Found problems: 913
Evaluate $\int_{0}^{\infty}\frac{\tan^{-1}(\pi x)-\tan^{-1}x}{x}dx$
The domain of the function $f(x) = \text{arcsin}(\log_{m}(nx))$ is a closed interval of length $\frac{1}{2013}$, where $m$ and $n$ are positive integers and $m > 1$. Find the remainder when the smallest possible sum $m+n$ is divided by $1000$.
Let $ f(x) \equal{} \log_{10} (\sin (\pi x)\cdot\sin (2\pi x)\cdot\sin (3\pi x) \cdots \sin (8\pi x))$. The intersection of the domain of $ f(x)$ with the interval $ [0,1]$ is a union of $ n$ disjoint open intervals. What is $ n$?
$ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 18 \qquad \textbf{(D)}\ 22 \qquad \textbf{(E)}\ 36$
Evaluate $ \int_0^1 \frac{(1\minus{}2x)e^{x}\plus{}(1\plus{}2x)e^{\minus{}x}}{(e^x\plus{}e^{\minus{}x})^3}\ dx.$
Determine all triples $(x,y,z)$ of real numbers such that the following system of equations holds true:
\begin{align*}2^{\sqrt[3]{x^2}}\cdot 4^{\sqrt[3]{y^2}}\cdot 16^{\sqrt[3]{z^2}}&=128\\
\left(xy^2+z^4\right)^2&=4+\left(xy^2-z^4\right)^2\mbox{.}\end{align*}
If $a+\log_2 3,a+\log_4 3,a+\log_8 3$ are a geometric series, then the common ratio is________.
For every positive integer $n$, let
\[x_n=\frac{(2n+1)(2n+3)\cdots (4n-1)(4n+1)}{(2n)(2n+2)\cdots (4n-2)(4n)}\]
Prove that $\frac{1}{4n}<x_n-\sqrt{2}<\frac{2}{n}$.
Find the mean value of the function $\ln r$ on the circle $(x - a)^2 + (y-b)^2 = R^2$ (of the function $1/r$ on the sphere).
Let $n$ be a positive integer. Let $\mathcal{F}$ be a family of sets that contains more than half of all subsets of an $n$-element set $X$. Prove that from $\mathcal{F}$ we can select $\lceil \log_2 n \rceil + 1$ sets that form a separating family on $X$, i.e., for any two distinct elements of $X$ there is a selected set containing exactly one of the two elements.
Moderator says: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=41&t=614827&hilit=Schweitzer+2014+separating
Verify that for every $ x > 0$, \[ \frac{\Gamma'(x\plus{}1)}{\Gamma (x\plus{}1)} > \log x.\]
[i]P. Medgyessy[/i]
For every $x \ge -\frac{1}{e}\,$, there is a unique number $W(x) \ge -1$ such that
\[
W(x) e^{W(x)} = x.
\]
The function $W$ is called Lambert's $W$ function. Let $y$ be the unique positive number such that
\[
\frac{y}{\log_{2} y} = - \frac{3}{5} \, .
\]
The value of $y$ is of the form $e^{-W(z \ln 2)}$ for some rational number $z$. What is the value of $z$?
Solve in $ \mathbb{Z} $ the system:
$$ \left\{ \begin{matrix} 2^x+\log_3 x=y^2 \\ 2^y+\log_3 y=x^2 \end{matrix} \right. . $$
Find the volume of the solid defined by the inequality $ x^2 \plus{} y^2 \plus{} \ln (1 \plus{} z^2)\leq \ln 2$.
Note that you may not directively use double integral here for Japanese high school students who don't study it.
Find all continuous functions $ f: \mathbb{R}\to [1,\infty)$ for wich there exists $ a\in\mathbb{R}$ and a positive integer $ k$ such that \[ f(x)f(2x)\cdot...\cdot f(nx)\leq an^k\]
for all real $ x$ and all positive integers $ n$.
