Olympiad problems

2010 Today's Calculation Of Integral, 656

Tags: calculus , integration , limit , trigonometry , function , calculus computations

Find $\lim_{n\to\infty} n\int_0^{\frac{\pi}{2}} \frac{1}{(1+\cos x)^n}dx\ (n=1,\ 2,\ \cdots).$

2012 Today's Calculation Of Integral, 827

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

Find $\lim_{n\to\infty}\sum_{k=0}^{\infty} \int_{2k\pi}^{(2k+1)\pi} xe^{-x}\sin x\ dx.$

1973 Miklós Schweitzer, 4

Tags: limit , number theory , logarithm

Let $ f(n)$ be that largest integer $ k$ such that $ n^k$ divides $ n!$, and let $ F(n)\equal{} \max_{2 \leq m \leq n} f(m)$. Show that \[ \lim_{n\rightarrow \infty} \frac{F(n) \log n}{n \log \log n}\equal{}1.\] [i]P. Erdos[/i]

2007 Today's Calculation Of Integral, 178

Tags: calculus , integration , function , trigonometry , limit , calculus computations

Let $f(x)$ be a differentiable function such that $f'(x)+f(x)=4xe^{-x}\sin 2x,\ \ f(0)=0.$ Find $\lim_{n\to\infty}\sum_{k=1}^{n}f(k\pi).$

1982 Putnam, A6

Tags: limit , real analysis

Let $\sigma$ be a bijection on the positive integers. Let $x_1,x_2,x_3,\ldots$ be a sequence of real numbers with the following three properties: $(\text i)$ $|x_n|$ is a strictly decreasing function of $n$; $(\text{ii})$ $|\sigma(n)-n|\cdot|x_n|\to0$ as $n\to\infty$; $(\text{iii})$ $\lim_{n\to\infty}\sum_{k=1}^nx_k=1$. Prove or disprove that these conditions imply that $$\lim_{n\to\infty}\sum_{k=1}^nx_{\sigma(k)}=1.$$

2000 District Olympiad (Hunedoara), 3

Tags: limit , calculus , real analysis , sequence

Let be two distinct natural numbers $ k_1 $ and $ k_2 $ and a sequence $ \left( x_n \right)_{n\ge 0} $ which satisfies $$ x_nx_m +k_1k_2\le k_1x_n +k_2x_m,\quad\forall m,n\in\{ 0\}\cup\mathbb{N}. $$ Calculate $ \lim_{n\to\infty}\frac{n!\cdot (-1)^{1+n}\cdot x_n^2}{n^n} . $

2020 LIMIT Category 1, 4

Tags: counting , limit , combinatorics

The total number of solutions of $xyz=2520$ (A)$2520$ (B)$2160$ (C)$540$ (D)None of these

2011 Today's Calculation Of Integral, 751

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

Find $\lim_{n\to\infty}\left(\frac{1}{n}\int_0^n (\sin ^ 2 \pi x)\ln (x+n)dx-\frac 12\ln n\right).$

2013 BMT Spring, 3

Tags: real analysis , limit

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

Today's calculation of integrals, 765

Tags: calculus , integration , function , limit , calculus computations

Define two functions $g(x),\ f(x)\ (x\geq 0)$ by $g(x)=\int_0^x e^{-t^2}dt,\ f(x)=\int_0^1 \frac{e^{-(1+s^2)x}}{1+s^2}ds.$ Now we know that $f'(x)=-\int_0^1 e^{-(1+s^2)x}ds.$ (1) Find $f(0).$ (2) Show that $f(x)\leq \frac{\pi}{4}e^{-x}\ (x\geq 0).$ (3) Let $h(x)=\{g(\sqrt{x})\}^2$. Show that $f'(x)=-h'(x).$ (4) Find $\lim_{x\rightarrow +\infty} g(x)$ Please solve the problem without using Double Integral or Jacobian for those Japanese High School Students who don't study them.

2011 Putnam, B3

Tags: function , limit

Let $f$ and $g$ be (real-valued) functions defined on an open interval containing $0,$ with $g$ nonzero and continuous at $0.$ If $fg$ and $f/g$ are differentiable at $0,$ must $f$ be differentiable at $0?$

1988 IMO Longlists, 9

Tags: limit , quadratic , algebra

If $a_0$ is a positive real number, consider the sequence $\{a_n\}$ defined by: \[ a_{n+1} = \frac{a^2_n - 1}{n+1}, n \geq 0. \] Show that there exist a real number $a > 0$ such that: [b]i.)[/b] for all $a_0 \geq a,$ the sequence $\{a_n\} \rightarrow \infty,$ [b]ii.)[/b] for all $a_0 < a,$ the sequence $\{a_n\} \rightarrow 0.$