Found problems: 2215
2024 ISI Entrance UGB, P4
Let $f: \mathbb R \to \mathbb R$ be a function which is differentiable at $0$. Define another function $g: \mathbb R \to \mathbb R$ as follows:
$$g(x) = \begin{cases}
f(x)\sin\left(\frac 1x\right) ~ &\text{if} ~ x \neq 0 \\
0 &\text{if} ~ x = 0.
\end{cases}$$
Suppose that $g$ is also differentiable at $0$. Prove that \[g'(0) = f'(0) = f(0) = g(0) = 0.\]
2004 Romania National Olympiad, 3
Let $n>2,n \in \mathbb{N}$ and $a>0,a \in \mathbb{R}$ such that $2^a + \log_2 a = n^2$. Prove that: \[ 2 \cdot \log_2 n>a>2 \cdot \log_2 n -\frac{1}{n} . \]
[i]Radu Gologan[/i]
2011 Today's Calculation Of Integral, 717
Let $a_n$ be the area of the part enclosed by the curve $y=x^n\ (n\geq 1)$, the line $x=\frac 12$ and the $x$ axis.
Prove that :
\[0\leq \ln 2-\frac 12-(a_1+a_2+\cdots\cdots+a_n)\leq \frac {1}{2^{n+1}}\]
2005 Today's Calculation Of Integral, 44
Evaluate
\[{\int_0^\frac{\pi}{2}} \frac{\sin 2005x}{\sin x}dx\]
2010 Today's Calculation Of Integral, 525
Let $ a,\ b$ be real numbers satisfying $ \int_0^1 (ax\plus{}b)^2dx\equal{}1$.
Determine the values of $ a,\ b$ for which $ \int_0^1 3x(ax\plus{}b)\ dx$ is maximized.
2014 Contests, 903
Let $\{a_n\}_{n\geq 1}$ be a sequence defined by $a_n=\int_0^1 x^2(1-x)^ndx$.
Find the real value of $c$ such that $\sum_{n=1}^{\infty} (n+c)(a_n-a_{n+1})=2.$
2017 VJIMC, 4
Let $f:(1,\infty) \to \mathbb{R}$ be a continuously differentiable function satisfying $f(x) \le x^2 \log(x)$ and $f'(x)>0$ for every $x \in (1,\infty)$. Prove that
\[\int_1^{\infty} \frac{1}{f'(x)} dx=\infty.\]
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}$.
1999 Harvard-MIT Mathematics Tournament, 2
A rectangle has sides of length $\sin x$ and $\cos x$ for some $x$. What is the largest possible area of such a rectangle?
2005 Today's Calculation Of Integral, 33
Evaluate
\[\int_{-\ln 2}^0\ \frac{dx}{\cos ^2 h x \cdot \sqrt{1-2a\tanh x +a^2}}\ (a>0)\]
2022 CMIMC Integration Bee, 3
\[\int_0^1 x\sqrt[4]{1-x}\,\mathrm dx\]
[i]Proposed by Connor Gordon[/i]
2010 Today's Calculation Of Integral, 658
Consider a parameterized curve $C: x=e^{-t}\cos t,\ y=e^{-t}\sin t\left (0\leq t\leq \frac{\pi}{2}\right).$
(1) Find the length $L$ of $C$.
(2) Find the area $S$ of the region enclosed by the $x,\ y$ axis and $C$.
Please solve the problem without using the formula of area for polar coordinate for Japanese High School Students who don't study it in High School.
