Found problems: 2215
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.
2007 Today's Calculation Of Integral, 200
Evaluate the following definite integral.
\[\int_{0}^{\pi}\frac{\cos nx}{2-\cos x}dx\ (n=0,\ 1,\ 2,\ \cdots)\]
2005 MOP Homework, 4
Consider an infinite array of integers. Assume that each integer is equal to the sum of the integers immediately above and immediately to the left. Assume that there exists a row $R_0$ such that all the number in the row are positive. Denote by $R_1$ the row below row $R_0$, by $R_2$ the row below row $R_1$, and so on. Show that for each positive integer $n$, row $R_n$ cannot contain more than $n$ zeros.
2004 IMO Shortlist, 6
Find all functions $f:\mathbb{R} \to \mathbb{R}$ satisfying the equation \[
f(x^2+y^2+2f(xy)) = (f(x+y))^2.
\] for all $x,y \in \mathbb{R}$.
2010 Harvard-MIT Mathematics Tournament, 2
Let $f$ be a function such that $f(0)=1$, $f^\prime (0)=2$, and \[f^{\prime\prime}(t)=4f^\prime(t)-3f(t)+1\] for all $t$. Compute the $4$th derivative of $f$, evaluated at $0$.
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$.
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 $f(2).$
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$.
2015 AMC 10, 24
For some positive integers $p$, there is a quadrilateral $ABCD$ with positive integer side lengths, perimeter $p$, right angles at $B$ and $C$, $AB=2$, and $CD=AD$. How many different values of $p<2015$ are possible?
$\textbf{(A) }30\qquad\textbf{(B) }31\qquad\textbf{(C) }61\qquad\textbf{(D) }62\qquad\textbf{(E) }63$
2012 Pre - Vietnam Mathematical Olympiad, 1
For $a,b,c>0: \; abc=1$ prove that
\[a^3+b^3+c^3+6 \ge (a+b+c)^2\]
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 \[\sum_{n \equal{} 0}^\infty \binom{2n}{n}\frac {1}{5^n}.\]
2003 IMO Shortlist, 6
Let $n$ be a positive integer and let $(x_1,\ldots,x_n)$, $(y_1,\ldots,y_n)$ be two sequences of positive real numbers. Suppose $(z_2,\ldots,z_{2n})$ is a sequence of positive real numbers such that $z_{i+j}^2 \geq x_iy_j$ for all $1\le i,j \leq n$.
Let $M=\max\{z_2,\ldots,z_{2n}\}$. Prove that \[
\left( \frac{M+z_2+\dots+z_{2n}}{2n} \right)^2
\ge
\left( \frac{x_1+\dots+x_n}{n} \right)
\left( \frac{y_1+\dots+y_n}{n} \right). \]
[hide="comment"]
[i]Edited by Orl.[/i]
[/hide]
[i]Proposed by Reid Barton, USA[/i]
1952 AMC 12/AHSME, 13
The function $ x^2 \plus{} px \plus{} q$ with $ p$ and $ q$ greater than zero has its minimum value when:
$ \textbf{(A)}\ x \equal{} \minus{} p \qquad\textbf{(B)}\ x \equal{} \frac {p}{2} \qquad\textbf{(C)}\ x \equal{} \minus{} 2p \qquad\textbf{(D)}\ x \equal{} \frac {p^2}{4q} \qquad\textbf{(E)}\ x \equal{} \frac { \minus{} p}{2}$
2010 Today's Calculation Of Integral, 639
Evaluate $\int_0^1 (x+3)\sqrt{xe^x}\ dx.$