Found problems: 963
1992 IMO Longlists, 55
For any positive integer $ x$ define $ g(x)$ as greatest odd divisor of $ x,$ and
\[ f(x) \equal{} \begin{cases} \frac {x}{2} \plus{} \frac {x}{g(x)} & \text{if \ \(x\) is even}, \\
2^{\frac {x \plus{} 1}{2}} & \text{if \ \(x\) is odd}. \end{cases}
\]
Construct the sequence $ x_1 \equal{} 1, x_{n \plus{} 1} \equal{} f(x_n).$ Show that the number 1992 appears in this sequence, determine the least $ n$ such that $ x_n \equal{} 1992,$ and determine whether $ n$ is unique.
2019 Dutch BxMO TST, 4
Do there exist a positive integer $k$ and a non-constant sequence $a_1, a_2, a_3, ...$ of positive integers such that $a_n = gcd(a_{n+k}, a_{n+k+1})$ for all positive integers $n$?
2017 Abels Math Contest (Norwegian MO) Final, 2
Let the sequence an be defined by $a_0 = 2, a_1 = 15$, and $a_{n+2 }= 15a_{n+1} + 16a_n$ for $n \ge 0$.
Show that there are infinitely many integers $k$ such that $269 | a_k$.
1959 Putnam, B2
Let $c$ be a positive real number. Prove that $c$ can be expressed in infinitely many ways as a sum of infinitely many distinct terms selected from the sequence $\left( \frac{1}{10n} \right)_{n\in \mathbb{N}}$
1988 IMO Shortlist, 24
Let $ \{a_k\}^{\infty}_1$ be a sequence of non-negative real numbers such that:
\[ a_k \minus{} 2 a_{k \plus{} 1} \plus{} a_{k \plus{} 2} \geq 0
\]
and $ \sum^k_{j \equal{} 1} a_j \leq 1$ for all $ k \equal{} 1,2, \ldots$. Prove that:
\[ 0 \leq a_{k} \minus{} a_{k \plus{} 1} < \frac {2}{k^2}
\]
for all $ k \equal{} 1,2, \ldots$.
2021 Brazil Team Selection Test, 6
Let $\mathcal{S}$ be a set consisting of $n \ge 3$ positive integers, none of which is a sum of two other distinct members of $\mathcal{S}$. Prove that the elements of $\mathcal{S}$ may be ordered as $a_1, a_2, \dots, a_n$ so that $a_i$ does not divide $a_{i - 1} + a_{i + 1}$ for all $i = 2, 3, \dots, n - 1$.
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 $n > 0$.
1979 Romania Team Selection Tests, 2.
For each $n\in \mathbb{Z}_{>0}$ let $a_n$ be the closest integer to $\sqrt{n}$. Compute the general term of the sequence: $(b_n)_{n\geqslant 1}$ with
\[b_n=\sum_{k=1}^{n^2} a_k.\]
[i]Pall Dalyay[/i]
1977 IMO Shortlist, 13
Let $B$ be a set of $k$ sequences each having $n$ terms equal to $1$ or $-1$. The product of two such sequences $(a_1, a_2, \ldots , a_n)$ and $(b_1, b_2, \ldots , b_n)$ is defined as $(a_1b_1, a_2b_2, \ldots , a_nb_n)$. Prove that there exists a sequence $(c_1, c_2, \ldots , c_n)$ such that the intersection of $B$ and the set containing all sequences from $B$ multiplied by $(c_1, c_2, \ldots , c_n)$ contains at most $\frac{k^2}{2^n}$ sequences.
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 $m>n$
1992 IMO Shortlist, 17
Let $ \alpha(n)$ be the number of digits equal to one in the binary representation of a positive integer $ n.$ Prove that:
(a) the inequality $ \alpha(n) (n^2 ) \leq \frac{1}{2} \alpha(n)(\alpha(n) + 1)$ holds;
(b) the above inequality is an equality for infinitely many positive integers, and
(c) there exists a sequence $ (n_i )^{\infty}_1$ such that $ \frac{\alpha ( n^2_i )}{\alpha (n_i }$
goes to zero as $ i$ goes to $ \infty.$
[i]Alternative problem:[/i] Prove that there exists a sequence a sequence $ (n_i )^{\infty}_1$ such that $ \frac{\alpha ( n^2_i )}{\alpha (n_i )}$
(d) $ \infty;$
(e) an arbitrary real number $ \gamma \in (0,1)$;
(f) an arbitrary real number $ \gamma \geq 0$;
as $ i$ goes to $ \infty.$
1998 Tuymaada Olympiad, 7
All possible sequences of numbers $-1$ and $+1$ of length $100$ are considered. For each of them, the square of the sum of the terms is calculated. Find the arithmetic average of the resulting values.
2007 Germany Team Selection Test, 1
The sequence of real numbers $a_0,a_1,a_2,\ldots$ is defined recursively by \[a_0=-1,\qquad\sum_{k=0}^n\dfrac{a_{n-k}}{k+1}=0\quad\text{for}\quad n\geq 1.\]Show that $ a_{n} > 0$ for all $ n\geq 1$.
