Found problems: 1687
2007 Today's Calculation Of Integral, 182
Find the area of the domain of the system of inequality
\[y(y-|x^{2}-5|+4)\leq 0,\ \ y+x^{2}-2x-3\leq 0. \]
1956 AMC 12/AHSME, 23
About the equation $ ax^2 \minus{} 2x\sqrt {2} \plus{} c \equal{} 0$, with $ a$ and $ c$ real constants, we are told that the discriminant is zero. The roots are necessarily:
$ \textbf{(A)}\ \text{equal and integral} \qquad\textbf{(B)}\ \text{equal and rational} \qquad\textbf{(C)}\ \text{equal and real}$
$ \textbf{(D)}\ \text{equal and irrational} \qquad\textbf{(E)}\ \text{equal and imaginary}$
2009 IMS, 5
Suppose that $ f: \mathbb R^2\rightarrow \mathbb R$ is a non-negative and continuous function that $ \iint_{\mathbb R^2}f(x,y)dxdy\equal{}1$. Prove that there is a closed disc $ D$ with the least radius possible such that $ \iint_D f(x,y)dxdy\equal{}\frac12$.
2011 Putnam, A3
Find a real number $c$ and a positive number $L$ for which
\[\lim_{r\to\infty}\frac{r^c\int_0^{\pi/2}x^r\sin x\,dx}{\int_0^{\pi/2}x^r\cos x\,dx}=L.\]
2012 Today's Calculation Of Integral, 846
For $a>0$, let $f(a)=\lim_{t\rightarrow +0} \int_{t}^{1} |ax+x\ln x|\ dx.$ Let $a$ vary in the range $0 <a< +\infty$, find the minimum value of $f(a)$.
2024 CMIMC Integration Bee, 11
\[\int_1^\infty \frac{\lfloor x^2\rfloor}{x^5}\mathrm dx\]
[i]Proposed by Robert Trosten[/i]
1983 Putnam, B1
Let $v$ be a vertex of a cube $C$ with edges of length $4$. Let $S$ be the largest sphere that can be inscribed in $C$. Let $R$ be the region consisting of all points $p$ between $S$ and $C$ such that $p$ is closer to $v$ than to any other vertex of the cube. Find the volume of $R$.
2012 Today's Calculation Of Integral, 772
Given are three points $A(2,\ 0,\ 2),\ B(1,\ 1,\ 0),\ C(0,\ 0,\ 3)$ in the coordinate space. Find the volume of the solid of a triangle $ABC$ generated by a rotation about $z$-axis.
2005 Alexandru Myller, 1
Let $f:[a,b]\to\mathbb R$ be a continous function with the property that there exists a constant $\lambda\in\mathbb R$ so that for every $x\in[a,b]$ there exists a $y\in[a,b]-\{x\}$ s.t. $\int_x^yf(x)dx=\lambda$. Prove that the function $f$ has at least two zeros in $(a,b)$.
[i]Eugen Paltanea[/i]
2007 Today's Calculation Of Integral, 200
Evaluate the following definite integral.
\[\int_{0}^{\pi}\frac{\cos nx}{2-\cos x}dx\ (n=0,\ 1,\ 2,\ \cdots)\]
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$
2009 Bulgaria National Olympiad, 6
Prove that if $ a_{1},a_{2},\ldots,a_{n}$, $ b_{1},b_{2},\ldots,b_{n}$ are arbitrary taken real numbers and $ c_{1},c_{2},\ldots,c_{n}$
are positive real numbers, than
$ \left(\sum_{i,j \equal{} 1}^{n}\frac {a_{i}a_{j}}{c_{i} \plus{} c_{j}}\right)\left(\sum_{i,j \equal{} 1}^{n}\frac {b_{i}b_{j}}{c_{i} \plus{} c_{j}}\right)\ge \left(\sum_{i,j \equal{} 1}^{n}\frac {a_{i}b_{j}}{c_{i} \plus{} c_{j}}\right)^{2}$.
2007 Today's Calculation Of Integral, 233
Find the minimum value of the following definite integral.
$ \int_0^{\pi} (a\sin x \plus{} b\sin 3x \minus{} 1)^2\ dx.$
2009 Today's Calculation Of Integral, 422
There are 10 cards, labeled from 1 to 10. Three cards denoted by $ a,\ b,\ c\ (a > b > c)$ are drawn from the cards at the same time.
Find the probability such that $ \int_0^a (x^2 \minus{} 2bx \plus{} 3c)\ dx \equal{} 0$.
2010 Contests, 2
Compute the sum of the series
$\sum_{k=0}^{\infty} \frac{1}{(4k+1)(4k+2)(4k+3)(4k+4)} = \frac{1}{1\cdot2\cdot3\cdot4} + \frac{1}{5\cdot6\cdot7\cdot8} + ...$
Today's calculation of integrals, 900
Find $\sum_{k=0}^n \frac{(-1)^k}{2k+1}\ _n C_k.$
2013 BMT Spring, 9
Evaluate the integral
$$\int^1_0\left(\sqrt{(x-1)^3+1}+x^{2/3}-(1-x)^{3/2}-\sqrt[3]{1-x^2}\right)dx$$
Today's calculation of integrals, 867
Express $\int_0^2 f(x)dx$ for any quadratic functions $f(x)$ in terms of $f(0),\ f(1)$ and $f(2).$
2000 IMC, 6
Let $f: \mathbb{R}\rightarrow ]0,+\infty[$ be an increasing differentiable function with $\lim_{x\rightarrow+\infty}f(x)=+\infty$ and $f'$ is bounded, and let $F(x)=\int^x_0 f(t) dt$.
Define the sequence $(a_n)$ recursively by $a_0=1,a_{n+1}=a_n+\frac1{f(a_n)}$
Define the sequence $(b_n)$ by $b_n=F^{-1}(n)$.
Prove that $\lim_{x\rightarrow+\infty}(a_n-b_n)=0$.
2009 Today's Calculation Of Integral, 435
Evaluate $ \int_{\frac{\pi}{4}}^{\frac {\pi}{2}} \frac {1}{(\sin x \plus{} \cos x \plus{} 2\sqrt {\sin x\cos x})\sqrt {\sin x\cos x}}dx$.
2009 Today's Calculation Of Integral, 408
Evaluate $ \int_1^e \{(1 \plus{} x)e^x \plus{} (1 \minus{} x)e^{ \minus{} x}\}\ln x\ dx$.
2007 Today's Calculation Of Integral, 227
Evaluate $ \frac{1}{\displaystyle \int _0^{\frac{\pi}{2}} \cos ^{2006}x \cdot \sin 2008 x\ dx}$
2012 Today's Calculation Of Integral, 791
Let $S$ be the domain in the coordinate plane determined by two inequalities:
\[y\geq \frac 12x^2,\ \ \frac{x^2}{4}+4y^2\leq \frac 18.\]
Denote by $V_1$ the volume of the solid by a rotation of $S$ about the $x$-axis and by $V_2$, by a rotation of $S$ about the $y$-axis.
(1) Find the values of $V_1,\ V_2$.
(2) Compare the size of the value of $\frac{V_2}{V_1}$ and 1.
2008 Harvard-MIT Mathematics Tournament, 10
Evaluate the infinite sum \[\sum_{n \equal{} 0}^\infty \binom{2n}{n}\frac {1}{5^n}.\]
2000 Baltic Way, 20
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 $\frac{1}{4n}<x_n-\sqrt{2}<\frac{2}{n}$.