Found problems: 1687
Today's calculation of integrals, 851
Let $T$ be a period of a function $f(x)=|\cos x|\sin x\ (-\infty,\ \infty).$
Find $\lim_{n\to\infty} \int_0^{nT} e^{-x}f(x)\ dx.$
2013 Romanian Masters In Mathematics, 1
Suppose two convex quadrangles in the plane $P$ and $P'$, share a point $O$ such that, for every line $l$ trough $O$, the segment along which $l$ and $P$ meet is longer then the segment along which $l$ and $P'$ meet. Is it possible that the ratio of the area of $P'$ to the area of $P$ is greater then $1.9$?
2018 Korea USCM, 3
$\Phi$ is a function defined on collection of bounded measurable subsets of $\mathbb{R}$ defined as
$$\Phi(S) = \iint_S (1-5x^2 + 4xy-5y^2 ) dx dy$$
Find the maximum value of $\Phi$.
2010 Today's Calculation Of Integral, 562
(1) Show the following inequality for every natural number $ k$.
\[ \frac {1}{2(k \plus{} 1)} < \int_0^1 \frac {1 \minus{} x}{k \plus{} x}dx < \frac {1}{2k}\]
(2) Show the following inequality for every natural number $ m,\ n$ such that $ m > n$.
\[ \frac {m \minus{} n}{2(m \plus{} 1)(n \plus{} 1)} < \log \frac {m}{n} \minus{} \sum_{k \equal{} n \plus{} 1}^{m} \frac {1}{k} < \frac {m \minus{} n}{2mn}\]
2007 Princeton University Math Competition, 6
Take the square with vertices $(0,0)$, $(1,0)$, $(0,1)$, and $(1,1)$. Choose a random point in this square and draw the line segment from it to $(0,0)$. Choose a second random point in this square and draw the line segment from it to $(1,0)$. What is the probability that the two line segments intersect?
Today's calculation of integrals, 866
Given a solid $R$ contained in a semi cylinder with the hight $1$ which has a semicircle with radius $1$ as the base. The cross section at the hight $x\ (0\leq x\leq 1)$ is the form combined with two right-angled triangles as attached figure as below. Answer the following questions.
(1) Find the cross-sectional area $S(x)$ at the hight $x$.
(2) Find the volume of $R$. If necessary, when you integrate, set $x=\sin t.$
2006 Putnam, B6
Let $k$ be an integer greater than $1.$ Suppose $a_{0}>0$ and define
\[a_{n+1}=a_{n}+\frac1{\sqrt[k]{a_{n}}}\]
for $n\ge 0.$ Evaluate
\[\lim_{n\to\infty}\frac{a_{n}^{k+1}}{n^{k}}.\]
1975 IMO Shortlist, 3
Find the integer represented by $\left[ \sum_{n=1}^{10^9} n^{-2/3} \right] $. Here $[x]$ denotes the greatest integer less than or equal to $x.$
2014 Dutch IMO TST, 5
Let $P(x)$ be a polynomial of degree $n \le 10$ with integral coefficients such that for every $k \in \{1, 2, \dots, 10\}$ there is an integer $m$ with $P(m) = k$. Furthermore, it is given that $|P(10) - P(0)| < 1000$. Prove that for every integer $k$ there is an integer $m$ such that $P(m) = k.$
2011 Romania National Olympiad, 2
Let be a continuous function $ f:[0,1]\longrightarrow\left( 0,\infty \right) $ having the property that, for any natural number $ n\ge 2, $ there exist $ n-1 $ real numbers $ 0<t_1<t_2<\cdots <t_{n-1}<1, $ such that
$$ \int_0^{t_1} f(t)dt=\int_{t_1}^{t_2} f(t)dt=\int_{t_2}^{t_3} f(t)dt=\cdots =\int_{t_{n-2}}^{t_{n-1}} f(t)dt=\int_{t_{n-1}}^{1} f(t)dt. $$
Calculate $ \lim_{n\to\infty } \frac{n}{\frac{1}{f(0)} +\sum_{i=1}^{n-1} \frac{1}{f\left( t_i \right)} +\frac{1}{f(1)}} . $
2006 Turkey Team Selection Test, 2
How many ways are there to divide a $2\times n$ rectangle into rectangles having integral sides, where $n$ is a positive integer?
