Found problems: 2215
2012 Today's Calculation Of Integral, 784
Define for positive integer $n$, a function $f_n(x)=\frac{\ln x}{x^n}\ (x>0).$ In the coordinate plane, denote by $S_n$ the area of the figure enclosed by $y=f_n(x)\ (x\leq t)$, the $x$-axis and the line $x=t$ and denote by $T_n$ the area of the rectagle with four vertices $(1,\ 0),\ (t,\ 0),\ (t,\ f_n(t))$ and $(1,\ f_n(t))$.
(1) Find the local maximum $f_n(x)$.
(2) When $t$ moves in the range of $t>1$, find the value of $t$ for which $T_n(t)-S_n(t)$ is maximized.
(3) Find $S_1(t)$ and $S_n(t)\ (n\geq 2)$.
(4) For each $n\geq 2$, prove that there exists the only $t>1$ such that $T_n(t)=S_n(t)$.
Note that you may use $\lim_{x\to\infty} \frac{\ln x}{x}=0.$
2016 BMT Spring, 9
Suppose $p''(x) = 4x^2 + 4x + 2$ where $$p(x) = a_0 + a_1(x - 1) + a_2(x -2)^2 + a_3(x- 3)^4 + a_4(x-4)^4.$$ We have $p'(-3) = -24$ and $p(x)$ has the unique property that the sum of the third powers of the roots of $p(x)$ is equal to the sum of the fourth powers of the roots of $p(x)$ . Find $a_0$.
2010 Today's Calculation Of Integral, 610
Evaluate $\int_2^a \frac{x^a-1-xa^x\ln a}{(x^a-1)^2}dx.$
proposed by kunny
2019 District Olympiad, 2
Let $n$ be a positive integer and $f:[0,1] \to \mathbb{R}$ be an integrable function. Prove that there exists a point $c \in \left[0,1- \frac{1}{n} \right],$ such that [center] $ \int\limits_c^{c+\frac{1}{n}}f(x)\mathrm{d}x=0$ or $\int\limits_0^c f(x) \mathrm{d}x=\int\limits_{c+\frac{1}{n}}^1f(x)\mathrm{d}x.$ [/center]
2007 Germany Team Selection Test, 1
The sequence of real numbers $a_0,a_1,a_2,\ldots$ is defined recursively by \[a_0=-1,\qquad\sum_{k=0}^n\dfrac{a_{n-k}}{k+1}=0\quad\text{for}\quad n\geq 1.\]Show that $ a_{n} > 0$ for all $ n\geq 1$.
[i]Proposed by Mariusz Skalba, Poland[/i]
2005 Korea National Olympiad, 4
Find all $f: \mathbb R \to\mathbb R$ such that for all real numbers $x$, $f(x) \geq 0$ and for all real numbers $x$ and $y$, \[ f(x+y)+f(x-y)-2f(x)-2y^2=0. \]
2007 Today's Calculation Of Integral, 174
Let $a$ be a positive number. Assume that the parameterized curve $C: \ x=t+e^{at},\ y=-t+e^{at}\ (-\infty <t< \infty)$ is touched to $x$ axis.
(1) Find the value of $a.$
(2) Find the area of the part which is surrounded by two straight lines $y=0, y=x$ and the curve $C.$
2005 Today's Calculation Of Integral, 71
Find the minimum value of $\int_{-1}^1 \sqrt{|t-x|}\ dt$
2004 Tuymaada Olympiad, 1
Do there exist a sequence $a_{1}, a_{2}, a_{3}, \ldots$ of real numbers and a non-constant polynomial $P(x)$ such that $a_{m}+a_{n}=P(mn)$ for every positive integral $m$ and $n?$
[i]Proposed by A. Golovanov[/i]
2011 Today's Calculation Of Integral, 697
Find the volume of the solid of the domain expressed by the inequality $x^2-x\leq y\leq x$, generated by a rotation about the line $y=x.$
2025 Romania EGMO TST, P1
The sequence of real numbers $a_0,a_1,a_2,\ldots$ is defined recursively by \[a_0=-1,\qquad\sum_{k=0}^n\dfrac{a_{n-k}}{k+1}=0\quad\text{for}\quad n\geq 1.\]Show that $ a_{n} > 0$ for all $ n\geq 1$.
