, . . . , 1982.\]
Prove that $P(1983) = F_{1983} - 1.$
2016 Saudi Arabia Pre-TST, 2.2
Given four numbers $x, y, z, t$, let $(a, b, c, d)$ be a permutation of $(x, y, z, t)$ and set $x_1 =|a- b|$, $y_1 = |b-c|$, $z_1 = |c-d|$, and $t_1 = |d -a|$. From $x_1, y_1, z_1, t_1$, form in the same fashion the numbers $x_2, y_2, z_2, t_2$, and so on. It is known that $x_n = x, y_n = y, z_n = z, t_n = t$ for some $n$. Find all possible values of $(x, y, z, t)$.
2020 Tournament Of Towns, 4
For an infinite sequence $a_1, a_2,. . .$ denote as it's [i]first derivative[/i] is the sequence $a'_n= a_{n + 1} - a_n$ (where $n = 1, 2,..$.), and her $k$- th derivative as the first derivative of its $(k-1)$-th derivative ($k = 2, 3,...$). We call a sequence [i]good[/i] if it and all its derivatives consist of positive numbers.
Prove that if $a_1, a_2,. . .$ and $b_1, b_2,. . .$ are good sequences, then sequence $a_1\cdot b_1, a_2 \cdot b_2,..$ is also a good one.
R. Salimov
2013 IMAR Test, 2
For every non-negative integer $n$ , let $s_n$ be the sum of digits in the decimal expansion of $2^n$. Is the sequence $(s_n)_{n \in \mathbb{N}}$ eventually increasing ?