Olympiad problems

2020 Jozsef Wildt International Math Competition, W38

Let $(a_n)_{n\in\mathbb N}$ be a sequence, given by the recurrence: $$ma_{n+1}+(m-2)a_n-a_{n-1}=0$$ where $m\in\mathbb R$ is a parameter and the first two terms of $a_n$ are fixed known real numbers. Find $m\in\mathbb R$, so that $$\lim_{n\to\infty}a_n=0$$ [i]Proposed by Laurențiu Modan[/i]

1995 VJIMC, Problem 4

Tags: limits , real analysis , trigonometry

Let $\{x_n\}_{n=1}^\infty$ be a sequence such that $x_1=25$, $x_n=\operatorname{arctan}(x_{n-1})$. Prove that this sequence has a limit and find it.

2017 Romania National Olympiad, 1

Tags: function , real analysis , Definite integral , Sequences , limits , calculus , integration

[b]a)[/b] Let be a continuous function $ f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R}_{>0} . $ Show that there exists a natural number $ n_0 $ and a sequence of positive real numbers $ \left( x_n \right)_{n>n_0} $ that satisfy the following relation. $$ n\int_0^{x_n} f(t)dt=1,\quad n_0<\forall n\in\mathbb{N} $$ [b]b)[/b] Prove that the sequence $ \left( nx_n \right)_{n> n_0} $ is convergent and find its limit.

1989 Bulgaria National Olympiad, Problem 2

Tags: Sequences , series , floor function , limits , algebra

Prove that the sequence $(a_n)$, where $$a_n=\sum_{k=1}^n\left\{\frac{\left\lfloor2^{k-\frac12}\right\rfloor}2\right\}2^{1-k},$$converges, and determine its limit as $n\to\infty$.

2004 Gheorghe Vranceanu, 1

Tags: Sequences , real analiysis , limits

Let be the sequence $ \left( x_n \right)_{n\ge 1} $ defined as $$ x_n= \frac{4009}{4018020} x_{n-1} -\frac{1}{4018020} x_{n-2} + \left( 1+\frac{1}{n} \right)^n. $$ Prove that $ \left( x_n \right)_{n\ge 1} $ is convergent and determine its limit.

1997 VJIMC, Problem 4-M

Tags: limits , real analysis , number theory , Digits

Find all real numbers $a>0$ for which the series $$\sum_{n=1}^\infty\frac{a^{f(n)}}{n^2}$$is convergent; $f(n)$ denotes the number of $0$'s in the decimal expansion of $f$.

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. $

1986 Traian Lălescu, 1.1

Tags: real analysis , limits

Let $ a $ be a positive real number. Calculate $ \lim_{n\to\infty} \frac{a^n}{(1+a)(1+a^2)\cdots (1+a^n)} . $

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.

2019 Ramnicean Hope, 1

Tags: factorial , trigonometric limit , limits , real analysis

Calculate $ \lim_{n\to\infty }\left(\lim_{x\to 0} \left( -\frac{n}{x}+1+\frac{1}{x}\sum_{r=2}^{n+1}\sqrt[r!]{1+\sin rx}\right)\right) . $ [i]Constantin Rusu[/i]

2003 District Olympiad, 4

Tags: function , limits , Integral , calculus , integration , real analysis

Consider the continuous functions $ f:[0,\infty )\longrightarrow\mathbb{R}, g: [0,1]\longrightarrow\mathbb{R} , $ where $ f $ has a finite limit at $ \infty . $ Show that: $$ \lim_{n \to \infty} \frac{1}{n}\int_0^n f(x) g\left( \frac{x}{n} \right) dx =\int_0^1 g(x)dx\cdot\lim_{x\to\infty} f(x) . $$

2019 Mathematical Talent Reward Programme, MCQ: P 6

Tags: limits

Find the limit $\lim \limits_{n \to \infty} \sin{n!}$ [list=1] [*] 1 [*] 0 [*] $\frac{\pi}{4}$ [*] None of the above [/list]

2014 SEEMOUS, Problem 4

Tags: integration , limits , calculus

a) Prove that $\lim_{n\to\infty}n\int^n_0\frac{\operatorname{arctan}\frac xn}{x(x^2+1)}dx=\frac\pi2$. b) Find the limit $\lim_{n\to\infty}n\left(m\int^n_0\frac{\operatorname{arctan}\frac xn}{x(x^2+1)}dx-\frac\pi2\right)$.

2015 Harvard-MIT Mathematics Tournament, 3

Tags: algebra , limits

Let $p$ be a real number and $c\neq 0$ such that \[c-0.1<x^p\left(\dfrac{1-(1+x)^{10}}{1+(1+x)^{10}}\right)<c+0.1\] for all (positive) real numbers $x$ with $0<x<10^{-100}$. (The exact value $10^{-100}$ is not important. You could replace it with any "sufficiently small number".) Find the ordered pair $(p,c)$.