[i]Proposed by Mariusz Skalba, Poland[/i]
2016 BMT Spring, 6
Amy is traveling on the $xy$-plane in a spaceship where her motion is described by the following equation $xe^y = ye^x$. Given that her $x$-component of velocity is a constant $3$ mph , the magnitude of her velocity as she approaches $(1,-1)$ can be expressed as $\sqrt{\frac{a + be^4}{ c + de^2}}$ . Find $\frac{ac}{bd}$ .
(You may assume that the initial conditions do allow her to approach $(1,-1)$)
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.
2010 Today's Calculation Of Integral, 609
Prove that for positive number $t$, the function $F(t)=\int_0^t \frac{\sin x}{1+x^2}dx$ always takes positive number.
1972 Tokyo University of Education entrance exam
2011 Today's Calculation Of Integral, 699
Find the volume of the part bounded by $z=x+y,\ z=x^2+y^2$ in the $xyz$ space.
1967 IMO Longlists, 3
Prove the trigonometric inequality $\cos x < 1 - \frac{x^2}{2} + \frac{x^4}{16},$ when $x \in \left(0, \frac{\pi}{2} \right).$
1969 Canada National Olympiad, 2
Determine which of the two numbers $\sqrt{c+1}-\sqrt{c}$, $\sqrt{c}-\sqrt{c-1}$ is greater for any $c\ge 1$.
2005 Today's Calculation Of Integral, 33
Evaluate
\[\int_{-\ln 2}^0\ \frac{dx}{\cos ^2 h x \cdot \sqrt{1-2a\tanh x +a^2}}\ (a>0)\]
2011 Canadian Open Math Challenge, 8
A group of n friends wrote a math contest consisting of eight short-answer problem $S_1, S_2, S_3, S_4, S_5, S_6, S_7, S_8$, and four full-solution problems $F_1, F_2, F_3, F_4$. Each person in the group correctly solved exactly 11 of the 12 problems. We create an 8 x 4 table. Inside the square located in the $i$th row and $j$th column, we write down the number of people who correctly solved both problem $S_i$ and $F_j$. If the 32 entries in the table sum to 256, what is the value of n?
1997 IMC, 1
Let $\{\epsilon_n\}^\infty_{n=1}$ be a sequence of positive reals with $\lim\limits_{n\rightarrow+\infty}\epsilon_n = 0$. Find \[ \lim\limits_{n\rightarrow\infty}\dfrac{1}{n}\sum\limits^{n}_{k=1}\ln\left(\dfrac{k}{n}+\epsilon_n\right) \]
2012 Today's Calculation Of Integral, 823
Let $C$ be the curve expressed by $x=\sin t,\ y=\sin 2t\ \left(0\leq t\leq \frac{\pi}{2}\right).$
(1) Express $y$ in terms of $x$.
(2) Find the area of the figure $D$ enclosed by the $x$-axis and $C$.
(3) Find the volume of the solid generated by a rotation of $D$ about the $y$-axis.
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 $|\alpha| = 1.$
2011 Today's Calculation Of Integral, 732
Let $a$ be parameter such that $0<a<2\pi$. For $0<x<2\pi$, find the extremum of $F(x)=\int_{x}^{x+a} \sqrt{1-\cos \theta}\ d\theta$.
2006 APMO, 2
Prove that every positive integer can be written as a finite sum of distinct integral powers of the golden ratio.
2014 AMC 12/AHSME, 20
For how many positive integers $x$ is $\log_{10}{(x-40)} + \log_{10}{(60-x)} < 2$?
${ \textbf{(A)}\ 10\qquad\textbf{(B)}\ 18\qquad\textbf{(C)}\ 19\qquad\textbf{(D)}}\ 20\qquad\textbf{(E)}\ \text{infinitely many} $