Olympiad problems

2004 VJIMC, Problem 3

Let $\sum_{n=1}^\infty a_n$ be a divergent series with positive nonincreasing terms. Prove that the series $$\sum_{n=1}^\infty\frac{a_n}{1+na_n}$$diverges.

1964 Putnam, B5

Tags: least common multiple , convergence

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

ICMC 7, 4

Tags: convergence , series , real analysis

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]

2010 SEEMOUS, Problem 1

Tags: function , convergence , integration , calculus , sequence

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 $\sum_{n=1}^\infty f_n(x),x\in[0,1]$.

2019 Teodor Topan, 3

Tags: density , mod 1 , real analysis , convergence , function , sequence , limit

Let $ \left( c_n \right)_{n\ge 1} $ be a sequence of real numbers. Prove that the sequences $ \left( c_n\sin n \right)_{n\ge 1} ,\left( c_n\cos n \right)_{n\ge 1} $ are both convergent if and only if $ \left( c_n \right)_{n\ge 1} $ converges to $ 0. $ [i]Mihai Piticari[/i] and [i]Vladimir Cerbu[/i]

2004 Alexandru Myller, 1

Tags: analysis , real analysis , convergence , sequence

[b]a)[/b] Let $ \left( x_n \right)_{n\ge 1} $ be a sequence of real numbers having the property that $ \left| x_{n+1} -x_n \right|\leqslant 1/2^n, $ for any $ n\geqslant 1. $ Show that $ \left( x_n \right)_{n\ge 1} $ is convergent. [b]b)[/b] Create a sequence $ \left( y_n \right)_{n\ge 1} $ of real numbers that has the following properties: $ \text{(i) } \lim_{n\to\infty } \left( y_{n+1} -y_n \right) = 0 $ $ \text{(ii) } $ is bounded $ \text{(iii) } $ is divergent [i]Eugen Popa[/i]

2017 Vietnamese Southern Summer School contest, Problem 1

Tags: analysis , sequence , real number , convergence

Given a real number $a$ and a sequence $(x_n)_{n=1}^\infty$ defined by: $$\left\{\begin{matrix} x_1=1 \\ x_2=0 \\ x_{n+2}=\frac{x_n^2+x_{n+1}^2}{4}+a\end{matrix}\right.$$ for all positive integers $n$. 1. For $a=0$, prove that $(x_n)$ converges. 2. Determine the largest possible value of $a$ such that $(x_n)$ converges.

1941 Putnam, B2

Tags: convergence , series

Find (i) $\lim_{n\to \infty} \sum_{i=1}^{n} \frac{1}{\sqrt{n^2 +i^{2}}}$. (ii) $\lim_{n\to \infty} \sum_{i=1}^{n} \frac{1}{\sqrt{n^2 +i}}$. (iii) $\lim_{n\to \infty} \sum_{i=1}^{n^{2}} \frac{1}{\sqrt{n^2 +i}}$.

2003 Gheorghe Vranceanu, 3

Tags: function , convergence , real analysis , series , floor function

Let be a sequence of functions $ a_n:\mathbb{R}\longrightarrow\mathbb{Z} $ defined as $ a_n(x)=\sum_{i=1}^n (-1)^i\lfloor xi\rfloor . $ [b]a)[/b] Find the real numbers $ y $ such that $ \left( a_n(y) \right)_{n\ge 1} $ converges to $ 1. $ [b]b)[/b] Find the real numbers $ z $ such that $ \left( a_n(z) \right)_{n\ge 1} $ converges.

1996 IMC, 7

Tags: function , real analysis , convergence

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 $$\lim_{n\to \infty}(x_{n+1}-x_{n})=0$$

1961 Putnam, B7

Tags: convergence , sequence

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.

1975 Putnam, B5

Tags: function , convergence

Define $f_{0}(x)=e^x$ and $f_{n+1}(x)=x f_{n}'(x)$. Show that $\sum_{n=0}^{\infty} \frac{f_{n}(1)}{n!}=e^e$.

2002 German National Olympiad, 4

Tags: inequalities , convergence , algebra , sequence , recursive

Given a positive real number $a_1$, we recursively define $a_{n+1} = 1+a_1 a_2 \cdots \cdot a_n.$ Furthermore, let $$b_n = \frac{1}{a_1 } + \frac{1}{a_2 } +\cdots + \frac{1}{a_n }.$$ Prove that $b_n < \frac{2}{a_1}$ for all positive integers $n$ and that this is the smallest possible bound.

