Found problems: 1239
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 $b_n\leq 2019a_n.$
1978 All Soviet Union Mathematical Olympiad, 268
Consider a sequence $$x_n=(1+\sqrt2+\sqrt3)^n$$ Each member can be represented as $$x_n=q_n+r_n\sqrt2+s_n\sqrt3+t_n\sqrt6$$ where $q_n, r_n, s_n, t_n$ are integers. Find the limits of the fractions $r_n/q_n, s_n/q_n, t_n/q_n$.
2017 Baltic Way, 1
Let $a_0,a_1,a_2,...$ be an infinite sequence of real numbers satisfying $\frac{a_{n-1}+a_{n+1}}{2}\geq a_n$ for all positive integers $n$. Show that $$\frac{a_0+a_{n+1}}{2}\geq \frac{a_1+a_2+...+a_n}{n}$$ holds for all positive integers $n$.
2017 Irish Math Olympiad, 5
Given a positive integer $m$, a sequence of real numbers $a= (a_1,a_2,a_3,...)$ is called $m$-powerful if it satisfies
$$(\sum_{k=1}^{n} a_k )^{m} = \sum_{k=1}^{n} a_k^{m}$$for all positive integers $n$.
(a) Show that a sequence is $30$-powerful if and only if at most one of its terms is non-zero.
(b) Find a sequence none of whose terms are zero but which is $2017$-powerful.
2007 Mathematics for Its Sake, 2
Let be a natural number $ k $ and let be two infinite sequences $ \left( x_n \right)_{n\ge 1} ,\left( y_n \right)_{n\ge 1} $ such that
$$ \{1\}\cap\{ x_1,x_2,\ldots ,x_k\}=\{1\}\cap\{ y_1,y_2,\ldots ,y_k\} =\{ x_1,x_2,\ldots ,x_k\}\cap\{ y_1,y_2,\ldots ,y_k\} =\emptyset , $$
and defined by the following recurrence relations:
$$ x_{n+k}=\frac{y_n}{x_n} ,\quad y_{n+k} =\frac{y_n-1}{x_n-1} $$
Prove that $ \left( x_n \right)_{n\ge 1} $ and $ \left( y_n \right)_{n\ge 1} $ are periodic.
[i]Dumitru Acu[/i]
1975 IMO Shortlist, 14
Let $x_0 = 5$ and $x_{n+1} = x_n + \frac{1}{x_n} \ (n = 0, 1, 2, \ldots )$. Prove that
\[45 < x_{1000} < 45. 1.\]
2025 China National Olympiad, 1
Let $\alpha > 1$ be an irrational number and $L$ be a integer such that $L > \frac{\alpha^2}{\alpha - 1}$. A sequence $x_1, x_2, \cdots$ satisfies that $x_1 > L$ and for all positive integers $n$, \[ x_{n+1} = \begin{cases} \left \lfloor \alpha x_n \right \rfloor & \textup{if} \; x_n \leqslant L \\\left \lfloor \frac{x_n}{\alpha} \right \rfloor & \textup{if} \; x_n > L \end{cases}. \]
Prove that
(i) $\left\{x_n\right\}$ is eventually periodic.
(ii) The eventual fundamental period of $\left\{x_n\right\}$ is an odd integer which doesn't depend on the choice of $x_1$.
2011 Germany Team Selection Test, 1
A sequence $x_1, x_2, \ldots$ is defined by $x_1 = 1$ and $x_{2k}=-x_k, x_{2k-1} = (-1)^{k+1}x_k$ for all $k \geq 1.$ Prove that $\forall n \geq 1$ $x_1 + x_2 + \ldots + x_n \geq 0.$
[i]Proposed by Gerhard Wöginger, Austria[/i]
2021 Thailand Online MO, P3
Let $a_1,a_2,\cdots$ be an infinity sequence of positive integers such that $a_1=2021$ and
$$a_{n+1}=(a_1+a_2+\cdots+a_n)^2-1$$
for all positive integers $n$. Prove that for any integer $n\ge 2$, $a_n$ is the product of at least $2n$ (not necessarily distinct) primes.
1994 Czech And Slovak Olympiad IIIA, 4
Let $a_1,a_2,...$ be a sequence of natural numbers such that for each $n$, the product $(a_n - 1)(a_n- 2)...(a_n - n^2)$ is a positive integral multiple of $n^{n^2-1}$. Prove that for any finite set $P$ of prime numbers the following inequality holds:
$$\sum_{p\in P}\frac{1}{\log_p a_p}< 1$$
2010 Dutch IMO TST, 1
Consider sequences $a_1, a_2, a_3,...$ of positive integers. Determine the smallest possible value of $a_{2010}$ if
(i) $a_n < a_{n+1}$ for all $n\ge 1$,
(ii) $a_i + a_l > a_j + a_k$ for all quadruples $ (i, j, k, l)$ which satisfy $1 \le i < j \le k < l$.
