Olympiad problems

2017 Korea USCM, 3

Sequence $\{a_n\}$ defined by recurrence relation $a_{n+1} = 1+\frac{n^2}{a_n}$. Given $a_1>1$, find the value of $\lim\limits_{n\to\infty} \frac{a_n}{n}$ with proof.

2001 VJIMC, Problem 3

Tags: limit , calculus , integration , function

Let $f:(0,+\infty)\to(0,+\infty)$ be a decreasing function which satisfies $\int^\infty_0f(x)\text dx<+\infty$. Prove that $\lim_{x\to+\infty}xf(x)=0$.

2010 Romania National Olympiad, 4

Tags: function , calculus , derivative , integration , limit , real analysis

Let $f:[-1,1]\to\mathbb{R}$ be a continuous function having finite derivative at $0$, and \[I(h)=\int^h_{-h}f(x)\text{ d}x,\ h\in [0,1].\] Prove that a) there exists $M>0$ such that $|I(h)-2f(0)h|\le Mh^2$, for any $h\in [0,1]$. b) the sequence $(a_n)_{n\ge 1}$, defined by $a_n=\sum_{k=1}^n\sqrt{k}|I(1/k)|$, is convergent if and only if $f(0)=0$. [i]Calin Popescu[/i]

1977 Miklós Schweitzer, 10

Tags: limit , probability and stats

Let the sequence of random variables $ \{ X_m, \; m \geq 0\ \}, \; X_0=0$, be an infinite random walk on the set of nonnegative integers with transition probabilities \[ p_i=P(X_{m+1}=i+1 \mid X_m=i) >0, \; i \geq 0 \,\] \[ q_i=P(X_{m+1}=i-1 \mid X_m=i ) >0, \; i>0.\] Prove that for arbitrary $ k >0$ there is an $ \alpha_k > 1$ such that \[ P_n(k)=P \left ( \max_{0 \leq j \leq n} X_j =k \right)\] satisfies the limit relation \[ \lim_{L \rightarrow \infty} \frac 1L \sum_{n=1}^L P_n(k) \alpha_k ^n < \infty.\] [i]J. Tomko[/i]

2012 Today's Calculation Of Integral, 837

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

Let $f_n(x)=\sum_{k=1}^n (-1)^{k+1} \left(\frac{x^{2k-1}}{2k-1}+\frac{x^{2k}}{2k}\right).$ Find $\lim_{n\to\infty} f_n(1).$

1958 November Putnam, B1

Tags: binomial coefficient , limit

Given $$b_n = \sum_{k=0}^{n} \binom{n}{k}^{-1}, \;\; n\geq 1,$$ prove that $$b_n = \frac{n+1}{2n} b_{n-1} +1, \;\; n \geq 2.$$ Hence, as a corollary, show $$ \lim_{n \to \infty} b_n =2.$$

1963 Putnam, B5

Tags: inequalities , limit , series

Let $(a_n )$ be a sequence of real numbers satisfying the inequalities $$ 0 \leq a_k \leq 100a_n \;\; \text{for} \;\, n \leq k \leq 2n \;\; \text{and} \;\; n=1,2,\ldots,$$ and such that the series $$\sum_{n=0}^{\infty} a_n $$ converges. Prove that $$\lim_{n\to \infty} n a_n = 0.$$

Today's calculation of integrals, 871

Tags: calculus , integration , trigonometry , limit , calculus computations , definite integral , logarithm

Define sequences $\{a_n\},\ \{b_n\}$ by \[a_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}d\theta,\ b_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}\cos \theta d\theta\ (n=1,\ 2,\ 3,\ \cdots).\] (1) Find $b_n$. (2) Prove that for each $n$, $b_n\leq a_n\leq \frac 2{\sqrt{3}}b_n.$ (3) Find $\lim_{n\to\infty} \frac 1{n}\ln (na_n).$

2006 Pre-Preparation Course Examination, 7

Tags: function , limit , combinatorics , logarithm

Suppose that for every $n$ the number $m(n)$ is chosen such that $m(n)\ln(m(n))=n-\frac 12$. Show that $b_n$ is asymptotic to the following expression where $b_n$ is the $n-$th Bell number, that is the number of ways to partition $\{1,2,\ldots,n\}$: \[ \frac{m(n)^ne^{m(n)-n-\frac 12}}{\sqrt{\ln n}}. \] Two functions $f(n)$ and $g(n)$ are asymptotic to each other if $\lim_{n\rightarrow \infty}\frac{f(n)}{g(n)}=1$.

2008 Romania National Olympiad, 1

Tags: function , integration , calculus , derivative , limit , real analysis

Let $ a>0$ and $ f: [0,\infty) \to [0,a]$ be a continuous function on $ (0,\infty)$ and having Darboux property on $ [0,\infty)$. Prove that if $ f(0)\equal{}0$ and for all nonnegative $ x$ we have \[ xf(x) \geq \int^x_0 f(t) dt ,\] then $ f$ admits primitives on $ [0,\infty)$.

2013 USAMTS Problems, 3

Tags: geometry , induction , ratio , inequalities , limit , arithmetic sequence

An infinite sequence of positive real numbers $a_1,a_2,a_3,\dots$ is called [i]territorial[/i] if for all positive integers $i,j$ with $i<j$, we have $|a_i-a_j|\ge\tfrac1j$. Can we find a territorial sequence $a_1,a_2,a_3,\dots$ for which there exists a real number $c$ with $a_i<c$ for all $i$?

