Found problems: 1239
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 $a^n$.
1984 Dutch Mathematical Olympiad, 3
For $n = 1,2,3,...$. $a_n$ is defined by:
$$a_n =\frac{1 \cdot 4 \cdot 7 \cdot ... (3n-2)}{2 \cdot 5 \cdot 8 \cdot ... (3n-1)}$$
Prove that for every $n$ holds that
$$\frac{1}{\sqrt{3n+1}}\le a_n \le \frac{1}{\sqrt[3]{3n+1}}$$
2013 IMO Shortlist, A4
Let $n$ be a positive integer, and consider a sequence $a_1 , a_2 , \dotsc , a_n $ of positive integers. Extend it periodically to an infinite sequence $a_1 , a_2 , \dotsc $ by defining $a_{n+i} = a_i $ for all $i \ge 1$. If \[a_1 \le a_2 \le \dots \le a_n \le a_1 +n \] and \[a_{a_i } \le n+i-1 \quad\text{for}\quad i=1,2,\dotsc, n, \] prove that \[a_1 + \dots +a_n \le n^2. \]
1999 Mongolian Mathematical Olympiad, Problem 3
Does there exist a sequence $(a_n)_{n\in\mathbb N}$ of distinct positive integers such that:
(i) $a_n<1999n$ for all $n$;
(ii) none of the $a_n$ contains three decimal digits $1$?
1965 Swedish Mathematical Competition, 4
Find constants $A > B$ such that $\frac{f\left( \frac{1}{1+2x}\right) }{f(x)}$ is independent of $x$, where $f(x) = \frac{1 + Ax}{1 + Bx}$ for all real $x \ne - \frac{1}{B}$. Put $a_0 = 1$, $a_{n+1} = \frac{1}{1 + 2a_n}$. Find an expression for an by considering $f(a_0), f(a_1), ...$.
2016 Thailand TSTST, 1
Let $a_1, a_2, a_3, \dots$ be a sequence of integers such that
$\text{(i)}$ $a_1=0$
$\text{(ii)}$ for all $i\geq 1$, $a_{i+1}=a_i+1$ or $-a_i-1$.
Prove that $\frac{a_1+a_2+\cdots+a_n}{n}\geq-\frac{1}{2}$ for all $n\geq 1$.
2021 Azerbaijan IMO TST, 1
Given a positive integer $k$ show that there exists a prime $p$ such that one can choose distinct integers $a_1,a_2\cdots, a_{k+3} \in \{1, 2, \cdots ,p-1\}$ such that p divides $a_ia_{i+1}a_{i+2}a_{i+3}-i$ for all $i= 1, 2, \cdots, k$.
[i]South Africa [/i]
1991 ITAMO, 6
We say that each positive number $x$ has two sons: $x+1$ and $\frac{x}{x+1}$. Characterize all the descendants of number $1$.
2004 Germany Team Selection Test, 1
Consider pairs of the sequences of positive real numbers \[a_1\geq a_2\geq a_3\geq\cdots,\qquad b_1\geq b_2\geq b_3\geq\cdots\] and the sums \[A_n = a_1 + \cdots + a_n,\quad B_n = b_1 + \cdots + b_n;\qquad n = 1,2,\ldots.\] For any pair define $c_n = \min\{a_i,b_i\}$ and $C_n = c_1 + \cdots + c_n$, $n=1,2,\ldots$.
(1) Does there exist a pair $(a_i)_{i\geq 1}$, $(b_i)_{i\geq 1}$ such that the sequences $(A_n)_{n\geq 1}$ and $(B_n)_{n\geq 1}$ are unbounded while the sequence $(C_n)_{n\geq 1}$ is bounded?
(2) Does the answer to question (1) change by assuming additionally that $b_i = 1/i$, $i=1,2,\ldots$?
Justify your answer.
2021 Saudi Arabia IMO TST, 10
Given a positive integer $k$ show that there exists a prime $p$ such that one can choose distinct integers $a_1,a_2\cdots, a_{k+3} \in \{1, 2, \cdots ,p-1\}$ such that p divides $a_ia_{i+1}a_{i+2}a_{i+3}-i$ for all $i= 1, 2, \cdots, k$.
[i]South Africa [/i]
2011 Ukraine Team Selection Test, 8
Is there an increasing sequence of integers $ 0 = {{a} _{0}} <{{a} _{1}} <{{a} _{2}} <\ldots $ for which the following two conditions are satisfied simultaneously:
1) any natural number can be given as $ {{a} _{i}} + {{a} _{j}} $ for some (possibly equal) $ i \ge 0 $, $ j \ge 0$ ,
2) $ {{a} _ {n}}> \tfrac {{{n} ^ {2}}} {16} $ for all natural $ n $?