2008 AIME Problems, 3
Ed and Sue bike at equal and constant rates. Similarly, they jog at equal and constant rates, and they swim at equal and constant rates. Ed covers $ 74$ kilometers after biking for $ 2$ hours, jogging for $ 3$ hours, and swimming for $ 4$ hours, while Sue covers $ 91$ kilometers after jogging for $ 2$ hours, swimming for $ 3$ hours, and biking for $ 4$ hours. Their biking, jogging, and swimming rates are all whole numbers of kilometers per hour. Find the sum of the squares of Ed's biking, jogging, and swimming rates.
2023 Bangladesh Mathematical Olympiad, P5
Consider an integrable function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $af(a)+bf(b)=0$ when $ab=1$. Find the value of the following integration:
$$ \int_{0}^{\infty} f(x) \,dx $$
2011 Today's Calculation Of Integral, 692
Evaluate $\int_0^{\frac{\pi}{12}} \frac{\tan ^ 2 x-3}{3\tan ^ 2 x-1}dx$.
created by kunny
1993 AMC 12/AHSME, 30
Given $0 \le x_0 <1$, let
\[ x_n=
\begin{cases}
2x_{n-1} & \text{if}\ 2x_{n-1} <1 \\
2x_{n-1}-1 & \text{if}\ 2x_{n-1} \ge 1
\end{cases} \] for all integers $n>0$. For how many $x_0$ is it true that $x_0=x_5$?
$ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 31 \qquad\textbf{(E)}\ \text{infinitely many} $
1996 South africa National Olympiad, 3
The sides of triangle $ABC$ has integer lengths. Given that $AC=6$ and $\angle BAC=120^\circ$, determine the lengths of the other two sides.
2005 Brazil Undergrad MO, 5
Prove that
\[ \sum_{n=1}^\infty {1\over n^n} = \int_0^1 x^{-x}\,dx. \]
Today's calculation of integrals, 868
In the coordinate space, define a square $S$, defined by the inequality $|x|\leq 1,\ |y|\leq 1$ on the $xy$-plane, with four vertices $A(-1,\ 1,\ 0),\ B(1,\ 1,\ 0),\ C(1,-1,\ 0), D(-1,-1,\ 0)$. Let $V_1$ be the solid by a rotation of the square $S$ about the line $BD$ as the axis of rotation, and let $V_2$ be the solid by a rotation of the square $S$ about the line $AC$ as the axis of rotation.
(1) For a real number $t$ such that $0\leq t<1$, find the area of cross section of $V_1$ cut by the plane $x=t$.
(2) Find the volume of the common part of $V_1$ and $V_2$.
1958 February Putnam, B6
A projectile moves in a resisting medium. The resisting force is a function of the velocity and is directed along the velocity vector. The equation $x=f(t)$ (where $f(t)$ is not constant) gives the horizontal distance in terms of the time $t$. Show that the vertical distance $y$ is given by
$$y=-gf(t) \int \frac{dt}{f'(t)} + g \int \frac{f(t)}{f'(t)} \, dt +Af(t)+B$$
where $A$ and $B$ are constants and $g$ is the acceleration due to gravity.
2011 Today's Calculation Of Integral, 753
Find $\lim_{n\to\infty} \sum_{k=1}^{2n} \frac{n}{2n^2+3nk+k^2}.$
2010 Today's Calculation Of Integral, 579
Let $ a$ be a positive real number. Find $ \lim_{n\to\infty} \frac{(n\plus{}1)^a\plus{}(n\plus{}2)^a\plus{}\cdots \plus{}(n\plus{}n)^a}{1^{a}\plus{}2^{a}\plus{}\cdots \plus{}n^{a}}$
2009 Today's Calculation Of Integral, 411
Find the area bounded by $ y\equal{}x^2\minus{}|x^2\minus{}1|\plus{}|2|x|\minus{}2|\plus{}2|x|\minus{}7$ and the $ x$ axis.
2014 Indonesia MO, 2
For some positive integers $m,n$, the system $x+y^2 = m$ and $x^2+y = n$ has exactly one integral solution $(x,y)$. Determine all possible values of $m-n$.
PEN A Problems, 96
Find all positive integers $n$ that have exactly $16$ positive integral divisors $d_{1},d_{2} \cdots, d_{16}$ such that $1=d_{1}<d_{2}<\cdots<d_{16}=n$, $d_6=18$, and $d_{9}-d_{8}=17$.
2005 Today's Calculation Of Integral, 2
Calculate the following indefinite integrals.
[1] $\int \cos \left(2x-\frac{\pi}{3}\right)dx$
[2]$\int \frac{dx}{\cos ^2 (3x+4)}$
[3]$\int (x-1)\sqrt[3]{x-2}dx$
[4]$\int x\cdot 3^{x^2+1}dx$
[5]$\int \frac{dx}{\sqrt{1-x}}dx$