Found problems: 53
1940 Putnam, A7
If $\sum_{i=1}^{\infty} u_{i}^{2}$ and $\sum_{i=1}^{\infty} v_{i}^{2}$ are convergent series of real numbers, prove that
$$\sum_{i=1}^{\infty}(u_{i}-v_{i})^{p}$$
is convergent, where $p\geq 2$ is an integer.
2019 IMC, 7
Let $C=\{4,6,8,9,10,\ldots\}$ be the set of composite positive integers. For each $n\in C$ let $a_n$ be the smallest positive integer $k$ such that $k!$ is divisible by $n$. Determine whether the following series converges:
$$\sum_{n\in C}\left(\frac{a_n}{n}\right)^n.$$
[i]Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan[/i]
2019 District Olympiad, 1
Let $(a_n)_{n \ge 1}$ be a sequence of positive real numbers such that the sequence $(a_{n+1}-a_n)_{n \ge 1}$ is convergent to a non-zero real number. Evaluate the limit $$ \lim_{n \to \infty} \left( \frac{a_{n+1}}{a_n} \right)^n.$$
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 $\lim_{n \to \infty} a_n =1$.
1996 IMC, 7
Prove that if $f:[0,1]\rightarrow[0,1]$ is a continuous function, then the sequence of iterates $x_{n+1}=f(x_{n})$ converges if and only if
$$\lim_{n\to \infty}(x_{n+1}-x_{n})=0$$
2025 VJIMC, 3
Let us call a sequence $(b_1, b_2, \ldots)$ of positive integers fast-growing if $b_{n+1} \geq b_n + 2$ for all $n \geq 1$. Also, for a sequence $a = (a(1), a(2), \ldots)$ of real numbers and a sequence $b = (b_1, b_2, \ldots)$ of positive integers, let us denote
\[
S(a, b) = \sum_{n=1}^{\infty} \left| a(b_n) + a(b_n + 1) + \cdots + a(b_{n+1} - 1) \right|.
\]
a) Do there exist two fast-growing sequences $b = (b_1, b_2, \ldots)$, $c = (c_1, c_2, \ldots)$ such that for every sequence $a = (a(1), a(2), \ldots)$, if all the series
\[
\sum_{n=1}^{\infty} a(n), \quad S(a, b) \quad \text{and} \quad S(a, c)
\]
are convergent, then the series $\sum_{n=1}^{\infty} |a(n)|$ is also convergent?
b) Do there exist three fast-growing sequences $b = (b_1, b_2, \ldots)$, $c = (c_1, c_2, \ldots)$, $d = (d_1, d_2, \ldots)$ such that for every sequence $a = (a(1), a(2), \ldots)$, if all the series
\[
S(a, b), \quad S(a, c) \quad \text{and} \quad S(a, d)
\]
are convergent, then the series $\sum_{n=1}^{\infty} |a(n)|$ is also convergent?
Kvant 2020, M365
[list=a]
[*]The sum of several numbers is equal to one. Can the sum of their cubes be greater than one?
[*]The same question as before, for numbers not exceeding one.
[*]Can it happen that the series $a_1+a_2+\cdots$ converges, but the series $a_1^3+a_2^3+\cdots$ diverges?
[/list]
2019 Centers of Excellency of Suceava, 2
Let be two real numbers $ b>a>0, $ and a sequence $ \left( x_n \right)_{n\ge 1} $ with $ x_2>x_1>0 $ and such that
$$ ax_{n+2}+bx_n\ge (a+b)x_{n+1} , $$
for any natural numbers $ n. $
Prove that $ \lim_{n\to\infty } x_n=\infty . $
[i]Dan Popescu[/i]
1988 Putnam, B4
Prove that if $\sum_{n=1}^\infty a_n$ is a convergent series of positive real numbers, then so is $\sum_{n=1}^\infty (a_n)^{n/(n+1)}$.
ICMC 7, 4
Let $(t_n)_{n\geqslant 1}$ be the sequence defined by $t_1=1, t_{2k}=-t_k$ and $t_{2k+1}=t_{k+1}$ for all $k\geqslant 1.$ Consider the series \[\sum_{n=1}^\infty\frac{t_n}{n^{1/2024}}.\]Prove that this series converges to a positive real number.
[i]Proposed by Dylan Toh[/i]
1964 Putnam, B5
Let $u_n$ denote the least common multiple of the first $n$ terms of a strictly increasing sequence of positive integers.
Prove that the series
$$\sum_{n=1}^{\infty} \frac{1}{ u_n }$$
is convergent
1953 Putnam, B1
Is the infinite series
$$\sum_{n=1}^{\infty} \frac{1}{n^{1+\frac{1}{n}}}$$
convergent?
2021 SEEMOUS, Problem 4
For $p \in \mathbb{R}$, let $(a_n)_{n \ge 1}$ be the sequence defined by
\[ a_n=\frac{1}{n^p} \int_0^n |\sin( \pi x)|^x \mathrm dx. \]
Determine all possible values of $p$ for which the series $\sum_{n=1}^\infty a_n$ converges.
1964 Putnam, A3
Let $P_1 , P_2 , \ldots$ be a sequence of distinct points which is dense in the interval $(0,1)$. The points $P_1 , \ldots , P_{n-1}$ decompose the interval into $n$ parts, and $P_n$ decomposes one of these into two parts. Let $a_n$ and $b_n$ be the length of these two intervals. Prove that
$$\sum_{n=1}^{\infty} a_n b_n (a_n +b_n) =1 \slash 3.$$
1952 Putnam, B5
If the terms of a sequence $a_{1}, a_{2}, \ldots$ are monotonic, and if $\sum_{n=1}^{\infty} a_n$ converges, show that $\sum_{n=1}^{\infty} n(a_{n} -a_{n+1 })$ converges.
