Found problems: 913
Let $a,\ b$ be positive constants.
Evaluate \[\int_0^1 \frac{\ln \frac{(x+a)^{x+a}}{(x+b)^{x+b}}}{(x+a)(x+b)\ln (x+a)\ln (x+b)}\ dx.\]
Evaluate $ \int_{2}^{6}\ln\frac{\minus{}1\plus{}\sqrt{1\plus{}4x}}{2}\ dx$.
$ \lim_{n\to\infty }\frac{1}{\sqrt[n]{n!}}\left\lfloor \log_5 \sum_{k=2}^{1+5^n} \sqrt[5^n]{k} \right\rfloor $
[i]Taclit Daniela Nadia[/i]
Let $ \alpha,\beta$ be real numbers satisfying $ 1 < \alpha < \beta.$ Find the greatest positive integer $ r$ having the following property: each of positive integers is colored by one of $ r$ colors arbitrarily, there always exist two integers $ x,y$ having the same color such that $ \alpha\le \frac {x}{y}\le\beta.$
Determine graphically the number of roots of the equation $\sin x = \lg x$.
[b](a) [/b] Find the minimal distance between the points of the graph of the function $y=\ln x$ from the line $y=x$.
[b](b)[/b] Find the minimal distance between two points, one of the point is in the graph of the function $y=e^x$ and the other point in the graph of the function $y=ln x$.
Evaluate
\[\int_{e^e}^{e^{e^{e}}}\left\{\ln (\ln (\ln x))+\frac{1}{(\ln x)\ln (\ln x)}\right\}dx.\]
Own
Consider an orange and black coloring of a $20 \times 14$ square grid. Let $n$ be the number of colorings such that every row and column has an even number of orange squares. Evaluate $\log_2 n$.
Let $a,\ b$ be given positive constants.
Evaluate
\[\int_0^1 \frac{\ln\ (x+a)^{x+a}(x+b)^{x+b}}{(x+a)(x+b)}dx.\]
Own
If $x$ is a real number, let $\lfloor x \rfloor$ be the greatest integer that is less than or equal to $x$. If $n$ is a positive integer, let $S(n)$ be defined by
\[
S(n)
= \left\lfloor \frac{n}{10^{\lfloor \log n \rfloor}} \right\rfloor
+ 10 \left( n - 10^{\lfloor \log n \rfloor}
\cdot \left\lfloor \frac{n}{10^{\lfloor \log n \rfloor}} \right\rfloor
\right) \, .
\]
(All the logarithms are base 10.) How many integers $n$ from 1 to 2011 (inclusive) satisfy $S(S(n)) = n$?
On the coordinate plane, Let $C$ be the graph of $y=(\ln x)^2\ (x>0)$ and for $\alpha >0$, denote $L(\alpha)$ be the tangent line of $C$ at the point $(\alpha ,\ (\ln \alpha)^2).$
(1) Draw the graph.
(2) Let $n(\alpha)$ be the number of the intersection points of $C$ and $L(\alpha)$. Find $n(\alpha)$.
(3) For $0<\alpha <1$, let $S(\alpha)$ be the area of the region bounded by $C,\ L(\alpha)$ and the $x$-axis. Find $S(\alpha)$.
2010 Tokyo Institute of Technology entrance exam, Second Exam.
Let $x$, $y$, and $z$ all exceed 1 and let $w$ be a positive number such that \[\log_x w = 24,\quad \log_y w = 40 \quad\text{and}\quad \log_{xyz} w = 12.\] Find $\log_z w$.
Prove that :
\[\int_1^{\sqrt{e}} (\ln x)^n\ dx=(-1)^{n-1}n!+\sqrt{e}\sum_{m=0}^{n} (-1)^{n-m}\frac{n!}{m!}\left(\frac 12\right)^{m}\]
If $x$ is real and positive and grows beyond all bounds, then $\log_3{(6x-5)}-\log_3{(2x+1)}$ approaches:
$\textbf{(A)}\ 0\qquad
\textbf{(B)}\ 1\qquad
\textbf{(C)}\ 3\qquad
\textbf{(D)}\ 4\qquad
\textbf{(E)}\ \text{no finite number}$
Let $\mathcal{F}$ be the set of functions $f: [0,1]\to\mathbb{R}$ that are differentiable, with continuous derivative, and $f(0)=0$, $f(1)=1$. Find the minimum of $\int_{0}^{1}\sqrt{1+x^{2}}\cdot \big(f'(x)\big)^{2}\ dx$ (where $f\in\mathcal{F}$) and find all functions $f\in\mathcal{F}$ for which this minimum is attained.
