Found problems: 1239
2022 Thailand TST, 3
Let $a_1,a_2,a_3,\ldots$ be an infinite sequence of positive integers such that $a_{n+2m}$ divides $a_{n}+a_{n+m}$ for all positive integers $n$ and $m.$ Prove that this sequence is eventually periodic, i.e. there exist positive integers $N$ and $d$ such that $a_n=a_{n+d}$ for all $n>N.$
1993 IMO Shortlist, 1
Define a sequence $\langle f(n)\rangle^{\infty}_{n=1}$ of positive integers by $f(1) = 1$ and \[f(n) = \begin{cases} f(n-1) - n & \text{ if } f(n-1) > n;\\ f(n-1) + n & \text{ if } f(n-1) \leq n, \end{cases}\]
for $n \geq 2.$ Let $S = \{n \in \mathbb{N} \;\mid\; f(n) = 1993\}.$
[b](i)[/b] Prove that $S$ is an infinite set.
[b](ii)[/b] Find the least positive integer in $S.$
[b](iii)[/b] If all the elements of $S$ are written in ascending order as \[ n_1 < n_2 < n_3 < \ldots , \] show that \[ \lim_{i\rightarrow\infty} \frac{n_{i+1}}{n_i} = 3. \]
Russian TST 2019, P1
Let $a_0, a_1, \ldots , a_n$ and $b_0, b_1, \ldots , b_n$ be sequences of real numbers such that $a_0 = b_0 \geqslant 0$, $a_n = b_n > 0$ and \[a_i=\sqrt{\frac{a_{i+1}+a_{i-1}}{2}},\quad b_i=\sqrt{\frac{b_{i+1}+b_{i-1}}{2}},\]for all $i=1,\ldots,n-1$. Prove that $a_1 = b_1$.
2012 Estonia Team Selection Test, 2
For a given positive integer $n$ one has to choose positive integers $a_0, a_1,...$ so that the following conditions hold:
(1) $a_i = a_{i+n}$ for any $i$,
(2) $a_i$ is not divisible by $n$ for any $i$,
(3) $a_{i+a_i}$ is divisible by $a_i$ for any $i$.
For which positive integers $n > 1$ is this possible only if the numbers $a_0, a_1, ...$ are all equal?
2010 Indonesia TST, 2
Let $ a_0$, $ a_1$, $ a_2$, $ \ldots$ be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, $ \gcd (a_i, a_{i \plus{} 1}) > a_{i \minus{} 1}$. Prove that $ a_n\ge 2^n$ for all $ n\ge 0$.
[i]Proposed by Morteza Saghafian, Iran[/i]
2022 Thailand Mathematical Olympiad, 7
Let $d \geq 2$ be a positive integer. Define the sequence $a_1,a_2,\dots$ by
$$a_1=1 \ \text{and} \ a_{n+1}=a_n^d+1 \ \text{for all }n\geq 1.$$
Determine all pairs of positive integers $(p, q)$ such that $a_p$ divides $a_q$.
2015 IMC, 3
Let $F(0)=0$, $F(1)=\frac32$, and $F(n)=\frac{5}{2}F(n-1)-F(n-2)$
for $n\ge2$.
Determine whether or not $\displaystyle{\sum_{n=0}^{\infty}\,
\frac{1}{F(2^n)}}$ is a rational number.
(Proposed by Gerhard Woeginger, Eindhoven University of Technology)
1961 All Russian Mathematical Olympiad, 011
Prove that for three arbitrary infinite sequences, of natural numbers $a_1,a_2,...,a_n,... $ , $b_1,b_2,...,b_n,... $, $c_1,c_2,...,c_n,...$ there exist numbers $p$ and $q$ such, that $a_p \ge a_q$, $b_p \ge b_q$ and $c_p \ge c_q$.
2010 VTRMC, Problem 6
Define a sequence by $a_1=1,a_2=\frac12$, and $a_{n+2}=a_{n+1}-\frac{a_na_{n+1}}2$ for $n$ a positive integer. Find $\lim_{n\to\infty}na_n$.
2024 Francophone Mathematical Olympiad, 4
Find all integers $n \ge 2$ for which there exists $n$ integers $a_1,a_2,\dots,a_n \ge 2$ such that for all indices $i \ne j$, we have $a_i \mid a_j^2+1$.
2014 Taiwan TST Round 2, 2
Let $r$ be a positive integer, and let $a_0 , a_1 , \cdots $ be an infinite sequence of real numbers. Assume that for all nonnegative integers $m$ and $s$ there exists a positive integer $n \in [m+1, m+r]$ such that
\[ a_m + a_{m+1} +\cdots +a_{m+s} = a_n + a_{n+1} +\cdots +a_{n+s} \]
Prove that the sequence is periodic, i.e. there exists some $p \ge 1 $ such that $a_{n+p} =a_n $ for all $n \ge 0$.
2020 Thailand TSTST, 2
For any positive integer $m \geq 2$, let $p(m)$ be the smallest prime dividing $m$ and $P(m)$ be the largest prime dividing $m$. Let $C$ be a positive integer. Define sequences $\{a_n\}$ and $\{b_n\}$ by $a_0 = b_0 = C$ and, for each positive integer $k$ such that $a_{k-1}\geq 2$,
$$a_k=a_{k-1}-\frac{a_{k-1}}{p(a_{k-1})};$$
and, for each positive integer $k$ such that $b_{k-1}\geq 2$,
$$b_k=b_{k-1}-\frac{b_{k-1}}{P(b_{k-1})}$$
It is easy to see that both $\{a_n\}$ and $\{b_n\}$ are finite sequences which terminate when they reach the number $1$.