2022 SEEMOUS, 4
Let $\mathcal{F}$ be the family of all nonempty finite subsets of $\mathbb{N} \cup \{0\}.$ Find all real numbers $a$ for which the series
$$\sum_{A \in \mathcal{F}} \frac{1}{\sum_{k \in A}a^k}$$
is convergent.
1952 Putnam, B7
Given any real number $N_0,$ if $N_{j+1}= \cos N_j ,$ prove that $\lim_{j\to \infty} N_j$ exists and is independent of $N_0.$
2024 CIIM, 6
Given a real number $x$, define the series
\[
S(x) = \sum_{n=1}^{\infty} \{n! \cdot x\},
\]
where $\{s\} = s - \lfloor s \rfloor$ is the fractional part of the number $s$. Determine if there exists an irrational number $x$ for which the series $S(x)$ converges.
2018 District Olympiad, 3
Let $(a_n)_{n\ge 1}$ be a sequence such that $a_n > 1$ and $a_{n+1}^2 \ge a_n a_{n + 2}$, for any $n\ge 1$. Show that the sequence $(x_n)_{n\ge 1}$ given by $x_n = \log_{a_n} a_{n + 1}$ for $n\ge 1$ is convergent and compute its limit.
2006 VTRMC, Problem 5
Let $\{a_n\}$ be a monotonically decreasing sequence of positive real numbers with limit $0$. Let $\{b_n\}$ be a rearrangement of the sequence such that for every non-negative integer $m$, the terms $b_{3m+1}$, $b_{3m+2}$, $b_{3m+3}$ are a rearrangement of the terms $a_{3m+1}$, $a_{3m+2}$, $a_{3m+3}$. Prove or give a counterexample to the following statement: the series $\sum_{n=1}^\infty(-1)^nb_n$ is convergent.
1961 Putnam, B7
Given a sequence $(a_n)$ of non-negative real numbers such that $a_{n+m}\leq a_{n} a_{m} $ for all pairs of positive integers $m$ and $n,$ prove that the sequence $(\sqrt[n]{a_n })$ converges.
2001 IMC, 2
Let $a_{0}=\sqrt{2}, b_{0}=2,a_{n+1}=\sqrt{2-\sqrt{4-a_{n}^{2}}},b_{n+1}=\frac{2b_{n}}{2+\sqrt{4+b_{n}^{2}}}$.
a) Prove that the sequences $(a_{n})$ and $(b_{n})$ are decreasing and converge to $0$.
b) Prove that the sequence $(2^{n}a_{n})$ is increasing, the sequence $(2^{n}b_{n})$ is decreasing and
both converge to the same limit.
c) Prove that there exists a positive constant $C$ such that for all $n$ the following inequality holds: $0 <b_{n}-a_{n} <\frac{C}{8^{n}}$.
2023 SEEMOUS, P4
Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous, strictly decreasing function such that $f([0,1])\subseteq[0,1]$.
[list=i]
[*]For all positive integers $n{}$ prove that there exists a unique $a_n\in(0,1)$, solution of the equation $f(x)=x^n$. Moreover, if $(a_n){}$ is the sequence defined as above, prove that $\lim_{n\to\infty}a_n=1$.
[*]Suppose $f$ has a continuous derivative, with $f(1)=0$ and $f'(1)<0$. For any $x\in\mathbb{R}$ we define \[F(x)=\int_x^1f(t) \ dt.\]Let $\alpha{}$ be a real number. Study the convergence of the series \[\sum_{n=1}^\infty F(a_n)^\alpha.\]
[/list]
2021 Alibaba Global Math Competition, 7
A subset $Q \subset H^s(\mathbb{R})$ is said to be equicontinuous if for any $\varepsilon>0$, $\exists \delta>0$ such that
\[\|f(x+h)-f(x)\|_{H^s}<\varepsilon, \quad \forall \vert h\vert<\delta, \quad f \in Q.\]
Fix $r<s$, given a bounded sequence of functions $f_n \in H^s(\mathbb{R}$. If $f_n$ converges in $H^r(\mathbb{R})$ and equicontinuous in $H^s(\mathbb{R})$, show that it also converges in $H^s(\mathbb{R})$.
2018 VTRMC, 7
A continuous function $f : [a,b] \to [a,b]$ is called piecewise monotone if $[a, b]$ can be subdivided into finitely many subintervals
$$I_1 = [c_0,c_1], I_2 = [c_1,c_2], \dots , I_\ell = [ c_{\ell - 1},c_\ell ]$$
such that $f$ restricted to each interval $I_j$ is strictly monotone, either increasing or decreasing. Here we are assuming that $a = c_0 < c_1 < \cdots < c_{\ell - 1} < c_\ell = b$. We are also assuming that each $I_j$ is a maximal interval on which $f$ is strictly monotone. Such a maximal interval is called a lap of the function $f$, and the number $\ell = \ell (f)$ of distinct laps is called the lap number of $f$. If $f : [a,b] \to [a,b]$ is a continuous piecewise-monotone function, show that the sequence $( \sqrt[n]{\ell (f^n )})$ converges; here $f^n$ means $f$ composed with itself $n$-times, so $f^2 (x) = f(f(x))$ etc.