[hide="Comment"]
In the contest, this was the b) point of the problem. The a) point was simply ``Prove the Cauchy inequality in integral form''.
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Determine all pairs $(a, b)$ of positive real numbers with $a \neq 1$ such that
\[\log_a b < \log_{a+1} (b + 1).\]
Powers of $2$ in base $10$ start with $3$ or $4$ more frequently? What is their state in base $3$? First write down an exact form of the question.
Let $x = \left( 1 + \frac{1}{n}\right)^n$ and $y = \left( 1 + \frac{1}{n}\right)^{n+1}$ where $n \in \mathbb{N}$. Which one of the numbers $x^y$, $y^x$ is bigger ?
Let $n$ be a positive integer. Let $P_n=\{2^n,2^{n-1}\cdot 3, 2^{n-2}\cdot 3^2, \dots, 3^n \}.$ For each subset $X$ of $P_n$, we write $S_X$ for the sum of all elements of $X$, with the convention that $S_{\emptyset}=0$ where $\emptyset$ is the empty set. Suppose that $y$ is a real number with $0 \leq y \leq 3^{n+1}-2^{n+1}.$
Prove that there is a subset $Y$ of $P_n$ such that $0 \leq y-S_Y < 2^n$
A circle has a radius of $ \log_{10}(a^2)$ and a circumference of $ \log_{10}(b^4)$. What is $ \log_ab$?
$ \textbf{(A)}\ \frac {1}{4\pi} \qquad \textbf{(B)}\ \frac {1}{\pi} \qquad \textbf{(C)}\ \pi \qquad \textbf{(D)}\ 2\pi \qquad \textbf{(E)}\ 10^{2\pi}$
Let $d$ be a positive integer and let $\varepsilon$ be any positive real. Prove that for all sufficiently large primes $p$ with $\gcd(p-1,d) \neq 1$, there exists an positive integer less than $p^r$ which is not a $d$th power modulo $p$, where $r$ is defined by \[ \log r = \varepsilon - \frac{1}{\gcd(d,p-1)}. \][i]Proposed by Shashwat Kishore[/i]
Calculate the following indefinite integrals.
[1] $\int x(x^2+3)^2 dx$
[2] $\int \ln (x+2) dx$
[3] $\int x\cos x dx$
[4] $\int \frac{dx}{(x+2)^2}dx$
[5] $\int \frac{x-1}{x^2-2x+3}dx$
For any positive integer $ n$, let \[f(n) \equal{} \begin{cases} \log_8{n}, & \text{if }\log_8{n}\text{ is rational,} \\
0, & \text{otherwise.} \end{cases}\] What is $ \sum_{n \equal{} 1}^{1997}{f(n)}$?
$ \textbf{(A)}\ \log_8{2047}\qquad \textbf{(B)}\ 6\qquad \textbf{(C)}\ \frac {55}{3}\qquad \textbf{(D)}\ \frac {58}{3}\qquad \textbf{(E)}\ 585$
If $3 = k \cdot 2^r$ and $15 = k \cdot 4^r$, then $r =$
$\text{(A)}\ - \log_2 5 \qquad \text{(B)}\ \log_5 2 \qquad \text{(C)}\ \log_{10} 5 \qquad \text{(D)}\ \log_2 5 \qquad \text{(E)}\ \displaystyle \frac{5}{2}$
Calculate the following indefinite integrals.
[1] $\int (x^2+4x-3)^2(x+2)dx$
[2] $\int \frac{\ln x}{x(\ln x+1)}dx$
[3] $\int \frac{\sin \ (\pi \log _2 x)}{x}dx$
[4] $\int \frac{dx}{\sin x\cos ^ 2 x}$
[5] $\int \sqrt{1-3x}\ dx$