Found problems: 2215
2012 Today's Calculation Of Integral, 839
Evaluate $\int_{\frac 12}^1 \sqrt{1-x^2}\ dx.$
2008 USAPhO, 4
Two beads, each of mass $m$, are free to slide on a rigid, vertical hoop of mass $m_h$. The beads are threaded on the hoop so that they cannot fall off of the hoop. They are released with negligible velocity at the top of the hoop and slide down to the bottom in opposite directions. The hoop remains vertical at all times. What is the maximum value of the ratio $m/m_h$ such that the hoop always remains in contact with the ground? Neglect friction.
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2024 CMIMC Integration Bee, 1
\[\int_1^e \frac{\log(x^{2024})}{x} \mathrm dx\]
[i]Proposed by Connor Gordon[/i]
2007 Today's Calculation Of Integral, 247
Evaluate $ \int_{\frac{\pi}{8}}^{\frac{3}{8}\pi} \frac{11\plus{}4\cos 2x \plus{}\cos 4x}{1\minus{}\cos 4x}\ dx.$
2021 ISI Entrance Examination, 8
A pond has been dug at the Indian Statistical Institute as an inverted truncated pyramid with a square base (see figure below). The depth of the pond is 6m. The square at the bottom has side length 2m and the top square has side length 8m. Water is filled in at a rate of $\tfrac{19}{3}$ cubic meters per hour. At what rate is the water level rising exactly $1$ hour after the water started to fill the pond?
[img]https://cdn.artofproblemsolving.com/attachments/0/9/ff8cac4bb4596ec6c030813da7e827e9a09dfd.png[/img]
2012 Grigore Moisil Intercounty, 2
$ \int_0^{\pi^2/4} \frac{dx}{1+\sin\sqrt x +\cos\sqrt x} $
2013 Online Math Open Problems, 20
A positive integer $n$ is called [i]mythical[/i] if every divisor of $n$ is two less than a prime. Find the unique mythical number with the largest number of divisors.
[i]Proposed by Evan Chen[/i]
2012 Today's Calculation Of Integral, 805
Prove the following inequalities:
(1) For $0\leq x\leq 1$,
\[1-\frac 13x\leq \frac{1}{\sqrt{1+x^2}}\leq 1.\]
(2) $\frac{\pi}{3}-\frac 16\leq \int_0^{\frac{\sqrt{3}}{2}} \frac{1}{\sqrt{1-x^4}}dx\leq \frac{\pi}{3}.$
2011 Today's Calculation Of Integral, 769
In $xyz$ space, find the volume of the solid expressed by $x^2+y^2\leq z\le \sqrt{3}y+1.$
1967 IMO Shortlist, 4
Find values of the parameter $u$ for which the expression
\[y = \frac{ \tan(x-u) + \tan(x) + \tan(x+u)}{ \tan(x-u)\tan(x)\tan(x+u)}\]
does not depend on $x.$
Today's calculation of integrals, 892
Evaluate $\int_0^{\frac{\pi}{2}} \frac{\sin x-\cos x}{1+\cos x}\ dx.$
2004 Bundeswettbewerb Mathematik, 4
A cube is decomposed in a finite number of rectangular parallelepipeds such that the volume of the cube's circum sphere volume equals the sum of the volumes of all parallelepipeds' circum spheres. Prove that all these parallelepipeds are cubes.
2011 Today's Calculation Of Integral, 719
Compute $\int_0^x \sin t\cos t\sin (2\pi\cos t)\ dt$.
2010 Today's Calculation Of Integral, 596
Find the minimum value of $\int_0^{\frac{\pi}{2}} |a\sin 2x-\cos ^ 2 x|dx\ (a>0).$
2009 Shimane University entrance exam/Medicine
2012 Today's Calculation Of Integral, 845
Consider for a real number $t>1$, $I(t)=\int_{-4}^{4t-4} (x-4)\sqrt{x+4}\ dx.$
Find the minimum value of $I(t)\ (t>1).$
2010 Contests, 525
Let $ a,\ b$ be real numbers satisfying $ \int_0^1 (ax\plus{}b)^2dx\equal{}1$.