[i]Proposed by Mariusz Skalba, Poland[/i]
2016 Latvia Baltic Way TST, 6
Given a natural number $n$, for which we can find a prime number less than $\sqrt{n}$ that is not a divisor of $n$. The sequence $a_1, a_2,... ,a_n$ is the numbers $1, 2,... ,n$ arranged in some order. For this sequence, we will find the longest ascending subsequense $a_{i_1} < a_{i_2} < ... < a_{i_k}$, ($i_1 <...< i_k$) and the longest decreasing substring $a_{j_1} > ... > a_{j_l}$, ($j_1 < ... < j_l$) . Prove that at least one of these two subsequnsces $a_{i_1} , . . . , a_{i_k}$ and $a_{j_1} > ... > a_{j_l}$ contains a number that is not a divisor of $n$.
1989 Dutch Mathematical Olympiad, 1
For a sequence of integers $a_1,a_2,a_3,...$ with $0<a_1<a_2<a_3<...$ applies:
$$a_n=4a_{n-1}-a_{n-2} \,\,\, for \,\,\, n > 2$$
It is further given that $a_4 = 194$. Calculate $a_5$.
2017 Ukraine Team Selection Test, 9
There're two positive inegers $a_1<a_2$. For every positive integer $n \geq 3$ let $a_n$ be the smallest integer that bigger than $a_{n-1}$ and such that there's unique pair $1\leq i< j\leq n-1$ such that this number equals to $a_i+a_j$. Given that there're finitely many even numbers in this sequence. Prove that sequence $\{a_{n+1}-a_n \}$ is periodic starting from some element.
2014 Hanoi Open Mathematics Competitions, 5
The first two terms of a sequence are $2$ and $3$. Each next term thereafter is the sum of the nearestly previous two terms if their sum is not greather than $10, 0$ otherwise. The $2014$th term is:
(A): $0$, (B): $8$, (C): $6$, (D): $4$, (E) None of the above.
2016 Saudi Arabia Pre-TST, 2.1
Given three numbers $x, y, z$, and set $x_1 = |x - y|, y_1 = | y -z|, z_1 = |z- x|$.
From $x_1, y_1, z_1$, form in the same fashion the numbers $x_2, y_2, z_2$, and so on.
It is known that $x_n = x, y_n = y, z_n = z$ for some $n$. Find all possible values of $(x, y, z)$.
2022 Iran MO (2nd round), 5
define $(a_n)_{n \in \mathbb{N}}$ such that $a_1=2$ and $$a_{n+1}=\left(1+\frac{1}{n}\right)^n \times a_{n}$$
Prove that there exists infinite number of $n$ such that $\frac{a_1a_2 \ldots a_n}{n+1}$ is a square of an integer.
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 $\frac{t}{a_t}=k$.
2019 Auckland Mathematical Olympiad, 4
Suppose $a_1 =\frac16$ and $a_n = a_{n-1} - \frac{1}{n}+ \frac{2}{n + 1} - \frac{1}{n + 2}$ for $n > 1$. Find $a_{100}$.
2015 Peru IMO TST, 16
Let $c \ge 1$ be an integer. Define a sequence of positive integers by $a_1 = c$ and \[a_{n+1}=a_n^3-4c\cdot a_n^2+5c^2\cdot a_n+c\] for all $n\ge 1$. Prove that for each integer $n \ge 2$ there exists a prime number $p$ dividing $a_n$ but none of the numbers $a_1 , \ldots , a_{n -1}$ .
[i]Proposed by Austria[/i]
2019 South East Mathematical Olympiad, 5
For positive integer n, define $a_n$ as the number of the triangles with integer length of every side and the length of the longest side being $2n.$
(1) Find $a_n$ in terms of $n;$
(2)If the sequence $\{ b_n\}$ satisfying for any positive integer $n,$ $\sum_{k=1}^n(-1)^{n-k}\binom {n}{k} b_k=a_n.$ Find the number of positive integer $n$ satisfying that $b_n\leq 2019a_n.$
1983 IMO Longlists, 19
Let $a$ be a positive integer and let $\{a_n\}$ be defined by $a_0 = 0$ and
\[a_{n+1 }= (a_n + 1)a + (a + 1)a_n + 2 \sqrt{a(a + 1)a_n(a_n + 1)} \qquad (n = 1, 2 ,\dots ).\]
Show that for each positive integer $n$, $a_n$ is a positive integer.
2018 Hanoi Open Mathematics Competitions, 7
Let $\{u_n\}_ {n\ge 1}$ be given sequence satisfying the conditions: $u_1 = 0$, $u_2 = 1$, $u_{n+1} = u_{n-1} + 2n - 1$ for $n \ge 2$.
1) Calculate $u_5$.
2) Calculate $u_{100} + u_{101}$.