Found problems: 280
1986 Bulgaria National Olympiad, Problem 6
Let $0<k<1$ be a given real number and let $(a_n)_{n\ge1}$ be an infinite sequence of real numbers which satisfies $a_{n+1}\le\left(1+\frac kn\right)a_n-1$. Prove that there is an index $t$ such that $a_t<0$.
Mathematical Minds 2024, P6
Consider the sequence $a_1, a_2, \dots$ of positive integers such that $a_1=2$ and $a_{n+1}=a_n^4+a_n^3-3a_n^2-a_n+2$, for all $n\geqslant 1$. Prove that there exist infinitely many prime numbers that don't divide any term of the sequence.
[i]Proposed by Pavel Ciurea[/i]
2012 IFYM, Sozopol, 5
We are given the following sequence: $a_1=8,a_2=20,a_{n+2}=a_{n+1}^2+12a_n a_{n+1}+11a_n$. Prove that none of the members of the sequence can be presented as a sum of three seventh powers of natural numbers.
2016 Miklós Schweitzer, 4
Prove that there exists a sequence $a(1),a(2),\dots,a(n),\dots$ of real numbers such that
\[
a(n+m)\le a(n)+a(m)+\frac{n+m}{\log (n+m)}
\]
for all integers $m,n\ge 1$, and such that the set $\{a(n)/n:n\ge 1\}$ is everywhere dense on the real line.
[i]Remark.[/i] A theorem of de Bruijn and Erdős states that if the inequality above holds with $f(n + m)$ in place of the last term on the right-hand side, where $f(n)\ge 0$ is nondecreasing and $\sum_{n=2}^\infty f(n)/n^2<\infty$, then $a(n)/n$ converges or tends to $(-\infty)$.
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]
2016 Thailand TSTST, 2
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.
2023 Simon Marais Mathematical Competition, A4
Let $x_0, x_1, x_2 \dots$ be a sequence of positive real numbers such that for all $n \geq 0$, $$x_{n+1} = \dfrac{(n^2+1)x_n^2}{x_n^3+n^2}$$ For which values of $x_0$ is this sequence bounded?
2020 China Northern MO, P1
The function $f(x)=x^2+ \sin x$ and the sequence of positive numbers $\{ a_n \}$ satisfy $a_1=1$, $f(a_n)=a_{n-1}$, where $n \geq 2$. Prove that there exists a positive integer $n$ such that $a_1+a_2+ \dots + a_n > 2020$.
2009 VJIMC, Problem 4
Let $(a_n)_{n=1}^\infty$ be a sequence of real numbers. We say that the sequence $(a_n)_{n=1}^\infty$ covers the set of
positive integers if for any positive integer $m$ there exists a positive integer $k$ such that $\sum_{n=1}^\infty a_n^k=m$.
a) Does there exist a sequence of real positive numbers which covers the set of positive integers?
b) Does there exist a sequence of real numbers which covers the set of positive integers?
2025 SEEMOUS, P4
Let $(a_n)_{n\geq 1}$ be a monotone decreasing sequence of real numbers that converges to $0$. Prove that $\sum_{n=1}^{\infty}\frac{a_n}{n}$ is convergent if and only if the sequence $(a_n\ln n)_{n\geq 1}$ is bounded and $\sum_{n=1}^{\infty} (a_n-a_{n+1})\ln n$ is convergent.
2004 Croatia National Olympiad, Problem 3
The sequences $(x_n),(y_n),(z_n),n\in\mathbb N$, are defined by the relations
$$x_{n+1}=\frac{2x_n}{x_n^2-1},\qquad y_{n+1}=\frac{2y_n}{y_n^2-1},\qquad z_{n+1}=\frac{2z_n}{z_n^2-1},$$where $x_1=2$, $y_1=4$, and $x_1y_1z_1=x_1+y_1+z_1$.
(a) Show that $x_n^2\ne1$, $y_n^2\ne1$, $z_n^2\ne1$ for all $n$;
(b) Does there exist a $k\in\mathbb N$ for which $x_k+y_k+z_k=0$?
2015 Romania National Olympiad, 3
Let be two nonnegative real numbers $ a,b $ with $ b>a, $ and a sequence $ \left( x_n \right)_{n\ge 1} $ of real numbers such that the sequence $ \left( \frac{x_1+x_2+\cdots +x_n}{n^a} \right)_{n\ge 1} $ is bounded.
Show that the sequence $ \left( x_1+\frac{x_2}{2^b} +\frac{x_3}{3^b} +\cdots +\frac{x_n}{n^b} \right)_{n\ge 1} $ is convergent.
1978 Bulgaria National Olympiad, Problem 1
We are given the sequence $a_1,a_2,a_3,\ldots$, for which:
$$a_n=\frac{a^2_{n-1}+c}{a_{n-2}}\enspace\text{for all }n>2.$$
Prove that the numbers $a_1$, $a_2$ and $\frac{a_1^2+a_2^2+c}{a_1a_2}$ are whole numbers.