1986 Iran MO (2nd round), 1

Tags: function , trigonometry , limit , algebra

Let $f$ be a function such that \[f(x)=\frac{(x^2-2x+1) \sin \frac{1}{x-1}}{\sin \pi x}.\] Find the limit of $f$ in the point $x_0=1.$

2009 IMS, 4

Tags: ratio , vector , algebra , polynomial , limit , induction , absolute value

In this infinite tree, degree of each vertex is equal to 3. A real number $ \lambda$ is given. We want to assign a real number to each node in such a way that for each node sum of numbers assigned to its neighbors is equal to $ \lambda$ times of the number assigned to this node. Find all $ \lambda$ for which this is possible.

2004 Unirea, 3

Tags: limit , monotone , inequalities , real analysis , sequence

[b]a)[/b] Prove that for any natural numbers $ n, $ the inequality $$ e^{2-1/n} >\prod_{k=1}^n (1+1/k^2) $$ holds. [b]b)[/b] Prove that the sequence $ \left( a_n \right)_{n\ge 1} $ with $ a_1=1 $ and defined by the recursive relation $ a_{n+1}=\frac{2}{n^2}\sum_{k=1}^n ka_k $ is nondecreasing. Is it convergent?

2010 Paenza, 3

Tags: limit , trigonometry , real analysis

Let $(x_n)_{n \in \mathbb{N}}$ be the sequence defined as $x_n = \sin(2 \pi n! e)$ for all $n \in \mathbb{N}$. Compute $\lim_{n \to \infty} x_n$.

2007 Today's Calculation Of Integral, 193

Tags: calculus , integration , parabola , geometry , limit , analytic geometry , conic

For $a>0$, let $l$ be the line created by rotating the tangent line to parabola $y=x^{2}$, which is tangent at point $A(a,a^{2})$, around $A$ by $-\frac{\pi}{6}$. Let $B$ be the other intersection of $l$ and $y=x^{2}$. Also, let $C$ be $(a,0)$ and let $O$ be the origin. (1) Find the equation of $l$. (2) Let $S(a)$ be the area of the region bounded by $OC$, $CA$ and $y=x^{2}$. Let $T(a)$ be the area of the region bounded by $AB$ and $y=x^{2}$. Find $\lim_{a \to \infty}\frac{T(a)}{S(a)}$.

2020 LIMIT Category 2, 4

Tags: limit , sequence , number theory

Define the sequence $\{a_n\}_{n\geq 1}$ as $a_n=n-1$, $n\leq 2$ and $a_n=$ remainder left by $a_{n-1}+a_{n-2}$ when divided by $3$ $\forall n\geq 2$. Then $\sum_{i=2018}^{2025}a_i=$? (A)$6$ (B)$7$ (C)$8$ (D)$9$

2010 Contests, 2

Tags: trigonometry , limit , real analysis

Calculate the sum of the series $\sum_{-\infty}^{\infty}\frac{\sin^33^k}{3^k}$.

1999 VJIMC, Problem 2

Tags: limit , real analysis , function

Let $a,b\in\mathbb R$, $a\le b$. Assume that $f:[a,b]\to[a,b]$ satisfies $f(x)-f(y)\le|x-y|$ for every $x,y\in[a,b]$. Choose an $x_1\in[a,b]$ and define $$x_{n+1}=\frac{x_n+f(x_n)}2,\qquad n=1,2,3,\ldots.$$Show that $\{x_n\}^\infty_{n=1}$ converges to some fixed point of $f$.

2020 LIMIT Category 2, 16

Tags: limit , derivative , calculus

The $n^{th}$ derivative of a function $f(x)$ (if it exists) is denoted by $f^{(n)}(x) $. Let $f(x)=\frac{e^x}{x}$. Suppose $f$ is differentiable infinitely many times in $(0,\infty) $. Then find $\lim_{n \to \infty}\frac{f^{(2n)}1}{(2n)!}$

2007 China Team Selection Test, 3

Tags: algebra , polynomial , function , induction , quadratic , limit , integration

Prove that for any positive integer $ n$, there exists only $ n$ degree polynomial $ f(x),$ satisfying $ f(0) \equal{} 1$ and $ (x \plus{} 1)[f(x)]^2 \minus{} 1$ is an odd function.

2013 District Olympiad, 1

Tags: limit , integration , calculus , calculus computations

Calculate: $\underset{n\to \infty }{\mathop{\lim }}\,\int_{0}^{1}{{{e}^{{{x}^{n}}}}dx}$

2011 Today's Calculation Of Integral, 730

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

Let $a_n$ be the local maximum of $f_n(x)=\frac{x^ne^{-x+n\pi}}{n!}\ (n=1,\ 2,\ \cdots)$ for $x>0$. Find $\lim_{n\to\infty} \ln \left(\frac{a_{2n}}{a_n}\right)^{\frac{1}{n}}$.

2008 Harvard-MIT Mathematics Tournament, 9

Tags: limit , integration , hmmt , calculus , real analysis , calculus computations , logarithm

([b]7[/b]) Evaluate the limit $ \lim_{n\rightarrow\infty} n^{\minus{}\frac{1}{2}\left(1\plus{}\frac{1}{n}\right)} \left(1^1\cdot2^2\cdot\cdots\cdot n^n\right)^{\frac{1}{n^2}}$.

2006 VTRMC, Problem 5

Tags: limit , convergence , summation , sequence

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.