Found problems: 894
2011 Today's Calculation Of Integral, 694
Prove the following inequality:
\[\int_1^e \frac{(\ln x)^{2009}}{x^2}dx>\frac{1}{2010\cdot 2011\cdot2012}\]
created by kunny
2010 Today's Calculation Of Integral, 607
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.
2014 Harvard-MIT Mathematics Tournament, 17
Let $f:\mathbb{N}\to\mathbb{N}$ be a function satisfying the following conditions:
(a) $f(1)=1$.
(b) $f(a)\leq f(b)$ whenever $a$ and $b$ are positive integers with $a\leq b$.
(c) $f(2a)=f(a)+1$ for all positive integers $a$.
How many possible values can the $2014$-tuple $(f(1),f(2),\ldots,f(2014))$ take?
2011 Today's Calculation Of Integral, 751
Find $\lim_{n\to\infty}\left(\frac{1}{n}\int_0^n (\sin ^ 2 \pi x)\ln (x+n)dx-\frac 12\ln n\right).$
2005 Today's Calculation Of Integral, 82
Let $0<a<b$.Prove the following inequaliy.
\[\frac{1}{b-a}\int_a^b \left(\ln \frac{b}{x}\right)^2 dx<2\]
PEN G Problems, 15
Prove that for any $ p, q\in\mathbb{N}$ with $ q > 1$ the following inequality holds:
\[ \left\vert\pi\minus{}\frac{p}{q}\right\vert\ge q^{\minus{}42}.\]
2007 Today's Calculation Of Integral, 237
Calculate $ \int \frac {dx}{x^{2008}(1 \minus{} x)}$
2011 Today's Calculation Of Integral, 724
Find $\lim_{n\to\infty}\left\{\left(1+n\right)^{\frac{1}{n}}\left(1+\frac{n}{2}\right)^{\frac{2}{n}}\left(1+\frac{n}{3}\right)^{\frac{3}{n}}\cdots\cdots 2\right\}^{\frac{1}{n}}$.
2011 China Second Round Olympiad, 3
Let $a,b$ be positive reals such that $\frac{1}{a}+\frac{1}{b}\leq2\sqrt2$ and $(a-b)^2=4(ab)^3$. Find $\log_a b$.
2023 India IMO Training Camp, 2
For a positive integer $k$, let $s(k)$ denote the sum of the digits of $k$. Show that there are infinitely many natural numbers $n$ such that $s(2^n) > s(2^{n+1})$.
2005 Today's Calculation Of Integral, 16
Calculate the following indefinite integrals.
[1] $\int \sin (\ln x)dx$
[2] $\int \frac{x+\sin ^ 2 x}{x\sin ^ 2 x}dx$
[3] $\int \frac{x^3}{x^2+1}dx$
[4] $\int \frac{x^2}{\sqrt{2x-1}}dx$
[5] $\int \frac{x+\cos 2x +1}{x\cos ^ 2 x}dx$
2012 Putnam, 4
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 $\log n=\log_en=\ln n.$)
2008 ITest, 58
Finished with rereading Isaac Asimov's $\textit{Foundation}$ series, Joshua asks his father, "Do you think somebody will build small devices that run on nuclear energy while I'm alive?"
"Honestly, Josh, I don't know. There are a lot of very different engineering problems involved in designing such devices. But technology moves forward at an amazing pace, so I won't tell you we can't get there in time for you to see it. I $\textit{did}$ go to a graduate school with a lady who now works on $\textit{portable}$ nuclear reactors. They're not small exactly, but they aren't nearly as large as most reactors. That might be the first step toward a nuclear-powered pocket-sized video game.
Hannah adds, "There are already companies designing batteries that are nuclear in the sense that they release energy from uranium hydride through controlled exoenergetic processes. This process is not the same as the nuclear fission going on in today's reactors, but we can certainly call it $\textit{nuclear energy}$."
"Cool!" Joshua's interest is piqued.
Hannah continues, "Suppose that right now in the year $2008$ we can make one of these nuclear batteries in a battery shape that is $2$ meters $\textit{across}$. Let's say you need that size to be reduced to $2$ centimeters $\textit{across}$, in the same proportions, in order to use it to run your little video game machine. If every year we reduce the necessary volume of such a battery by $1/3$, in what year will the batteries first get small enough?"
Joshua asks, "The battery shapes never change? Each year the new batteries are similar in shape - in all dimensions - to the bateries from previous years?"
"That's correct," confirms Joshua's mother. "Also, the base $10$ logarithm of $5$ is about $0.69897$ and the base $10$ logarithm of $3$ is around $0.47712$." This makes Joshua blink. He's not sure he knows how to use logarithms, but he does think he can compute the answer. He correctly notes that after $13$ years, the batteries will already be barely more than a sixth of their original width.