Prove that the numbers of terms in the two sequences are always equal.
1974 Bundeswettbewerb Mathematik, 3
A circle $K_1$ of radius $r_1 = 1\slash 2$ is inscribed in a semi-circle $H$ with diameter $AB$ and radius $1.$ A sequence of different circles $K_2, K_3, \ldots$ with radii $r_2, r_3, \ldots$ respectively are drawn so that for each $n\geq 1$, the circle $K_{n+1}$ is tangent to $H$, $K_n$ and $AB.$ Prove that $a_n = 1\slash r_n$ is an integer for each $n$, and that it is a perfect square for $n$ even and twice a perfect square for $n$ odd.
2016 Peru IMO TST, 2
Determine how many $100$-positive integer sequences satisfy the two conditions following:
- At least one term of the sequence is equal to $4$ or $5$.
- Any two adjacent terms differ as a maximum in $2$.
2018 China Northern MO, 6
For $a_1 = 3$, define the sequence $a_1, a_2, a_3, \ldots$ for $n \geq 1$ as $$na_{n+1}=2(n+1)a_n-n-2.$$
Prove that for any odd prime $p$, there exist positive integer $m,$ such that $p|a_m$ and $p|a_{m+1}.$
2016 AIME Problems, 10
A strictly increasing sequence of positive integers $a_1, a_2, a_3, \ldots$ has the property that for every positive integer $k$, the subsequence $a_{2k-1}, a_{2k}, a_{2k+1}$ is geometric and the subsequence $a_{2k}, a_{2k+1}, a_{2k+2}$ is arithmetic. Suppose that $a_{13} = 2016$. Find $a_1$.
2015 Silk Road, 2
Let $\left\{ {{a}_{n}} \right\}_{n \geq 1}$ and $\left\{ {{b}_{n}} \right\}_{n \geq 1}$ be two infinite arithmetic progressions, each of which the first term and the difference are mutually prime natural numbers. It is known that for any natural $n$, at least one of the numbers $\left( a_n^2+a_{n+1}^2 \right)\left( b_n^2+b_{n+1}^2 \right) $ or $\left( a_n^2+b_n^2 \right) \left( a_{n+1}^2+b_{n+1}^2 \right)$ is an perfect square. Prove that ${{a}_{n}}={{b}_{n}}$, for any natural $n$ .
2011 Indonesia TST, 4
Let $a, b$, and $c$ be positive integers such that $gcd(a, b) = 1$. Sequence $\{u_k\}$, is given such that $u_0 = 0$, $u_1 = 1$, and u$_{k+2} = au_{k+1} + bu_k$ for all $k \ge 0$. Let $m$ be the least positive integer such that $c | u_m$ and $n$ be an arbitrary positive integer such that $c | u_n$. Show that $m | n$.
[hide=PS.] There was a typo in the last line, as it didn't define what n does. Wording comes from [b]tst-2011-1.pdf[/b] from [url=https://sites.google.com/site/imoidn/idntst/2011tst]here[/url]. Correction was made according to #2[/hide]
2004 Unirea, 3
[b]a)[/b] Prove that for any natural numbers $ n, $ the inequality
$$ e^{2-1/n} >\prod_{k=1}^n (1+1/k^2) $$
holds.
[b]b)[/b] Prove that the sequence $ \left( a_n \right)_{n\ge 1} $ with $ a_1=1 $ and defined by the recursive relation $ a_{n+1}=\frac{2}{n^2}\sum_{k=1}^n ka_k $ is nondecreasing. Is it convergent?
1985 IMO Longlists, 63
Let $x_n = \sqrt[2]{2+\sqrt[3]{3+\cdots+\sqrt[n]{n}}}.$ Prove that
\[x_{n+1}-x_n <\frac{1}{n!} \quad n=2,3,\cdots\]
2012 Bogdan Stan, 2
Let be a bounded sequence of positive real numbers $ \left( x_n \right)_{n\ge 1} $ satisfying the recurrence:
$$ x_{n+3} =\sqrt[3]{3x_n-2} . $$
Prove that $ \left( x_n \right)_{n\ge 1} $ is convergent.
[i]Cristinel Mortici[/i]
1976 All Soviet Union Mathematical Olympiad, 223
The natural numbers $x_1$ and $x_2$ are less than $1000$. We construct a sequence:
$$x_3 = |x_1 - x_2|$$
$$x_4 = min \{ |x_1 - x_2|, |x_1 - x_3|, |x_2 - x_3|\}$$
$$...$$
$$x_k = min \{ |x_i - x_j|, 0 <i < j < k\}$$
$$...$$
Prove that $x_{21} = 0$.
1992 Yugoslav Team Selection Test, Problem 2
Periodic sequences $(a_n),(b_n),(c_n)$ and $(d_n)$ satisfy the following conditions:
$$a_{n+1}=a_n+b_n,\enspace\enspace b_{n+1}=b_n+c_n,$$
$$c_{n+1}=c_n+d_n,\enspace\enspace d_{n+1}=d_n+a_n,$$
for $n=1,2,\ldots$. Prove that $a_2=b_2=c_2=d_2=0$.
1969 IMO Shortlist, 28
$(GBR 5)$ Let us define $u_0 = 0, u_1 = 1$ and for $n\ge 0, u_{n+2} = au_{n+1}+bu_n, a$ and $b$ being positive integers. Express $u_n$ as a polynomial in $a$ and $b.$ Prove the result. Given that $b$ is prime, prove that $b$ divides $a(u_b -1).$
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]