[i]1997 Kyoto University entrance exam/Science[/i]
2000 USA Team Selection Test, 4
Let $n$ be a positive integer. Prove that
\[ \binom{n}{0}^{-1} + \binom{n}{1}^{-1} + \cdots + \binom{n}{n}^{-1} = \frac{n+1}{2^{n+1}} \left( \frac{2}{1} + \frac{2^2}{2} + \cdots + \frac{2^{n+1}}{n+1} \right). \]
2021 JHMT HS, 1
The value of $x$ in the interval $[0, 2\pi]$ that minimizes the value of $x + 2\cos x$ can be written in the form $a\pi/b,$ where $a$ and $b$ are relatively prime positive integers. Compute $a + b.$
Today's calculation of integrals, 886
Find the functions $f(x),\ g(x)$ such that
$f(x)=e^{x}\sin x+\int_0^{\pi} ug(u)\ du$
$g(x)=e^{x}\cos x+\int_0^{\pi} uf(u)\ du$
2022 CMIMC Integration Bee, 8
\[\int_{-\infty}^{0} \frac{1}{e^{-x}+2e^{x}+e^{3x}}\,\mathrm dx\]
[i]Proposed by Connor Gordon[/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, $\lim_{x \to \infty} \frac{\ln(x)}{g(x)}$
2011 South East Mathematical Olympiad, 1
If $\min \left \{ \frac{ax^2+b}{\sqrt{x^2+1}} \mid x \in \mathbb{R}\right \} = 3$, then (1) Find the range of $b$; (2) for every given $b$, find $a$.
2008 Bosnia Herzegovina Team Selection Test, 1
Prove that in an isosceles triangle $ \triangle ABC$ with $ AC\equal{}BC\equal{}b$ following inequality holds $ b> \pi r$, where $ r$ is inradius.
2005 ISI B.Stat Entrance Exam, 2
Let
\[f(x)=\int_0^1 |t-x|t \, dt\]
for all real $x$. Sketch the graph of $f(x)$. What is the minimum value of $f(x)$?
2015 Romania National Olympiad, 3
Let $\mathcal{C}$ be the set of all twice differentiable functions $f:[0,1] \to \mathbb{R}$ with at least two (not necessarily distinct) zeros and $|f''(x)| \le 1,$ for all $x \in [0,1].$ Find the greatest value of the integral $$\int\limits_0^1 |f(x)| \mathrm{d}x$$ when $f$ runs through the set $\mathcal{C},$ as well as the functions that achieve this maximum.
[i]Note: A differentiable function $f$ has two zeros in the same point $a$ if $f(a)=f'(a)=0.$[/i]
2011 Today's Calculation Of Integral, 686
Let $L$ be a positive constant. For a point $P(t,\ 0)$ on the positive part of the $x$ axis on the coordinate plane, denote $Q(u(t),\ v(t))$ the point at which the point reach starting from $P$ proceeds by distance $L$ in counter-clockwise on the perimeter of a circle passing the point $P$ with center $O$.
(1) Find $u(t),\ v(t)$.
(2) For real number $a$ with $0<a<1$, find $f(a)=\int_a^1 \sqrt{\{u'(t)\}^2+\{v'(t)\}^2}\ dt$.
(3) Find $\lim_{a\rightarrow +0} \frac{f(a)}{\ln a}$.
[i]2011 Tokyo University entrance exam/Science, Problem 3[/i]
2005 Today's Calculation Of Integral, 56
Evaluate
\[\lim_{n\to\infty} \sum_{k=1}^n \frac{[\sqrt{2n^2-k^2}\ ]}{n^2}\]
$[x]$ is the greatest integer $\leq x$.
2011 China Team Selection Test, 2
Let $\ell$ be a positive integer, and let $m,n$ be positive integers with $m\geq n$, such that $A_1,A_2,\cdots,A_m,B_1,\cdots,B_m$ are $m+n$ pairwise distinct subsets of the set $\{1,2,\cdots,\ell\}$. It is known that $A_i\Delta B_j$ are pairwise distinct, $1\leq i\leq m, 1\leq j\leq n$, and runs over all nonempty subsets of $\{1,2,\cdots,\ell\}$. Find all possible values of $m,n$.
2024 UMD Math Competition Part II, #4
Prove for every positive integer $n{:}$
\[ \frac {1 \cdot 3 \cdots (2n - 1)}{2 \cdot 4 \cdots (2n)} < \frac 1{\sqrt{3n}}\]