Assuming Hannah's prediction of volume reduction is correct and effects are compounded continuously, compute the first year that the nuclear batteries get small enough for pocket video game machines. Assume also that the year $2008$ is $7/10$ complete.
2002 Baltic Way, 3
Find all sequences $0\le a_0\le a_1\le a_2\le \ldots$ of real numbers such that
\[a_{m^2+n^2}=a_m^2+a_n^2 \]
for all integers $m,n\ge 0$.
2003 China Team Selection Test, 1
Find all functions $f: \mathbb{Z}^+\to \mathbb{R}$, which satisfies $f(n+1)\geq f(n)$ for all $n\geq 1$ and $f(mn)=f(m)f(n)$ for all $(m,n)=1$.
1951 AMC 12/AHSME, 34
The value of $ 10^{\log_{10}7}$ is:
$ \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 10 \qquad\textbf{(D)}\ \log_{10} 7 \qquad\textbf{(E)}\ \log_7 10$
2006 IMC, 3
Compare $\tan(\sin x)$ with $\sin(\tan x)$, for $x\in \left]0,\frac{\pi}{2}\right[$.
2012 Stanford Mathematics Tournament, 10
Let $X_1$, $X_2$, ..., $X_{2012}$ be chosen independently and uniformly at random from the interval $(0,1]$. In other words, for each $X_n$, the probability that it is in the interval $(a,b]$ is $b-a$. Compute the probability that $\lceil\log_2 X_1\rceil+\lceil\log_4 X_2\rceil+\cdots+\lceil\log_{1024} X_{2012}\rceil$ is even. (Note: For any real number $a$, $\lceil a \rceil$ is defined as the smallest integer not less than $a$.)
2007 Putnam, 4
A [i]repunit[/i] is a positive integer whose digits in base $ 10$ are all ones. Find all polynomials $ f$ with real coefficients such that if $ n$ is a repunit, then so is $ f(n).$
2011 Today's Calculation Of Integral, 711
Evaluate $\int_e^{e^2} \frac{4(\ln x)^2+1}{(\ln x)^{\frac 32}}\ dx.$
1950 Miklós Schweitzer, 7
Examine the behavior of the expression
$ \sum_{\nu\equal{}1}^{n\minus{}1}\frac{\log(n\minus{}\nu)}{\nu}\minus{}\log^2 n$
as $ n\rightarrow \infty$
1958 AMC 12/AHSME, 17
If $ x$ is positive and $ \log{x} \ge \log{2} \plus{} \frac{1}{2}\log{x}$, then:
$ \textbf{(A)}\ {x}\text{ has no minimum or maximum value}\qquad \\
\textbf{(B)}\ \text{the maximum value of }{x}\text{ is }{1}\qquad \\
\textbf{(C)}\ \text{the minimum value of }{x}\text{ is }{1}\qquad \\
\textbf{(D)}\ \text{the maximum value of }{x}\text{ is }{4}\qquad \\
\textbf{(E)}\ \text{the minimum value of }{x}\text{ is }{4}$
2005 Iran MO (3rd Round), 3
$f(n)$ is the least number that there exist a $f(n)-$mino that contains every $n-$mino.
Prove that $10000\leq f(1384)\leq960000$.
Find some bound for $f(n)$
2010 Today's Calculation Of Integral, 576
For a function $ f(x)\equal{}(\ln x)^2\plus{}2\ln x$, let $ C$ be the curve $ y\equal{}f(x)$. Denote $ A(a,\ f(a)),\ B(b,\ f(b))\ (a<b)$ the points of tangency of two tangents drawn from the origin $ O$ to $ C$ and the curve $ C$. Answer the following questions.
(1) Examine the increase and decrease, extremal value and inflection point , then draw the approximate garph of the curve $ C$.
(2) Find the values of $ a,\ b$.
(3) Find the volume by a rotation of the figure bounded by the part from the point $ A$ to the point $ B$ and line segments $ OA,\ OB$ around the $ y$-axis.
2010 Today's Calculation Of Integral, 530
Answer the following questions.
(1) By setting $ x\plus{}\sqrt{x^2\minus{}1}\equal{}t$, find the indefinite integral $ \int \sqrt{x^2\minus{}1}\ dx$.
(2) Given two points $ P(p,\ q)\ (p>1,\ q>0)$ and $ A(1,\ 0)$ on the curve $ x^2\minus{}y^2\equal{}1$. Find the area $ S$ of the figure bounded by two lines $ OA,\ OP$ and the curve in terms of $ p$.
(3) Let $ S\equal{}\frac{\theta}{2}$. Express $ p,\ q$ in terms of $ \theta$.