2022 SEEMOUS, 4

Tags: subset , convergence , binary representation , real analysis

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.

2022 Miklós Schweitzer, 8

Tags: dirichlet series , convergence

Original in Hungarian; translated with Google translate; polished by myself. Prove that, the signs $\varepsilon_n = \pm 1$ can be chosen such that the function $f(s) = \sum_{n = 1}^\infty\frac{\varepsilon_n}{n^s}\colon \{s\in\Bbb C:\operatorname{Re}s > 1\}\to \Bbb C$ converges to every complex value at every point $\xi \in \{s\in\Bbb C:\operatorname{Re}s = 1\}$ (i.e. for every $\xi \in \{s\in\Bbb C:\operatorname{Re}s = 1\}$ and every $z \in \Bbb C$, there exists a sequence $s_n \to \xi$, $\operatorname{Re}s_n > 1$, for which $f(s_n) \to z$).

1981 Putnam, A1

Tags: convergence

Let $E(n)$ denote the largest integer $k$ such that $5^k$ divides $1^{1}\cdot 2^{2} \cdot 3^{3} \cdot \ldots \cdot n^{n}.$ Calculate $$\lim_{n\to \infty} \frac{E(n)}{n^2 }.$$

1964 Putnam, B1

Tags: convergence

Let $u_k$ be a sequence of integers, and let $V_n$ be the number of those which are less than or equal to $n$. Show that if $$\sum_{k=1}^{\infty} \frac{1}{u_k } < \infty,$$ then $$\lim_{n \to \infty} \frac{ V_{n}}{n}=0.$$

2020 LIMIT Category 2, 20

Tags: countable , convergence , calculus , sequence

Let $\{a_n \}_n$ be a sequence of real numbers such there there are countably infinite distinct subsequences converging to the same point. We call two subsequences distinct if they do not have a common term. Which of the following statements always holds: (A) $\{a_n \}_n$ is bounded (B) $\{a_n \}_n$ is unbounded (C) The set of convergent subsequence $\{a_n \}_n$ is countable (D) None of these

1969 Putnam, A6

Tags: convergence

Let a sequence $(x_n)$ be given and let $y_n = x_{n-1} +2 x_n $ for $n>1.$ Suppose that the sequence $(y_n)$ converges. Prove that the sequence $(x_n)$ converges, too.

2001 IMC, 2

Tags: inequalities , convergence , real analysis

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: $0 <b_{n}-a_{n} <\frac{C}{8^{n}}$.

ICMC 6, 6

Tags: real analysis , convergence , floor function , sequence

Consider the sequence defined by $a_1 = 2022$ and $a_{n+1} = a_n + e^{-a_n}$ for $n \geq 1$. Prove that there exists a positive real number $r$ for which the sequence $$\{ra_1\}, \{ra_{10}\}, \{ra_{100}\}, . . . $$converges. [i]Note[/i]: $\{x \} = x - \lfloor x \rfloor$ denotes the part of $x$ after the decimal point. [i]Proposed by Ethan Tan[/i]

2015 VJIMC, 3

Tags: summation , convergence , infinite sum , real analysis

[b]Problem 3[/b] Determine the set of real values of $x$ for which the following series converges, and find its sum: $$\sum_{n=1}^{\infty} \left(\sum_{\substack{k_1, k_2,\ldots , k_n \geq 0\\ 1\cdot k_1 + 2\cdot k_2+\ldots +n\cdot k_n = n}} \frac{(k_1+\ldots+k_n)!}{k_1!\cdot \ldots \cdot k_n!} x^{k_1+\ldots +k_n} \right) \ . $$

2006 VJIMC, Problem 2

Tags: convergence , limit

Suppose that $(a_n)$ is a sequence of real numbers such that the series $$\sum_{n=1}^\infty\frac{a_n}n$$is convergent. Show that the sequence $$b_n=\frac1n\sum^n_{j=1}a_j$$is convergent and find its limit.

2021 Alibaba Global Math Competition, 7

Tags: convergence , sobolev space , continuity

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 $H^s(\mathbb{R})$.

2006 VTRMC, Problem 5

Tags: convergence , summation , sequence , limit

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.