Found problems: 963
VMEO III 2006 Shortlist, A10
Let ${a_n}$ be a sequence defined by $a_1=2$, $a_{n+1}=\left[ \frac {3a_n}{2}\right]$ $\forall n \in \mathbb N$
$0.a_1a_2...$ rational or irrational?
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 $a^n$.
2009 Ukraine Team Selection Test, 7
Let $ a_1$, $ a_2$, $ \ldots$, $ a_n$ be distinct positive integers, $ n\ge 3$. Prove that there exist distinct indices $ i$ and $ j$ such that $ a_i \plus{} a_j$ does not divide any of the numbers $ 3a_1$, $ 3a_2$, $ \ldots$, $ 3a_n$.
[i]Proposed by Mohsen Jamaali, Iran[/i]
1976 IMO, 3
A sequence $(u_{n})$ is defined by \[ u_{0}=2 \quad u_{1}=\frac{5}{2}, u_{n+1}=u_{n}(u_{n-1}^{2}-2)-u_{1} \quad \textnormal{for } n=1,\ldots \] Prove that for any positive integer $n$ we have \[ [u_{n}]=2^{\frac{(2^{n}-(-1)^{n})}{3}} \](where $[x]$ denotes the smallest integer $\leq x)$
2018 Estonia Team Selection Test, 10
A sequence of positive real numbers $a_1, a_2, a_3, ... $ satisfies $a_n = a_{n-1} + a_{n-2}$ for all $n \ge 3$. A sequence $b_1, b_2, b_3, ...$ is defined by equations
$b_1 = a_1$ ,
$b_n = a_n + (b_1 + b_3 + ...+ b_{n-1})$ for even $n > 1$ ,
$b_n = a_n + (b_2 + b_4 + ... +b_{n-1})$ for odd $n > 1$.
Prove that if $n\ge 3$, then $\frac13 < \frac{b_n}{n \cdot a_n} < 1$
1979 Bundeswettbewerb Mathematik, 4
An infinite sequence $p_1, p_2, p_3, \ldots$ of natural numbers in the decimal system has the following property: For every $i \in \mathbb{N}$ the last digit of $p_{i+1}$ is different from $9$, and by omitting this digit one obtains number $p_i$. Prove that this sequence contains infinitely many composite numbers.
2020 Canadian Mathematical Olympiad Qualification, 3
Let $N$ be a positive integer and $A = a_1, a_2, ... , a_N$ be a sequence of real numbers.
Define the sequence $f(A)$ to be
$$f(A) = \left( \frac{a_1 + a_2}{2},\frac{a_2 + a_3}{2}, ...,\frac{a_{N-1} + a_N}{2},\frac{a_N + a_1}{2}\right)$$
and for $k$ a positive integer define $f^k (A)$ to be$ f$ applied to $A$ consecutively $k$ times (i.e. $f(f(... f(A)))$)
Find all sequences $A = (a_1, a_2,..., a_N)$ of integers such that $f^k (A)$ contains only integers for all $k$.
2015 Saudi Arabia BMO TST, 1
Prove that for any integer $n \ge 2$, there exists a unique finite sequence $x_0, x_1,..., x_n$ of real numbers which satisfies $x_0 = x_n = 0$ and $x_{i+1} - 8x_i^3 -4x_i + 3x_{i-1} + 1 = 0$ for all $i = 1,2,...,n - 1$. Prove moreover that $ |x_i| \le \frac12$ for all $i = 1,2,...,n - 1$.
Nguyễn Duy Thái Sơn
1973 Bulgaria National Olympiad, Problem 2
Let the numbers $a_1,a_2,a_3,a_4$ form an arithmetic progression with difference $d\ne0$. Prove that there are no exists geometric progressions $b_1,b_2,b_3,b_4$ and $c_1,c_2,c_3,c_4$ such that:
$$a_1=b_1+c_1,a_2=b_2+c_2,a_3=b_3+c_3,a_4=b_4+c_4.$$
1983 IMO Shortlist, 19
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 $P(1983) = F_{1983} - 1.$
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 $a^n$.
2024 Romania EGMO TST, P1
Define sequence $a_{0}, a_{1}, a_{2}, \ldots, a_{2018}, a_{2019}$ as below:
$
a_{0}=1
$
$a_{n+1}=a_{n}-\frac{a_{n}^{2}}{2019}$, $n=0,1,2, \ldots, 2018$
Prove $a_{2019} < \frac{1}{2} < a_{2018}$
2020 Kosovo National Mathematical Olympiad, 4
Let $a_0$ be a fixed positive integer. We define an infinite sequence of positive integers $\{a_n\}_{n\ge 1}$ in an inductive way as follows: if we are given the terms $a_0,a_1,...a_{n-1}$ , then $a_n$ is the smallest positive integer such that $\sqrt[n]{a_0\cdot a_1\cdot ...\cdot a_n}$ is a positive integer. Show that the sequence $\{a_n\}_{n\ge 1}$ is eventually constant.
