Olympiad problems

2010 Paenza, 4

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function with the following property: for all $\alpha \in \mathbb{R}_{>0}$, the sequence $(a_n)_{n \in \mathbb{N}}$ defined as $a_n = f(n\alpha)$ satisfies $\lim_{n \to \infty} a_n = 0$. Is it necessarily true that $\lim_{x \to +\infty} f(x) = 0$?

2006 Moldova MO 11-12, 6

Tags: linear algebra , matrix , limit , vector , algebra

Sequences $(x_n)_{n\ge1}$, $(y_n)_{n\ge1}$ satisfy the relations $x_n=4x_{n-1}+3y_{n-1}$ and $y_n=2x_{n-1}+3y_{n-1}$ for $n\ge1$. If $x_1=y_1=5$ find $x_n$ and $y_n$. Calculate $\lim_{n\rightarrow\infty}\frac{x_n}{y_n}$.

2005 Harvard-MIT Mathematics Tournament, 5

Tags: limit , logarithm , calculus

Calculate \[ \lim_{x \to 0^+} \left( x^{x^x} - x^x \right). \]

1987 Austrian-Polish Competition, 3

Tags: functional , limit , functional equation , sequence

A function $f: R \to R$ satisfies $f (x + 1) = f (x) + 1$ for all $x$. Given $a \in R$, define the sequence $(x_n)$ recursively by $x_0 = a$ and $x_{n+1} = f (x_n)$ for $n \ge 0$. Suppose that, for some positive integer m, the difference $x_m - x_0 = k$ is an integer. Prove that the limit $\lim_{n\to \infty}\frac{x_n}{n}$ exists and determine its value.

2007 Romania Team Selection Test, 2

Tags: number theory , limit , probability

The world-renowned Marxist theorist [i]Joric[/i] is obsessed with both mathematics and social egalitarianism. Therefore, for any decimal representation of a positive integer $n$, he tries to partition its digits into two groups, such that the difference between the sums of the digits in each group be as small as possible. Joric calls this difference the [i]defect[/i] of the number $n$. Determine the average value of the defect (over all positive integers), that is, if we denote by $\delta(n)$ the defect of $n$, compute \[\lim_{n \rightarrow \infty}\frac{\sum_{k = 1}^{n}\delta(k)}{n}.\] [i]Iurie Boreico[/i]

2004 Vietnam National Olympiad, 1

Tags: trigonometry , algebra , limit

The sequence $ (x_n)^{\infty}_{n\equal{}1}$ is defined by $ x_1 \equal{} 1$ and $ x_{n\plus{}1} \equal{}\frac{(2 \plus{} \cos 2\alpha)x_n \minus{} \cos^2\alpha}{(2 \minus{} 2 \cos 2\alpha)x_n \plus{} 2 \minus{} \cos 2\alpha}$, for all $ n \in\mathbb{N}$, where $ \alpha$ is a given real parameter. Find all values of $ \alpha$ for which the sequence $ (y_n)$ given by $ y_n \equal{} \sum_{k\equal{}1}^{n}\frac{1}{2x_k\plus{}1}$ has a finite limit when $ n \to \plus{}\infty$ and find that limit.

2019 Jozsef Wildt International Math Competition, W. 9

Tags: limit , sequence

Let $\alpha > 0$ be a real number. Compute the limit of the sequence $\{x_n\}_{n\geq 1}$ defined by $$x_n=\begin{cases} \sum \limits_{k=1}^n \sinh \left(\frac{k}{n^2}\right),& \text{when}\ n>\frac{1}{\alpha}\\ 0,& \text{when}\ n\leq \frac{1}{\alpha}\end{cases}$$

2005 IberoAmerican Olympiad For University Students, 3

Tags: limit , algebra

Consider the sequence defined recursively by $(x_1,y_1)=(0,0)$, $(x_{n+1},y_{n+1})=\left(\left(1-\frac{2}{n}\right)x_n-\frac{1}{n}y_n+\frac{4}{n},\left(1-\frac{1}{n}\right)y_n-\frac{1}{n}x_n+\frac{3}{n}\right)$. Find $\lim_{n\to \infty}(x_n,y_n)$.

1970 IMO Longlists, 52

Tags: integration , calculus , limit , inequality , area

The real numbers $a_0,a_1,a_2,\ldots$ satisfy $1=a_0\le a_1\le a_2\le\ldots. b_1,b_2,b_3,\ldots$ are defined by $b_n=\sum_{k=1}^n{1-{a_{k-1}\over a_k}\over\sqrt a_k}$. [b]a.)[/b] Prove that $0\le b_n<2$. [b]b.)[/b] Given $c$ satisfying $0\le c<2$, prove that we can find $a_n$ so that $b_n>c$ for all sufficiently large $n$.

2008 Teodor Topan, 4

Tags: limit , real analysis

Let $ (a_n)_{n \in \mathbb{N}^*}$ be a sequence of real positive numbers such that $ a_n>a_0,n\in \mathbb{N}$. Prove that $ \displaystyle\lim_{n\to\infty}\displaystyle\sum_{k\equal{}0}^{n}(\frac{a_k}{a_{n\minus{}k}})^k\equal{}\infty$.

1949 Miklós Schweitzer, 1

Tags: limit , real analysis

Let an infinite sequence of measurable sets be given on the interval $ (0,1)$ the measures of which are $ \geq \alpha>0$. Show that there exists a point of $ (0,1)$ which belongs to infinitely many terms of the sequence.