2016 Saint Petersburg Mathematical Olympiad, 1
In the sequence of integers $(a_n)$, the sum $a_m + a_n$ is divided by $m + n$ with any different $m$ and $n$. Prove that $a_n$ is a multiple of $n$ for any $n$.
1980 IMO Longlists, 19
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 $i =1,2,\ldots,n-1$: \[x_{1}x_{2}\ldots x_{n}= 1980 \quad \text{and}\quad x_{i}+\frac{1980}{x_{i}}= a_{i}.\]
2015 South East Mathematical Olympiad, 1
Suppose that the sequence $\{a_n\}$ satisfy $a_1=1$ and $a_{2k}=a_{2k-1}+a_k, \quad a_{2k+1}=a_{2k}$ for $k=1,2, \ldots$ \\Prove that $a_{2^n}< 2^{\frac{n^2}{2}}$ for any integer $n \geq 3$.
1978 IMO Shortlist, 9
Let $0<f(1)<f(2)<f(3)<\ldots$ a sequence with all its terms positive$.$ The $n-th$ positive integer which doesn't belong to the sequence is $f(f(n))+1.$ Find $f(240).$
2020 Thailand TSTST, 5
Let $\{a_n\}$ be a sequence of positive integers such that $a_{n+1} = a_n^2+1$ for all $n \geq 1$. Prove that there is no positive integer $N$ such that $$\prod_{k=1}^N(a_k^2+a_k+1)$$ is a perfect square.
2016 ISI Entrance Examination, 8
Suppose that $(a_n)_{n\geq 1}$ is a sequence of real numbers satisfying $a_{n+1} = \frac{3a_n}{2+a_n}$.
(i) Suppose $0 < a_1 <1$, then prove that the sequence $a_n$ is increasing and hence show that $\lim_{n \to \infty} a_n =1$.
(ii) Suppose $ a_1 >1$, then prove that the sequence $a_n$ is decreasing and hence show that $\lim_{n \to \infty} a_n =1$.
1975 IMO Shortlist, 11
Let $a_{1}, \ldots, a_{n}$ be an infinite sequence of strictly positive integers, so that $a_{k} < a_{k+1}$ for any $k.$ Prove that there exists an infinity of terms $ a_{m},$ which can be written like $a_m = x \cdot a_p + y \cdot a_q$ with $x,y$ strictly positive integers and $p \neq q.$
2010 SEEMOUS, Problem 1
Let $f_0:[0,1]\to\mathbb R$ be a continuous function. Define the sequence of functions $f_n:[0,1]\to\mathbb R$ by
$$f_n(x)=\int^x_0f_{n-1}(t)dt$$
for all integers $n\ge1$.
a) Prove that the series $\sum_{n=1}^\infty f_n(x)$ is convergent for every $x\in[0,1]$.
b) Find an explicit formula for the sum of the series $\sum_{n=1}^\infty f_n(x),x\in[0,1]$.
1993 ITAMO, 3
Consider an infinite chessboard whose rows and columns are indexed by positive integers. At most one coin can be put on any cell of the chessboard. Let be given two arbitrary sequences ($a_n$) and ($b_n$) of positive integers ($n \in N$). Assuming that infinitely many coins are available, prove that they can be arranged on the chessboard so that there are $a_n$ coins in the $n$-th row and $b_n$ coins in the $n$-th column for all $n$.
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 $a_5$.
2004 Federal Math Competition of S&M, 4
The sequence $(a_n)$ is given by $a_1 = x \in \mathbb{R}$ and $3a_{n+1} = a_n+1$ for $n \geq 1$. Set
$A = \sum_{n=1}^\infty \Big[ a_n - \frac{1}{6}\Big]$, $B = \sum_{n=1}^\infty \Big[ a_n + \frac{1}{6}\Big]$.
Compute the sum $A+B$ in terms of $x$.
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 $0^0=1$
2018 India IMO Training Camp, 3
Let $a_n, b_n$ be sequences of positive reals such that,$$a_{n+1}= a_n + \frac{1}{2b_n}$$ $$b_{n+1}= b_n + \frac{1}{2a_n}$$ for all $n\in\mathbb N$.
Prove that, $\text{max}\left(a_{2018}, b_{2018}\right) >44$.
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 $a_k+a_m=2a_l$