[b]Note:[/b] The sequence $\{a_n\}_{n\ge 1}$ is eventually constant if there exists a positive integer $k$ such that $a_n=c$, for every $n\ge k$.
2023 Brazil Cono Sur TST, 4
Let $p$ be a prime number. Determine all positive integers $a$ such that the sequence $(a_n)_{n\geq 0}$ defined by $a_0=a$ and $a_{n+1}=pa_n-(p-1)\lfloor \sqrt[p]{a_ n} \rfloor^p$, for every $n\geq0$, is eventually constant.
1980 Austrian-Polish Competition, 7
Find the greatest natural number $n$ such there exist natural numbers $x_{1}, x_{2}, \ldots, x_{n}$ and natural $a_{1}< a_{2}< \ldots < a_{n-1}$ satisfying the following equations for $i =1,2,\ldots,n-1$: \[x_{1}x_{2}\ldots x_{n}= 1980 \quad \text{and}\quad x_{i}+\frac{1980}{x_{i}}= a_{i}.\]
1977 IMO Shortlist, 15
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.
1985 All Soviet Union Mathematical Olympiad, 405
The sequence $a_1, a_2, ... , a_k, ...$ is constructed according to the rules: $$a_{2n} = a_n,a_{4n+1} = 1,a_{4n+3} = 0$$Prove that it is non-periodical sequence.
2024 Bulgarian Autumn Math Competition, 12.1
Let $a_0,a_1,a_2 \dots a_n, \dots$ be an infinite sequence of real numbers, defined by $$a_0 = c$$ $$a_{n+1} = {a_n}^2+\frac{a_n}{2}+c$$ for some real $c > 0$. Find all values of $c$ for which the sequence converges and the limit for those values.
1982 Austrian-Polish Competition, 4
Let $P(x)$ denote the product of all (decimal) digits of a natural number $x$. For any positive integer $x_1$, define the sequence $(x_n)$ recursively by $x_{n+1} = x_n + P(x_n)$. Prove or disprove that the sequence $(x_n)$ is necessarily bounded.
2022 International Zhautykov Olympiad, 6
Do there exist two bounded sequences $a_1, a_2,\ldots$ and $b_1, b_2,\ldots$ such that for each positive integers $n$ and $m>n$ at least one of the two inequalities $|a_m-a_n|>1/\sqrt{n},$ and $|b_m-b_n|>1/\sqrt{n}$ holds?
1957 Moscow Mathematical Olympiad, 363
Eight consecutive numbers are chosen from the Fibonacci sequence $1, 2, 3, 5, 8, 13, 21,...$. Prove that the sequence does not contain the sum of chosen numbers.
2017 Kazakhstan National Olympiad, 3
$\{a_n\}$ is an infinite, strictly increasing sequence of positive integers and $a_{a_n}\leq a_n+a_{n+3}$ for all $n\geq 1$. Prove that, there are infinitely many triples $(k,l,m)$ of positive integers such that $k<l<m$ and $a_k+a_m=2a_l$
2009 Germany Team Selection Test, 2
Let $ a_1$, $ a_2$, $ \ldots$, $ a_n$ be distinct positive integers, $ n\ge 3$. Prove that there exist distinct indices $ i$ and $ j$ such that $ a_i \plus{} a_j$ does not divide any of the numbers $ 3a_1$, $ 3a_2$, $ \ldots$, $ 3a_n$.
[i]Proposed by Mohsen Jamaali, Iran[/i]
2011 Peru IMO TST, 6
Let $a_1, a_2, \cdots , a_n$ be real numbers, with $n\geq 3,$ such that $a_1 + a_2 +\cdots +a_n = 0$ and $$ 2a_k\leq a_{k-1} + a_{k+1} \ \ \ \text{for} \ \ \ k = 2, 3, \cdots , n-1.$$ Find the least number $\lambda(n),$ such that for all $k\in \{ 1, 2, \cdots, n\} $ it is satisfied that $|a_k|\leq \lambda (n)\cdot \max \{|a_1|, |a_n|\} .$
1987 Austrian-Polish Competition, 3
A function $f: R \to R$ satisfies $f (x + 1) = f (x) + 1$ for all $x$. Given $a \in R$, define the sequence $(x_n)$ recursively by $x_0 = a$ and $x_{n+1} = f (x_n)$ for $n \ge 0$. Suppose that, for some positive integer m, the difference $x_m - x_0 = k$ is an integer. Prove that the limit $\lim_{n\to \infty}\frac{x_n}{n}$ exists and determine its value.