2020 Jozsef Wildt International Math Competition, W10

Tags: Sequences , limits , real analysis

Let there be $(a_n)_{n\ge1},(b_n)_{n\ge1},a_n,b_n\in\mathbb R^*_+=(0,\infty)$ such that $\lim_{n\to\infty}a_n=a\in\mathbb R^*_+$ and $(b_n)_{n\ge1}$ is a bounded sequence. If $(x_n)_{n\ge1}$, $x_n=\prod_{k=1}^n(ka_h+b_h)$ find: $$\lim_{n\to\infty}\left(\sqrt[n+1]{x_{n+1}}-\sqrt[n]{x_n}\right)$$ [i]Proposed by D.M. Bătinețu-Giurgiu and Daniel Sitaru[/i]

2005 VJIMC, Problem 3

Tags: calculus , integration , limits

Let $f:[0,1]\times[0,1]\to\mathbb R$ be a continuous function. Find the limit $$\lim_{n\to\infty}\left(\frac{(2n+1)!}{(n!)^2}\right)^2\int^1_0\int^1_0(xy(1-x)(1-y))^nf(x,y)\text dx\text dy.$$

Today's calculation of integrals, 877

Tags: calculus , integration , limit , trigonometry , calculus computations , Definite integral , limits

Let $f(x)=\lim_{n\to\infty} \frac{\sin^{n+2}x+\cos^{n+2}x}{\sin^n x+\cos^n x}$ for $0\leq x\leq \frac{\pi}2.$ Evaluate $\int_0^{\frac{\pi}2} f(x)\ dx.$

2005 Grigore Moisil Urziceni, 3

Tags: Sequences , limits , real analysis

Let be a sequence $ \left( a_n \right)_{n\ge 1} $ with $ a_1>0 $ and satisfying the equality $$ a_n=\sqrt{a_{n+1} -\sqrt{a_{n+1} +a_n}} , $$ for all natural numbers $ n. $ [b]a)[/b] Find a recurrence relation between two consecutive elements of $ \left( a_n \right)_{n\ge 1} . $ [b]b)[/b] Prove that $ \lim_{n\to\infty } \frac{\ln\ln a_n}{n} =\ln 2. $

2013 BMT Spring, 3

Tags: limits , real analysis

Evaluate $$\lim_{x\to0}\frac{\sin2x}{e^{3x}-e^{-3x}}$$

2005 Grigore Moisil Urziceni, 2

Tags: limits , real analysis

[b]a)[/b] Prove that $ \lim_{x\to\infty } \sqrt{x}\cdot\sum_{k=1}^{\lfloor \sqrt{x} \rfloor} \frac{1}{k+x}=1. $ [b]b)[/b] Show that $ \lim_{x\to\infty } \left( -\left\lfloor\sqrt{x}\right\rfloor +x\cdot\sum_{k=1}^{\lfloor \sqrt{x} \rfloor} \frac{1}{k+x} \right) =\frac{-1}{2} $ [b]c)[/b] What about $ \lim_{x\to\infty } \left( -\sqrt{x} +x\cdot\sum_{k=1}^{\lfloor \sqrt{x} \rfloor} \frac{1}{k+x} \right) ? $

2007 Nicolae Coculescu, 3

Tags: real analysis , limits , seuqence

Let be the sequence $ \left( a_n \right)_{n\ge 0} $ of positive real numbers defined by $$ a_n=1+\frac{a_{n-1}}{n} ,\quad\forall n\ge 1. $$ Calculate $ \lim_{n\to\infty } a_n ^n . $ [i]Florian Dumitrel[/i]

2015 Miklos Schweitzer, 4

Tags: number theory , number theory proposed , limits , limit , algebra and number theory , prime numbers

Let $a_n$ be a series of positive integers with $a_1=1$ and for any arbitrary prime number $p$, the set $\{a_1,a_2,\cdots,a_p\}$ is a complete remainder system modulo $p$. Prove that $\lim_{n\rightarrow \infty} \cfrac{a_n}{n}=1$.

2011 VJIMC, Problem 2

Tags: limits , Sequences

Let $(a_n)^\infty_{n=1}$ be an unbounded and strictly increasing sequence of positive reals such that the arithmetic mean of any four consecutive terms $a_n,a_{n+1},a_{n+2},a_{n+3}$ belongs to the same sequence. Prove that the sequence $\frac{a_{n+1}}{a_n}$ converges and find all possible values of its limit.

2014 BMT Spring, 5

Tags: limits , real analysis

Determine $$\lim_{x\to\infty}\frac{\sqrt{x+2014}}{\sqrt x+\sqrt{x+2014}}$$

2003 Romania National Olympiad, 3

Tags: function , continuity , limits , real analysis , romania

Let be two functions $ f,g:\mathbb{R}_{\ge 0 }\longrightarrow\mathbb{R} $ having that properties that $ f $ is continuous, $ g $ is nondecreasing and unbounded, and for any sequence of rational numbers $ \left( x_n \right)_{n\ge 1} $ that diverges to $ \infty , $ we have $$ 1=\lim_{n\to\infty } f\left( x_n \right) g\left( x_n \right) . $$ Prove that $1=\lim_{x\to\infty } f\left( x \right) g\left( x \right) . $ [i]Radu Gologan[/i]