Found problems: 88
2020 Korea National Olympiad, 5
For some positive integer $n$, there exists $n$ different positive integers $a_1, a_2, ..., a_n$ such that
$(1)$ $a_1=1, a_n=2000$
$(2)$ $\forall i\in \mathbb{Z}$ $s.t.$ $2\le i\le n, a_i -a_{i-1}\in \{-3,5\}$
Determine the maximum value of n.
2007 Korea Junior Math Olympiad, 1
A sequence $a_1,a_2,...,a_{2007}$ where $a_i \in\{2,3\}$ for $i = 1,2,...,2007$ and an integer sequence $x_1,x_2,...,x_{2007}$ satis fies the following: $a_ix_i + x_{i+2 }\equiv 0$ ($mod 5$) , where the indices are taken modulo $2007$. Prove that $x_1,x_2,...,x_{2007}$ are all multiples of $5$.
2021 China Girls Math Olympiad, 4
Call a sequence of positive integers $(a_n)_{n \ge 1}$ a "CGMO sequence" if $(a_n)_{n \ge 1}$ strictly increases, and for all integers $n \ge 2022$, $a_n$ is the smallest integer such that there exists a non-empty subset of $\{a_{1}, a_{2}, \cdots, a_{n-1} \}$ $A_n$ where $a_n \cdot \prod\limits_{a \in A_n} a$ is a perfect square.
Proof: there exists $c_1, c_2 \in \mathbb{R}^{+}$ s.t. for any "CGMO sequence" $(a_n)_{n \ge 1}$ , there is a positive integer $N$ that satisfies any $n \ge N$, $c_1 \cdot n^2 \le a_n \le c_2 \cdot n^2$.
2023 Indonesia TST, N
Given an integer $a>1$. Prove that there exists a sequence of positive integers
\[ n_1, n_2, n_3, \ldots \]
Such that
\[ \gcd(a^{n_i+1} + a^{n_i} - 1, \ a^{n_j + 1} + a^{n_j} - 1) =1 \] For every $i \neq j$.
1996 Poland - Second Round, 4
Let $a_1$, $a_2$ ,..., $a_{99}$ be a sequence of digits from the set ${0,...,9}$ such that if for some $n$ ∈ $N$, $a_n = 1$, then $a_{n+1} \ne 2$, and if $a_n = 3$ then $a_{n+1} \ne 4$. Prove that there exist indices $k,l$ ∈ ${1,...,98}$ such that $a_k = a_l$ and $a_{k+1} = a_{l+1}$.
1998 IMO Shortlist, 4
A sequence of integers $ a_{1},a_{2},a_{3},\ldots$ is defined as follows: $ a_{1} \equal{} 1$ and for $ n\geq 1$, $ a_{n \plus{} 1}$ is the smallest integer greater than $ a_{n}$ such that $ a_{i} \plus{} a_{j}\neq 3a_{k}$ for any $ i,j$ and $ k$ in $ \{1,2,3,\ldots ,n \plus{} 1\}$, not necessarily distinct. Determine $ a_{1998}$.
2024-IMOC, A3
Find all infinite integer sequences $a_1,a_2,\ldots$ satisfying
\[a_{n+2}^{a_{n+1}}=a_{n+1}+a_n\] holds for all $n\geq 1$. Define $0^0=1$
1996 IMO, 6
Let $ p,q,n$ be three positive integers with $ p \plus{} q < n$. Let $ (x_{0},x_{1},\cdots ,x_{n})$ be an $ (n \plus{} 1)$-tuple of integers satisfying the following conditions :
(a) $ x_{0} \equal{} x_{n} \equal{} 0$, and
(b) For each $ i$ with $ 1\leq i\leq n$, either $ x_{i} \minus{} x_{i \minus{} 1} \equal{} p$ or $ x_{i} \minus{} x_{i \minus{} 1} \equal{} \minus{} q$.
Show that there exist indices $ i < j$ with $ (i,j)\neq (0,n)$, such that $ x_{i} \equal{} x_{j}$.
2023 Estonia Team Selection Test, 4
A $\pm 1$-[i]sequence[/i] is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and $$\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.$$
2021 Korea Junior Math Olympiad, 2
Let $\{a_n\}$ be a sequence of integers satisfying the following conditions.
[list]
[*] $a_1=2021^{2021}$
[*] $0 \le a_k < k$ for all integers $k \ge 2$
[*] $a_1-a_2+a_3-a_4+ \cdots + (-1)^{k+1}a_k$ is multiple of $k$ for all positive integers $k$.
[/list]
Determine the $2021^{2022}$th term of the sequence $\{a_n\}$.
1994 Mexico National Olympiad, 1
The sequence $1, 2, 4, 5, 7, 9 ,10, 12, 14, 16, 17, ... $ is formed as follows. First we take one odd number, then two even numbers, then three odd numbers, then four even numbers, and so on. Find the number in the sequence which is closest to $1994$.
