Found problems: 280
2016 Belarus Team Selection Test, 1
Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \cdots$ defined by \[ a_0 = M + \frac{1}{2} \qquad \textrm{and} \qquad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots \] contains at least one integer term.
2014 Contests, 2
Consider the following sequence
$$(a_n)_{n=1}^{\infty}=(1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,1,\dots)$$
Find all pairs $(\alpha, \beta)$ of positive real numbers such that $\lim_{n\to \infty}\frac{\displaystyle\sum_{k=1}^n a_k}{n^{\alpha}}=\beta$.
(Proposed by Tomas Barta, Charles University, Prague)
2014 AMC 10, 24
A sequence of natural numbers is constructed by listing the first $4$, then skipping one, listing the next $5$, skipping $2$, listing $6$, skipping $3$, and, on the $n$th iteration, listing $n+3$ and skipping $n$. The sequence begins $1,2,3,4,6,7,8,9,10,13$. What is the $500,000$th number in the sequence?
$ \textbf{(A)}\ 996,506\qquad\textbf{(B)}\ 996507\qquad\textbf{(C)}\ 996508\qquad\textbf{(D)}\ 996509\qquad\textbf{(E)}\ 996510 $
2005 Alexandru Myller, 3
[b]a)[/b] Find the number of infinite sequences of integers $ \left( a_n \right)_{n\ge 1} $ that have the property that $ a_na_{n+2}a_{n+3}=-1, $ for any natural number $ n. $
[b]b)[/b] Prove that there is no infinite sequence of integers $ \left( b_n \right)_{n\ge 1} $ that have the property that $ b_nb_{n+2}b_{n+3}=2005, $ for any natural number $ n. $
2010 VTRMC, Problem 7
Let $\sum_{n=1}^\infty a_n$ be a convergent series of positive terms (so $a_i>0$ for all $i$) and set $b_n=\frac1{na_n^2}$ for $n\ge1$. Prove that $\sum_{n=1}^\infty\frac n{b_1+b_2+\ldots+b_n}$ is convergent.
1969 Bulgaria National Olympiad, Problem 2
Prove that
$$S_n=\frac1{1^2}+\frac1{2^2}+\ldots+\frac1{n^2}<2$$for every $n\in\mathbb N$.
2007 Grigore Moisil Intercounty, 4
Let $ \left( x_n \right)_{n\ge 1} $ be a sequence of positive real numbers, verifying the inequality $ x_n\le \frac{x_{n-1}+x_{n-2}}{2} , $ for any natural number $ n\ge 3. $
Show that $ \left( x_n \right)_{n\ge 1} $ is convergent.
1946 Putnam, B4
For each positive integer $n$, put
$$p_n =\left(1+\frac{1}{n}\right)^{n},\; P_n =\left(1+\frac{1}{n}\right)^{n+1}, \; h_n = \frac{2 p_n P_{n}}{ p_n + P_n }.$$
Prove that $h_1 < h_2 < h_3 <\ldots$
2011 Miklós Schweitzer, 7
prove that for any sequence of nonnegative numbers $(a_n)$, $$\liminf_{n\to\infty} (n^2(4a_n(1-a_{n-1})-1))\leq\frac{1}{4}$$
2013 BMT Spring, 5
Suppose that $c_n=(-1)^n(n+1)$. While the sum $\sum_{n=0}^\infty c_n$ is divergent, we can still attempt to assign a value to the sum using other methods. The Abel Summation of a sequence, $a_n$, is $\operatorname{Abel}(a_n)=\lim_{x\to1^-}\sum_{n=0}^\infty a_nx^n$. Find $\operatorname{Abel}(c_n)$.
2014 SEEMOUS, Problem 2
Consider the sequence $(x_n)$ given by
$$x_1=2,\enspace x_{n+1}=\frac{x_n+1+\sqrt{x_n^2+2x_n+5}}2,\enspace n\ge2.$$Prove that the sequence $y_n=\sum_{k=1}^n\frac1{x_k^2-1},\enspace n\ge1$ is convergent and find its limit.
2006 Mathematics for Its Sake, 3
Let be two positive real numbers $ a,b, $ and an infinite arithmetic sequence of natural numbers $ \left( x_n \right)_{n\ge 1} . $
Study the convergence of the sequences
$$ \left( \frac{1}{x_n}\sum_{i=1}^n\sqrt[x_i]{b} \right)_{n\ge 1}\text{ and } \left( \left(\sum_{i=1}^n \sqrt[x_i]{a}/\sqrt[x_i]{b} \right)^\frac{x_n}{\ln x_n} \right)_{n\ge 1} , $$
and calculate their limits.