2007 Denmark MO - Mohr Contest, 5
The sequence of numbers $a_0,a_1,a_2,...$ is determined by $a_0 = 0$, and
$$a_n= \begin{cases} 1+a_{n-1} \,\,\, when\,\,\, n \,\,\, is \,\,\, positive \,\,\, and \,\,\, odd \\
3a_{n/2} \,\,\,when \,\,\,n \,\,\,is \,\,\,positive \,\,\,and \,\,\,even\end{cases}$$
How many of these numbers are less than $2007$ ?
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 $\left( \frac{1}{10n} \right)_{n\in \mathbb{N}}$
2018 Latvia Baltic Way TST, P3
Let $a_1,a_2,...$ be an infinite sequence of integers that satisfies $a_{n+2}=a_{n+1}+a_n$ for all $n \ge 1$. There exists a positive integer $k$ such that $a_k=a_{k+2018}$. Prove that there exists a term of the sequence which is equal to zero.
2004 Federal Competition For Advanced Students, P2, 4
Show that there is an infinite sequence $a_1,a_2,...$ of natural numbers such that $a^2_1+a^2_2+ ...+a^2_N$ is a perfect square for all $N$. Give a recurrent formula for one such sequence.
2012 Balkan MO Shortlist, N1
A sequence $(a_n)_{n=1}^{\infty}$ of positive integers satisfies the condition $a_{n+1} = a_n +\tau (n)$ for all positive integers $n$ where $\tau (n)$ is the number of positive integer divisors of $n$. Determine whether two consecutive terms of this sequence can be perfect squares.
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 $u_{100} + u_{101}$.
2015 Singapore Junior Math Olympiad, 3
There are $30$ children, $a_1,a_2,...,a_{30}$ seated clockwise in a circle on the floor. The teacher walks behind the children in the clockwise direction with a box of $1000$ candies. She drops a candy behind the first child $a_1$. She then skips one child and drops a candy behind the third child, $a_3$. Now she skips two children and drops a candy behind the next child, $a_6$. She continues this way, at each stage skipping one child more than at the preceding stage before dropping a candy behind the next child. How many children will never receive a candy? Justify your answer.
2016 Saudi Arabia IMO TST, 1
Define the sequence $a_1, a_2,...$ as follows: $a_1 = 1$, and for every $n \ge 2$, $a_n = n - 2$ if $a_{n-1} = 0$ and $a_n = a_{n-1} - 1$, otherwise. Find the number of $1 \le k \le 2016$ such that there are non-negative integers $r, s$ and a positive integer $n$ satisfying $k = r + s$ and $a_{n+r} = a_n + s$.
2020 MMATHS, I2
Let $b$ and $c$ be real numbers not both equal to $1$ such that $1,b,c$ is an arithmetic progression and $1,c,b$ is a geometric progression. What is $100(b-c)$?
[i]Proposed by Noah Kravitz[/i]
2010 Korea Junior Math Olympiad, 4
Let there be a sequence $a_n$ such that $a_1 = 2,a_2 = 0, a_3 = 1, a_4 = 0$, and for $n \ge 1, a_{n+4}$ is the remainder when $a_n + 2a_{n+1} + 3a_{n+2} + 4a_{n+3}$ is divided by $9$. Prove that there are no positive integer $k$ such that $$a_k = 0, a_{k+1} = 1, a_{k+2} = 0,a_{k+3} = 2.$$
KoMaL A Problems 2019/2020, A. 776
Let $k > 1$ be a fixed odd number, and for non-negative integers $n$ let
$$f_n=\sum_{\substack{0\leq i\leq n\\ k\mid n-2i}}\binom{n}{i}.$$
Prove that $f_n$ satisfy the following recursion:
$$f_{n}^2=\sum_{i=0}^{n} \binom{n}{i}f_{i}f_{n-i}.$$
2015 Belarus Team Selection Test, 3
Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.)
[i]Proposed by Hong Kong[/i]
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.
2024 VJIMC, 3
Let $a_1>0$ and for $n \ge 1$ define
\[a_{n+1}=a_n+\frac{1}{a_1+a_2+\dots+a_n}.\]
Prove that
\[\lim_{n \to \infty} \frac{a_n^2}{\ln n}=2.\]