Olympiad problems

ICMC 6, 6

Tags: ICMC , college contests , real analysis , Convergence , Sequences , floor function

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]

1969 Putnam, A6

Tags: Putnam , 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.

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.

2010 SEEMOUS, Problem 1

Tags: function , Convergence , Sequences , integration , calculus

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

2012 SEEMOUS, Problem 2

Tags: Summation , Convergence

Let $a_n>0$, $n\ge1$. Consider the right triangles $\triangle A_0A_1A_2$, $\triangle A_0A_2A_3,\ldots$, $\triangle A_0A_{n-1}A_n,\ldots,$ as in the figure. (More precisely, for every $n\ge2$ the hypotenuse $A_0A_n$ of $\triangle A_0A_{n-1}A_n$ is a leg of $\triangle A_0A_nA_{n+1}$ with right angle $\angle A_0A_nA_{n+1}$, and the vertices $A_{n-1}$ and $A_{n+1}$ lie on the opposite sides of the straight line $A_0A_n$; also, $|A_{n-1}A_n|=a_n$ for every $n\ge1$.) [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvYi8yL2M1ZjAxM2I1ZWU0N2E4MzQyYWIzNmQ5OGM3NjJlZjljODdmMTliLnBuZw==&rn=U0VFTU9VUyAyMDEyLnBuZw==[/img] Is it possible for the set of points $\{A_n\mid n\ge0\}$ to be unbounded but the series $\sum_{n=2}^\infty m\angle A_{n-1}A_0A_n$ to be convergent? [i]Note.[/i] A subset $B$ of the plane is bounded if and only if there is a disk $D$ such that $B\subseteq D$.

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.

2006 VJIMC, Problem 2

Tags: limits , Convergence

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.

2001 SNSB Admission, 2

Tags: function , real analysis , series , Riemann sum , Sequences , Convergence

Let be a number $ a\in \left[ 1,\infty \right) $ and a function $ f\in\mathcal{C}^2(-a,a) . $ Show that the sequence $$ \left( \sum_{k=1}^n f\left( \frac{k}{n^2} \right) \right)_{n\ge 1} $$ is convergent, and determine its limit.

1941 Putnam, B2

Tags: Putnam , 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) $\lim_{n\to \infty} \sum_{i=1}^{n^{2}} \frac{1}{\sqrt{n^2 +i}}$.

2007 Grigore Moisil Intercounty, 4

Tags: inequalities , Sequences , real analysis , Convergence

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.

2003 IMC, 6

Tags: limit , Convergence , real analysis

Let $(a_{n})$ be the sequence defined by $a_{0}=1,a_{n+1}=\sum_{k=0}^{n}\dfrac{a_k}{n-k+2}$. Find the limit \[\lim_{n \rightarrow \infty} \sum_{k=0}^{n}\dfrac{a_{k}}{2^{k}},\] if it exists.

2015 VJIMC, 3

Tags: Summation , Convergence , infinite sum , college contests , real analysis

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

2005 Gheorghe Vranceanu, 4

Tags: real analysis , limits , Convergence

Let be a sequence of real numbers $ \left( x_n \right)_{n\geqslant 0} $ with $ x_0\neq 0,1 $ and defined as $ x_{n+1}=x_n+x_n^{-1/x_0} . $ [b]a)[/b] Show that the sequence $ \left( x_n\cdot n^{-\frac{x_0}{1+x_0}} \right)_{n\geqslant 0} $ is convergent. [b]b)[/b] Prove that $ \inf_{x_0\neq 0,1} \lim_{n\to\infty } x_n\cdot n^{-\frac{x_0}{1+x_0}} =1. $

2004 VJIMC, Problem 3

Tags: Convergence , Sequences , limits , real analysis

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.

1973 Putnam, A2

Tags: Putnam , Convergence , series

Consider an infinite series whose $n$-th term is $\pm (1\slash n)$, the $\pm$ signs being determined according to a pattern that repeats periodically in blocks of eight (there are $2^{8}$ possible patterns). (a) Show that a sufficient condition for the series to be conditionally convergent is that there are four "$+$" signs and four "$-$" signs in the block of eight signs. (b) Is this sufficient condition also necessary?

1981 Putnam, A1

Tags: Putnam , 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 $$\lim_{n\to \infty} \frac{E(n)}{n^2 }.$$

1975 Putnam, B5

Tags: Putnam , function , Convergence

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

2004 Alexandru Myller, 1

Tags: analysis , real analysis , Sequences , Convergence

[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]

2019 District Olympiad, 4

Tags: continuous function , Convergence , Sequences , calculus

Let $f: [0, \infty) \to [0, \infty)$ be a continuous function with $f(0)>0$ and having the property $$x-y<f(y)-f(x) \le 0~\forall~0 \le x<y.$$ Prove that: $a)$ There exists a unique $\alpha \in (0, \infty)$ such that $(f \circ f)(\alpha)=\alpha.$ $b)$ The sequence $(x_n)_{n \ge 1},$ defined by $x_1 \ge 0$ and $x_{n+1}=f(x_n)~\forall~n \in \mathbb{N}$ is convergent.

1964 Putnam, B1

Tags: Putnam , 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 $$\lim_{n \to \infty} \frac{ V_{n}}{n}=0.$$

2023 Brazil Undergrad MO, 2

Tags: function , real analysis , Convergence , binomial coefficients

Let $a_n = \frac{1}{\binom{2n}{n}}, \forall n \leq 1$. a) Show that $\sum\limits_{n=1}^{+\infty}a_nx^n$ converges for all $x \in (-4, 4)$ and that the function $f(x) = \sum\limits_{n=1}^{+\infty}a_nx^n$ satisfies the differential equation $x(x - 4)f'(x) + (x + 2)f(x) = -x$. b) Prove that $\sum\limits_{n=1}^{+\infty}\frac{1}{\binom{2n}{n}} = \frac{1}{3} + \frac{2\pi\sqrt{3}}{27}$.

2010 VJIMC, Problem 1

Tags: Summation , Convergence

a) Is it true that for every bijection $f:\mathbb N\to\mathbb N$ the series $$\sum_{n=1}^\infty\frac1{nf(n)}$$is convergent? b) Prove that there exists a bijection $f:\mathbb N\to\mathbb N$ such that the series $$\sum_{n=1}^\infty\frac1{n+f(n)}$$is convergent. ($\mathbb N$ is the set of all positive integers.)

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.

2021 Romanian Master of Mathematics Shortlist, A4

Tags: analysis , RMM Shortlist , Sequences , Convergence , algebra

Let $f: \mathbb{R} \to \mathbb{R}$ be a non-decreasing function such that $f(y) - f(x) < y - x$ for all real numbers $x$ and $y > x$. The sequence $u_1,u_2,\ldots$ of real numbers is such that $u_{n+2} = f(u_{n+1}) - f(u_n)$ for all $n\geq 1$. Prove that for any $\varepsilon > 0$ there exists a positive integer $N$ such that $|u_n| < \varepsilon$ for all $n\geq N$.

2022 Miklós Schweitzer, 8

Tags: college contests , 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 $f(s_n) \to z$).