Determine the values of $ a,\ b$ for which $ \int_0^1 3x(ax\plus{}b)\ dx$ is maximized.
2012 Today's Calculation Of Integral, 785
For a positive real number $x$, find the minimum value of $f(x)=\int_x^{2x} (t\ln t-t)dt.$
2014 IPhOO, 4
A rock is dropped off a cliff of height $ h $ As it falls, a camera takes several photographs, at random intervals. At each picture, I measure the distance the rock has fallen. Let the average (expected value) of all of these distances be $ kh $. If the number of photographs taken is huge, find $ k $. That is: what is the time-average of the distance traveled divided by $ h $, dividing by $h$?
$ \textbf {(A) } \dfrac{1}{4} \qquad \textbf {(B) } \dfrac{1}{3} \qquad \textbf {(C) } \dfrac{1}{\sqrt{2}} \qquad \textbf {(D) } \dfrac{1}{2} \qquad \textbf {(E) } \dfrac{1}{\sqrt{3}} $
[i]Problem proposed by Ahaan Rungta[/i]
2005 Today's Calculation Of Integral, 9
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] $\int \sqrt{1-3x}\ dx$
2012 Today's Calculation Of Integral, 809
For $a>0$, denote by $S(a)$ the area of the part bounded by the parabolas $y=\frac 12x^2-3a$ and $y=-\frac 12x^2+2ax-a^3-a^2$.
Find the maximum area of $S(a)$.
2009 Today's Calculation Of Integral, 456
Find $ \lim_{n\to\infty} \frac{\pi}{n}\left\{\frac{1}{\sin \frac{\pi (n\plus{}1)}{4n}}\plus{}\frac{1}{\sin \frac{\pi (n\plus{}2)}{4n}}\plus{}\cdots \plus{}\frac{1}{\sin \frac{\pi (n\plus{}n)}{4n}}\right\}$
1950 AMC 12/AHSME, 17
The formula which expresses the relationship between $x$ and $y$ as shown in the accompanying table is:
\[ \begin{tabular}[t]{|c|c|c|c|c|c|}\hline
x&0&1&2&3&4\\\hline
y&100&90&70&40&0\\\hline
\end{tabular}\]
$\textbf{(A)}\ y=100-10x \qquad
\textbf{(B)}\ y=100-5x^2 \qquad
\textbf{(C)}\ y=100-5x-5x^2 \qquad\\
\textbf{(D)}\ y=20-x-x^2 \qquad
\textbf{(E)}\ \text{None of these}$
2008 USAMO, 4
Let $ \mathcal{P}$ be a convex polygon with $ n$ sides, $ n\ge3$. Any set of $ n \minus{} 3$ diagonals of $ \mathcal{P}$ that do not intersect in the interior of the polygon determine a [i]triangulation[/i] of $ \mathcal{P}$ into $ n \minus{} 2$ triangles. If $ \mathcal{P}$ is regular and there is a triangulation of $ \mathcal{P}$ consisting of only isosceles triangles, find all the possible values of $ n$.
2019 Romania National Olympiad, 3
Let $f:[0, \infty) \to (0, \infty)$ be an increasing function and $g:[0, \infty) \to \mathbb{R}$ be a two times differentiable function such that $g''$ is continuous and $g''(x)+f(x)g(x) = 0, \: \forall x \geq 0.$
$\textbf{a)}$ Provide an example of such functions, with $g \neq 0.$
$\textbf{b)}$ Prove that $g$ is bounded.
2011 District Olympiad, 1
Prove the rationality of the number $ \frac{1}{\pi }\int_{\sin\frac{\pi }{13}}^{\cos\frac{\pi }{13}} \sqrt{1-x^2} dx. $