1952 Moscow Mathematical Olympiad, 216
A sequence of integers is constructed as follows: $a_1$ is an arbitrary three-digit number, $a_2$ is the sum of squares of the digits of $a_1, a_3$ is the sum of squares of the digits of $a_2$, etc. Prove that either $1$ or $4$ must occur in the sequence $a_1, a_2, a_3, ....$
1954 Moscow Mathematical Olympiad, 282
Given a sequence of numbers $a_1, a_2, ..., a_{15}$, one can always construct a new sequence $b_1,b_2, ..., b_{15}$, where $b_i$ is equal to the number of terms in the sequence $\{a_k\}^{15}_{k=1}$ less than $a_i$ ($i = 1, 2,..., 15$). Is there a sequence $\{a_k\}^{15}_{k=1}$ for which the sequence $\{b_k\}^{15}_{k=1}$ is $$1, 0, 3, 6, 9, 4, 7, 2, 5, 8, 8, 5, 10, 13, 13 \,?$$
1998 Nordic, 3
(a) For which positive numbers $n$ does there exist a sequence $x_1, x_2, ..., x_n$, which contains each of the numbers $1, 2, ..., n$ exactly once and for which $x_1 + x_2 +... + x_k$ is divisible by $k$ for each $k = 1, 2,...., n$?
(b) Does there exist an infinite sequence $x_1, x_2, x_3, ..., $ which contains every positive integer exactly once and such that $x_1 + x_2 +... + x_k$ is divisible by $k$ for every positive integer $k$?
2023 Brazil Team Selection Test, 1
A $\pm 1$-[i]sequence[/i] is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and $$\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.$$
2015 Indonesia MO Shortlist, C6
Let $k$ be a fixed natural number. In the infinite number of real line, each integer is colored with color ..., red, green, blue, red, green, blue, ... and so on. A number of flea settles at first at integer points. On each turn, a flea will jump over the other tick so that the distance $k$ is the original distance. Formally, we may choose $2$ tails $A, B$ that are spaced $n$ and move $A$ to the different side of $B$ so the current distance is $kn$. Some fleas may occupy the same point because we consider the size of fleas very small. Determine all the values of $k$ so that, whatever the initial position of the ticks, we always get a position where all ticks land on the same color.
1964 Vietnam National Olympiad, 4
Define the sequence of positive integers $f_n$ by $f_0 = 1, f_1 = 1, f_{n+2} = f_{n+1} + f_n$. Show that $f_n =\frac{ (a^{n+1} - b^{n+1})}{\sqrt5}$, where $a, b$ are real numbers such that $a + b = 1, ab = -1$ and $a > b$.
2016 Bosnia and Herzegovina Team Selection Test, 3
For an infinite sequence $a_1<a_2<a_3<...$ of positive integers we say that it is [i]nice[/i] if for every positive integer $n$ holds $a_{2n}=2a_n$. Prove the following statements:
$a)$ If there is given a [i]nice[/i] sequence and prime number $p>a_1$, there exist some term of the sequence which is divisible by $p$.
$b)$ For every prime number $p>2$, there exist a [i]nice[/i] sequence such that no terms of the sequence are divisible by $p$.
2023 Germany Team Selection Test, 2
A $\pm 1$-[i]sequence[/i] is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and $$\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.$$
1988 Spain Mathematical Olympiad, 1
A sequence of integers $(x_n)_{n=1}^{\infty}$ satisfies $x_1 = 1$ and $x_n < x_{n+1} \le 2n$ for all $n$.
Show that for every positive integer $k$ there exist indices $r, s$ such that $x_r-x_s = k$.
1994 Korea National Olympiad, Problem 2
Given a set $S \subset N$ and a positive integer n, let $S\oplus \{n\} = \{s+n / s \in S\}$. The sequence $S_k$ of sets is defined inductively as follows: $S_1 = {1}$, $S_k=(S_{k-1} \oplus \{k\}) \cup \{2k-1\}$ for $k = 2,3,4, ...$
(a) Determine $N - \cup _{k=1}^{\infty} S_k$.
(b) Find all $n$ for which $1994 \in S_n$.
1981 All Soviet Union Mathematical Olympiad, 313
Find all the sequences of natural $k_n$ with two properties:
a) $k_n \le n \sqrt {n}$ for all $n$
b) $(k_n - k_m)$ is divisible by $(m-n)$ for all $m>n$
2015 Dutch BxMO/EGMO TST, 2
Given are positive integers $r$ and $k$ and an infi nite sequence of positive integers $a_1 \le a_2 \le ...$ such that $\frac{r}{a_r}= k + 1$. Prove that there is a $t$ satisfying $\frac{t}{a_t}=k$.
2015 EGMO, 4
Determine whether there exists an infinite sequence $a_1, a_2, a_3, \dots$ of positive integers
which satisfies the equality \[a_{n+2}=a_{n+1}+\sqrt{a_{n+1}+a_{n}} \] for every positive integer $n$.
2017 International Zhautykov Olympiad, 1
Let $(a_n)$ be sequnce of positive integers such that first $k$ members $a_1,a_2,...,a_k$ are distinct positive integers, and for each $n>k$, number $a_n$ is the smallest positive integer that can't be represented as a sum of several (possibly one) of the numbers $a_1,a_2,...,a_{n-1}$. Prove that $a_n=2a_{n-1}$ for all sufficently large $n$.