KoMaL A Problems 2024/2025, A. 905
We say that a strictly increasing sequence of positive integers $n_1, n_2,\ldots$ is [i]non-decelerating[/i] if $n_{k+1}-n_k\le n_{k+2}-n_{k+1}$ holds for all positive integers $k$. We say that a strictly increasing sequence $n_1, n_2, \ldots$ is [i]convergence-inducing[/i], if the following statement is true for all real sequences $a_1, a_2, \ldots$: if subsequence $a_{m+n_1}, a_{m+n_2}, \ldots$ is convergent and tends to $0$ for all positive integers $m$, then sequence $a_1, a_2, \ldots$ is also convergent and tends to $0$. Prove that a non-decelerating sequence $n_1, n_2,\ldots$ is convergence-inducing if and only if sequence $n_2-n_1$, $n_3-n_2$, $\ldots$ is bounded from above.
[i]Proposed by András Imolay[/i]
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 $x$.
1969 Czech and Slovak Olympiad III A, 3
Let $p$ be a prime. How many different (infinite) sequences $\left(a_k\right)_{k\ge0}$ exist such that for every positive integer $n$ \[\frac{a_0}{a_1}+\frac{a_0}{a_2}+\cdots+\frac{a_0}{a_n}+\frac{p}{a_{n+1}}=1?\]
2017 Romania National Olympiad, 1
Let be a surjective function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that has the property that if the sequence $ \left( f\left( x_n \right) \right)_{n\ge 1} $ is convergent, then the sequence $ \left( x_n \right)_{n\ge 1} $ is convergent. Prove that it is continuous.
2018 IMC, 1
Let $(a_n)_{n=1}^{\infty}$ and $(b_n)_{n=1}^{\infty}$ be two sequences of positive numbers. Show that the following statements are equivalent:
[list=1]
[*]There is a sequence $(c_n)_{n=1}^{\infty}$ of positive numbers such that $\sum_{n=1}^{\infty}{\frac{a_n}{c_n}}$ and $\sum_{n=1}^{\infty}{\frac{c_n}{b_n}}$ both converge;[/*]
[*]$\sum_{n=1}^{\infty}{\sqrt{\frac{a_n}{b_n}}}$ converges.[/*]
[/list]
[i]Proposed by Tomáš Bárta, Charles University, Prague[/i]
2019 Jozsef Wildt International Math Competition, W. 33
Let $0 < \frac{1}{q} \leq \frac{1}{p} < 1$ and $\frac{1}{p}+\frac{1}{q}=1$. Let $u_k$, $v_k$, $a_k$ and $b_k$ be non-negative real sequences such as $u^2_k > a^p_k$ and $v_k > b^q_k$, where $k = 1, 2,\cdots , n$. If $0 < m_1\leq u_k \leq M_1$ and $0 < m_2 \leq v_k \leq M_2$ , then $$\left(\sum \limits_{k=1}^n\left(l^p\left(u_k+v_k\right)^2-\left(a_k+b_k\right)^p\right)\right)^{\frac{1}{p}}\geq \left(\sum \limits_{k=1}^n\left(u_k^2-a_k^p\right)\right)^{\frac{1}{p}}\left(\sum \limits_{k=1}^n\left(v_k^2-b_k^p\right)\right)^{\frac{1}{p}}$$where $$l=\frac{M_1M_2+m_1m_2}{2\sqrt{m_1M_1m_2M_2}}$$
1973 Bulgaria National Olympiad, Problem 1
Let the sequence $a_1,a_2,\ldots,a_n,\ldots$ is defined by the conditions: $a_1=2$ and $a_{n+1}=a_n^2-a_n+1$ $(n=1,2,\ldots)$. Prove that:
(a) $a_m$ and $a_n$ are relatively prime numbers when $m\ne n$.
(b) $\lim_{n\to\infty}\sum_{k=1}^n\frac1{a_k}=1$
[i]I. Tonov[/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 $n \geq 3$.
2019 Belarus Team Selection Test, 4.3
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 $a_{2018}-a_{2017}$.
2016 Azerbaijan 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.
2022 IMO Shortlist, A8
For a positive integer $n$, an [i]$n$-sequence[/i] is a sequence $(a_0,\ldots,a_n)$ of non-negative integers satisfying the following condition: if $i$ and $j$ are non-negative integers with $i+j \leqslant n$, then $a_i+a_j \leqslant n$ and $a_{a_i+a_j}=a_{i+j}$.
Let $f(n)$ be the number of $n$-sequences. Prove that there exist positive real numbers $c_1$, $c_2$, and $\lambda$ such that \[c_1\lambda^n<f(n)<c_2\lambda^n\] for all positive integers $n$.
2015 District Olympiad, 4
Let $ \left( x_n\right)_{n\ge 1} $ be a sequence of real numbers of the interval $ [1,\infty) . $ Suppose that the sequence $ \left( \left[ x_n^k\right]\right)_{n\ge 1} $ is convergent for all natural numbers $ k. $ Prove that $ \left( x_n\right)_{n\ge 1} $ is convergent.
Here, $ [\beta ] $ means the greatest integer smaller than $ \beta . $