2019 District Olympiad, 1

Tags: calculus , limit , convergence , sequence

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 $$ \lim_{n \to \infty} \left( \frac{a_{n+1}}{a_n} \right)^n.$$

2007 Today's Calculation Of Integral, 176

Tags: calculus computations , symmetry , integration , trigonometry , calculus , limit , induction

Let $f_{n}(x)=\sum_{k=1}^{n}\frac{\sin kx}{\sqrt{k(k+1)}}.$ Find $\lim_{n\to\infty}\int_{0}^{2\pi}\{f_{n}(x)\}^{2}dx.$

2010 Argentina Team Selection Test, 3

Tags: limit , algebra , function

Find all functions $f: \mathbb R \rightarrow \mathbb R$ such that \[f(x+xy+f(y)) = \left(f(x)+\frac{1}{2}\right) \left(f(y)+\frac{1}{2}\right)\] holds for all real numbers $x,y$.

1990 Dutch Mathematical Olympiad, 2

Tags: limit , induction , algebra

Consider the sequence $ a_1\equal{}\frac{3}{2}, a_{n\plus{}1}\equal{}\frac{3a_n^2\plus{}4a_n\minus{}3}{4a_n^2}.$ $ (a)$ Prove that $ 1<a_n$ and $ a_{n\plus{}1}<a_n$ for all $ n$. $ (b)$ From $ (a)$ it follows that $ \displaystyle\lim_{n\to\infty}a_n$ exists. Find this limit. $ (c)$ Determine $ \displaystyle\lim_{n\to\infty}a_1a_2a_3...a_n$.

2008 Grigore Moisil Intercounty, 1

Tags: real analysis , limit

Let be a sequence of positive real numbers $ \left( a_n\right)_{n\ge 1} $ defined by the recurrence relation $ a_{n+1}=\ln \left(1+a_n\right) . $ Show that: [b]1)[/b] $ \lim_{n\to\infty } a_n=0 $ [b]2)[/b] $ \lim_{n\to\infty } na_n=2 $ [b]3[/b] $ \lim_{n\to\infty } \frac{n(na_n-2)}{\ln n}=2/3 $ [i]Dorel Duca[/i] and [i]Dorian Popa[/i]

1994 Turkey Team Selection Test, 2

Tags: algebra , limit , logarithm

Show that positive integers $n_i,m_i$ $(i=1,2,3, \cdots )$ can be found such that $ \mathop{\lim }\limits_{i \to \infty } \frac{2^{n_i}}{3^{m_i }} = 1$

2006 IMS, 4

Tags: limit , function , topology , advanced fields

Assume that $X$ is a seperable metric space. Prove that if $f: X\longrightarrow\mathbb R$ is a function that $\lim_{x\rightarrow a}f(x)$ exists for each $a\in\mathbb R$. Prove that set of points in which $f$ is not continuous is countable.

2007 Nicolae Coculescu, 2

Tags: function , sequence , limit , real analysis

Let $ F:\mathbb{R}\longrightarrow\mathbb{R} $ be a primitive with $ F(0)=0 $ of the function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ defined by $ f(x)=\frac{x}{1+e^x} , $ and let be a sequence $ \left( x_n \right)_{n\ge 0} $ such that $ x_0>0 $ and defined as $ x_n=F\left( x_{n-1} \right) . $ Calculate $ \lim_{n\to\infty } \frac{1}{n}\sum_{k=1}^n \frac{x_k}{\sqrt{x_{k+1}}} $ [i]Florian Dumitrel[/i]

2010 Today's Calculation Of Integral, 584

Tags: limit , calculus , calculus computations , integration , logarithm

Find $ \lim_{x\rightarrow \infty} \left(\int_0^x \sqrt{1\plus{}e^{2t}}\ dt\minus{}e^x\right)$.

2024 ISI Entrance UGB, P5

Tags: limit , lhopital , polynomial limit , calculus

Let $P(x)$ be a polynomial with real coefficients. Let $\alpha_1 , \dots , \alpha_k$ be the distinct real roots of $P(x)=0$. If $P'$ is the derivative of $P$, show that for each $i=1,\dots , k$ \[\lim_{x\to \alpha_i} \frac{(x-\alpha_i)P'(x)}{P(x)} = r_i, \] for some positive integer $r_i$.

1989 Greece National Olympiad, 3

Tags: limit , sequence , real analysis

Find the limit of the sequence $x_n$ defined by recurrence relation $$x_{n+2}=\frac{1}{12}x_{n+1}+\frac{1}{2}x_{n}+1$$ where $n=0,1,2,...$ for any initial values $x_2,x_1$.

2006 VTRMC, Problem 5

Tags: sequence , limit , convergence , summation

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.

2011 N.N. Mihăileanu Individual, 4

Tags: sequence , limit , real analysis

[b]a)[/b] Prove that there exists an unique sequence of real numbers $ \left( x_n \right)_{n\ge 1} $ satisfying $$ -\text{ctg} x_n=x_n\in\left( (2n+1)\pi /2,(n+1)\pi \right) , $$ for any nonnegative integer $ n. $ [b]b)[/b] Show that $ \lim_{n\to\infty } \left( \frac{x_n}{(n+1)\pi } \right)^{n^2} =e^{-1/\pi^2} . $ [i]Cătălin Zârnă[/i]

2020 LIMIT Category 1, 5

Tags: counting , limit

Rohit is counting the minimum number of lines $m$, that can be drawn so that the number of distinct points of intersection exceeds $2020$. Find $m$. (A)$63$ (B)$64$ (C)$65$ (D)$66$