[i]Dumitru Acu[/i]
2019 Ukraine Team Selection Test, 3
Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.
2025 VJIMC, 1
Let $x_0=a, x_1= b, x_2 = c$ be given real numbers and let $x_{n+2} = \frac{x_n + x_{n-1}}{2}$ for all $n\geq 1$. Show that the sequence $(x_n)_{n\geq 0}$ converges and find its limit.
2024 Bundeswettbewerb Mathematik, 2
Determine the set of all real numbers $r$ for which there exists an infinite sequence $a_1,a_2,\dots$ of positive integers satisfying the following three properties:
(1) No number occurs more than once in the sequence.
(2) The sum of two different elements of the sequence is never a power of two.
(3) For all positive integers $n$, we have $a_n<r \cdot n$.
2016 Saudi Arabia BMO TST, 1
Let $ p $ and $ q $ be given primes and the sequence $ \{ p_n \}_{n = 1}^{\infty} $ defined recursively as follows:
$ p_1 = p $, $ p_2 = q $, and $ p_{n+2} $ is the largest prime divisor of the number $( p_n + p_{n + 1} + 2016) $ for all $ n \geq 1 $. Prove that this sequence is bounded. That is, there exists a positive real number $ M $ such that $ p_n < M $ for all positive integers $ n $.
2014 IMC, 1
For a positive integer $x$, denote its $n^{\mathrm{th}}$ decimal digit by $d_n(x)$, i.e. $d_n(x)\in \{ 0,1, \dots, 9\}$ and $x=\sum_{n=1}^{\infty} d_n(x)10^{n-1}$. Suppose that for some sequence $(a_n)_{n=1}^{\infty}$, there are only finitely many zeros in the sequence $(d_n(a_n))_{n=1}^{\infty}$. Prove that there are infinitely many positive integers that do not occur in the sequence $(a_n)_{n=1}^{\infty}$.
(Proposed by Alexander Bolbot, State University, Novosibirsk)
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 $n$.
1998 Brazil Team Selection Test, Problem 5
Consider $k$ positive integers $a_1,a_2,\ldots,a_k$ satisfying $1\le a_1<a_2<\ldots<a_k\le n$ and $\operatorname{lcm}(a_i,a_j)\le n$ for any $i,j$. Prove that
$$k\le2\lfloor\sqrt n\rfloor.$$
2020 Jozsef Wildt International Math Competition, W38
Let $(a_n)_{n\in\mathbb N}$ be a sequence, given by the recurrence:
$$ma_{n+1}+(m-2)a_n-a_{n-1}=0$$
where $m\in\mathbb R$ is a parameter and the first two terms of $a_n$ are fixed known real numbers. Find $m\in\mathbb R$, so that
$$\lim_{n\to\infty}a_n=0$$
[i]Proposed by Laurențiu Modan[/i]
2018 Turkey MO (2nd Round), 3
A sequence $a_1,a_2,\dots$ satisfy
$$
\sum_{i =1}^n a_{\lfloor \frac{n}{i}\rfloor }=n^{10},
$$
for every $n\in\mathbb{N}$.
Let $c$ be a positive integer. Prove that, for every positive integer $n$,
$$
\frac{c^{a_n}-c^{a_{n-1}}}{n}
$$
is an integer.
2011 SEEMOUS, Problem 4
Let $f:[0,1]\to\mathbb R$ be a twice continuously differentiable increasing function. Define the sequences given by $L_n=\frac1n\sum_{k=0}^{n-1}f\left(\frac kn\right)$ and $U_n=\frac1n\sum_{k=0}^nf\left(\frac kn\right)$, $n\ge1$. 1. The interval $[L_n,U_n]$ is divided into three equal segments. Prove that, for large enough $n$, the number $I=\int^1_0f(x)\text dx$ belongs to the middle one of these three segments.
2022 Iran Team Selection Test, 5
Find all $C\in \mathbb{R}$ such that every sequence of integers $\{a_n\}_{n=1}^{\infty}$ which is bounded from below and for all $n\geq 2$ satisfy $$0\leq a_{n-1}+Ca_n+a_{n+1}<1$$ is periodic.
Proposed by Navid Safaei
2017 Romania National Olympiad, 1
[b]a)[/b] Let be a continuous function $ f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R}_{>0} . $ Show that there exists a natural number $ n_0 $ and a sequence of positive real numbers $ \left( x_n \right)_{n>n_0} $ that satisfy the following relation.
$$ n\int_0^{x_n} f(t)dt=1,\quad n_0<\forall n\in\mathbb{N} $$
[b]b)[/b] Prove that the sequence $ \left( nx_n \right)_{n> n_0} $ is convergent and find its limit.
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]