[i]author :Radu Gologan[/i]
If $a_{1}$, $a_{2}$, $\ldots$, $a_{n}\geq 0$ are such that \[a_{1}^{2}+\cdots+a_{n}^{2}=1,\]
then find the maximum value of the product $(1-a_{1})\cdots (1-a_{n})$.
Let $n > 3$ be an integer. Let $\Omega$ be the set of all triples of distinct elements of
$\{1, 2, \ldots , n\}$. Let $m$ denote the minimal number of colours which suffice to colour $\Omega$ so that whenever
$1\leq a<b<c<d \leq n$, the triples $\{a,b,c\}$ and $\{b,c,d\}$ have different colours. Prove that $\frac{1}{100}\log\log n \leq m \leq100\log \log n$.
Calculate the following indefinite integrals.
[1] $\int \frac{x}{\sqrt{5-x}}dx$
[2] $\int \frac{\sin x \cos ^2 x}{1+\cos x}dx$
[3] $\int (\sin x+\cos x)^2dx$
[4] $\int \frac{x-\cos ^2 x}{x\cos^ 2 x}dx$
[5]$\int (\sin x+\sin 2x)^2 dx$
If $x > 10$, what is the greatest possible value of the expression
\[
{( \log x )}^{\log \log \log x} - {(\log \log x)}^{\log \log x} ?
\]
All the logarithms are base 10.
Let $ n\geq 2$ be natural number. Answer the following questions.
(1) Evaluate the definite integral $ \int_1^n x\ln x\ dx.$
(2) Prove the following inequality.
$ \frac 12n^2\ln n \minus{} \frac 14(n^2 \minus{} 1) < \sum_{k \equal{} 1}^n k\ln k < \frac 12n^2\ln n \minus{} \frac 14 (n^2 \minus{} 1) \plus{} n\ln n.$
(3) Find $ \lim_{n\to\infty} (1^1\cdot 2^2\cdot 3^3\cdots\cdots n^n)^{\frac {1}{n^2 \ln n}}.$
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 $\lim_{n\to\infty} \frac 1{n}\ln (na_n).$
Evaluate
\[\int_{\frac {\pi}{12}}^{\frac{\pi}{6}} \frac{\sin x-\cos x-x(\sin x+\cos x)+1}{x^2-x(\sin x+\cos x)+\sin x\cos x}\ dx.\]
Let $f:[0,\infty )\rightarrow\mathbb{R}$ be a periodical function, with period $1$, integrable on $[0,1]$. For a strictly increasing and unbounded sequence $(x_n)_{n\ge 0},\, x_0=0,$ with $\lim_{n\rightarrow\infty} (x_{n+1}-x_n)=0$, we denote $r(n)=\max \{ k\mid x_k\le n\}$.
a) Show that:
\[\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=1}^{r(n)}(x_k-x_{k+1})f(x_k)=\int_0^1 f(x)\, dx\]
b) Show that:
\[ \lim_{n\rightarrow\infty} \frac{1}{\ln n}\sum_{k=1}^{r(n)}\frac{f(\ln k)}{k}=\int_0^1f(x)\, dx\]
Let $f_n(x)=\sum_{k=1}^n (-1)^{k+1} \left(\frac{x^{2k-1}}{2k-1}+\frac{x^{2k}}{2k}\right).$
Find $\lim_{n\to\infty} f_n(1).$
Suppose that $a_0=1$ and that $a_{n+1}=a_n+e^{-a_n}$ for $n=0,1,2,\dots.$ Does $a_n-\log n$ have a finite limit as $n\to\infty?$ (Here $\log n=\log_en=\ln n.$)
Let $ n$ be natural number. Find the limit value of ${ \lim_{n\to\infty} \frac{1}{n}(\frac{1}{\sqrt{2}}+\frac{2}{\sqrt{5}}}+\cdots\cdots +\frac{n}{\sqrt{n^2+1}}).$