Found problems: 2215
2011 Today's Calculation Of Integral, 731
Let $C$ be the point of intersection of the tangent lines $l,\ m$ at $A(a,\ a^2),\ B(b,\ b^2)\ (a<b)$ on the parabola $y=x^2$ respectively.
When $C$ moves on the parabola $y=\frac 12 x^2-x-2$, find the minimum area bounded by 2 lines $l,\ m$ and the parabola $y=x^2$.
2005 Today's Calculation Of Integral, 62
For $a>1$, let $f(a)=\frac{1}{2}\int_0^1 |ax^n-1|dx+\frac{1}{2}\ (n=1,2,\cdots)$ and let $b_n$ be the minimum value of $f(a)$ at $a>1$.
Evaluate
\[\lim_{m\to\infty} b_m\cdot b_{m+1}\cdot \cdots\cdots b_{2m}\ (m=1,2,3,\cdots)\]
2007 Today's Calculation Of Integral, 217
Evaluate $ \int_{0}^{1}e^{\sqrt{e^{x}}}\ dx\plus{}2\int_{e}^{e^{\sqrt{e}}}\ln (\ln x)\ dx$.
2005 Today's Calculation Of Integral, 36
A sequence of polynomial $f_n(x)\ (n=0,1,2,\cdots)$ satisfies $f_0(x)=2,f_1(x)=x$,
\[f_n(x)=xf_{n-1}(x)-f_{n-2}(x),\ (n=2,3,4,\cdots)\]
Let $x_n\ (n\geqq 2)$ be the maximum real root of the equation $f_n(x)=0\ (|x|\leqq 2)$
Evaluate
\[\lim_{n\to\infty} n^2 \int_{x_n}^2 f_n(x)dx\]
2003 China Second Round Olympiad, 2
Let the three sides of a triangle be $\ell, m, n$, respectively, satisfying $\ell>m>n$ and $\left\{\frac{3^\ell}{10^4}\right\}=\left\{\frac{3^m}{10^4}\right\}=\left\{\frac{3^n}{10^4}\right\}$, where $\{x\}=x-\lfloor{x}\rfloor$ and $\lfloor{x}\rfloor$ denotes the integral part of the number $x$. Find the minimum perimeter of such a triangle.
2023 CMIMC Integration Bee, 1
\[\int_2^0 x^2+3\,\mathrm dx\]
[i]Proposed by Connor Gordon[/i]
2012 Today's Calculation Of Integral, 776
Evaluate $\int_{\frac{1-\sqrt{5}}{2}}^{\frac{1+\sqrt{5}}{2}} (2x^2-1)e^{2x}dx.$
2003 VJIMC, Problem 4
Let $f,g:[0,1]\to(0,+\infty)$ be two continuous functions such that $f$ and $\frac gf$ are increasing. Prove that
$$\int^1_0\frac{\int^x_0f(t)\text dt}{\int^x_0g(t)\text dt}\text dx\le2\int^1_0\frac{f(t)}{g(t)}\text dt.$$
1951 Miklós Schweitzer, 16
Let $ \mathcal{F}$ be a surface which is simply covered by two systems of geodesics such that any two lines belonging to different systems form angles of the same opening. Prove that $ \mathcal{F}$ can be developed (that is, isometrically mapped) into the plane.
2011 Today's Calculation Of Integral, 675
In the coordinate plane with the origin $O$, consider points $P(t+2,\ 0),\ Q(0, -2t^2-2t+4)\ (t\geq 0).$ If the $y$-coordinate of $Q$ is nonnegative, then find the area of the region swept out by the line segment $PQ$.
[i]2011 Ritsumeikan University entrance exam/Pharmacy[/i]
2024 CMI B.Sc. Entrance Exam, 2
$g(x) \colon \int_{10}^{x} \log_{10}(\log_{10}(t^2-1000t+10^{1000})) dt$
(a) Find the domain of $g(x)$
(b) Approximate the value of $g(1000)$
(c) Find $x \in [10, 1000]$ to maximize the slope of $g(x)$
(d) Find $x \in [10, 1000]$ to minimize the slope of $g(x)$
(e) Determine, if it exists, $\lim_{x \to \infty} \frac{\ln(x)}{g(x)}$
2004 Harvard-MIT Mathematics Tournament, 1
Let $f(x)=\sin(\sin(x))$. Evaluate \[ \lim_{h \to 0} \dfrac {f(x+h)-f(h)}{x} \] at $x=\pi$.
2010 Today's Calculation Of Integral, 603
Find the minimum value of $\int_0^1 \{\sqrt{x}-(a+bx)\}^2dx$.
Please solve the problem without using partial differentiation for those who don't learn it.
1961 Waseda University entrance exam/Science and Technology
2005 Vietnam Team Selection Test, 1
Let be given positive reals $a$, $b$, $c$. Prove that: $\frac{a^{3}}{\left(a+b\right)^{3}}+\frac{b^{3}}{\left(b+c\right)^{3}}+\frac{c^{3}}{\left(c+a\right)^{3}}\geq \frac{3}{8}$.
2005 Today's Calculation Of Integral, 24
Find the minimum value of $\int_0^{\pi} (x-y)^2 (\sin x)|\cos x|dx$.
2005 Putnam, A5
Evaluate $\int_0^1\frac{\ln(x+1)}{x^2+1}\,dx.$
2010 Today's Calculation Of Integral, 566
In the coordinate space, consider the cubic with vertices $ O(0,\ 0,\ 0),\ A(1,\ 0,\ 0),\ B(1,\ 1,\ 0),\ C(0,\ 1,\ 0),\ D(0,\ 0,\ 1),\ E(1,\ 0,\ 1),\ F(1,\ 1,\ 1),\ G(0,\ 1,\ 1)$. Find the volume of the solid generated by revolution of the cubic around the diagonal $ OF$ as the axis of rotation.
2019 Jozsef Wildt International Math Competition, W. 12
If $0 < a < b$ then: $$\frac{\int \limits^{\frac{a+b}{2}}_{a}\left(\tan^{-1}t\right)dt}{\int \limits_{a}^{b}\left(\tan^{-1}t\right)dt}<\frac{1}{2}$$
2015 VTRMC, Problem 5
Evaluate $\int^\infty_0\frac{\operatorname{arctan}(\pi x)-\operatorname{arctan}(x)}xdx$ (where $0\le\operatorname{arctan}(x)<\frac\pi2$ for $0\le x<\infty$).
2013 Today's Calculation Of Integral, 871
Define sequences $\{a_n\},\ \{b_n\}$ by
\[a_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}d\theta,\ b_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}\cos \theta d\theta\ (n=1,\ 2,\ 3,\ \cdots).\]
(1) Find $b_n$.
(2) Prove that for each $n$, $b_n\leq a_n\leq \frac 2{\sqrt{3}}b_n.$
(3) Find $\lim_{n\to\infty} \frac 1{n}\ln (na_n).$
2005 Today's Calculation Of Integral, 80
Let $S$ be the domain surrounded by the two curves $C_1:y=ax^2,\ C_2:y=-ax^2+2abx$ for constant positive numbers $a,b$.
Let $V_x$ be the volume of the solid formed by the revolution of $S$ about the axis of $x$, $V_y$ be the volume of the solid formed by the revolution of $S$
about the axis of $y$. Find the ratio of $\frac{V_x}{V_y}$.
1989 China Team Selection Test, 3
Find the greatest $n$ such that $(z+1)^n = z^n + 1$ has all its non-zero roots in the unitary circumference, e.g. $(\alpha+1)^n = \alpha^n + 1, \alpha \neq 0$ implies $|\alpha| = 1.$
2005 Georgia Team Selection Test, 3
Let $ x,y,z$ be positive real numbers,satisfying equality $ x^{2}\plus{}y^{2}\plus{}z^{2}\equal{}25$. Find the minimal possible value of the expression $ \frac{xy}{z} \plus{} \frac{yz}{x} \plus{} \frac{zx}{y}$.
2011 Baltic Way, 4
Let $a,b,c,d$ be non-negative reals such that $a+b+c+d=4$. Prove the inequality
\[\frac{a}{a^3+8}+\frac{b}{b^3+8}+\frac{c}{c^3+8}+\frac{d}{d^3+8}\le\frac{4}{9}\]
2010 Harvard-MIT Mathematics Tournament, 10
Let $f(n)=\displaystyle\sum_{k=1}^n \dfrac{1}{k}$. Then there exists constants $\gamma$, $c$, and $d$ such that \[f(n)=\ln(x)+\gamma+\dfrac{c}{n}+\dfrac{d}{n^2}+O\left(\dfrac{1}{n^3}\right),\] where the $O\left(\dfrac{1}{n^3}\right)$ means terms of order $\dfrac{1}{n^3}$ or lower. Compute the ordered pair $(c,d)$.