Found problems: 1687
2012 Today's Calculation Of Integral, 832
Find the limit
\[\lim_{n\to\infty} \frac{1}{n\ln n}\int_{\pi}^{(n+1)\pi} (\sin ^ 2 t)(\ln t)\ dt.\]
2010 Today's Calculation Of Integral, 600
Evaluate $\int_{-a}^a \left(x+\frac{1}{\sin x+\frac{1}{e^x-e^{-x}}}\right)dx\ (a>0)$.
created by kunny
1973 Miklós Schweitzer, 5
Verify that for every $ x > 0$, \[ \frac{\Gamma'(x\plus{}1)}{\Gamma (x\plus{}1)} > \log x.\]
[i]P. Medgyessy[/i]
2010 Today's Calculation Of Integral, 654
A function $f(x)$ defined in $x\geq 0$ satisfies $\lim_{x\to\infty} \frac{f(x)}{x}=1$.
Find $\int_0^{\infty} \{f(x)-f'(x)\}e^{-x}dx$.
[i]1997 Hokkaido University entrance exam/Science[/i]
2022 CMIMC Integration Bee, 4
\[\int_0^1 \sqrt{x}\log(x)\,\mathrm dx\]
[i]Proposed by Connor Gordon[/i]
2008 Harvard-MIT Mathematics Tournament, 7
Compute $ \sum_{n \equal{} 1}^\infty\sum_{k \equal{} 1}^{n \minus{} 1}\frac {k}{2^{n \plus{} k}}$.
2009 Today's Calculation Of Integral, 461
Let $ I_n\equal{}\int_0^{\sqrt{3}} \frac{1}{1\plus{}x^{n}}\ dx\ (n\equal{}1,\ 2,\ \cdots)$.
(1) Find $ I_1,\ I_2$.
(2) Find $ \lim_{n\to\infty} I_n$.
2006 AIME Problems, 3
Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is $\frac{1}{29}$ of the original integer.
2011 China National Olympiad, 3
Let $A$ be a set consist of finite real numbers,$A_1,A_2,\cdots,A_n$ be nonempty sets of $A$, such that
[b](a)[/b] The sum of the elements of $A$ is $0,$
[b](b)[/b] For all $x_i \in A_i(i=1,2,\cdots,n)$,we have $x_1+x_2+\cdots+x_n>0$.
Prove that there exist $1\le k\le n,$ and $1\le i_1<i_2<\cdots<i_k\le n$, such that
\[|A_{i_1}\bigcup A_{i_2} \bigcup \cdots \bigcup A_{i_k}|<\frac{k}{n}|A|.\]
Where $|X|$ denote the numbers of the elements in set $X$.
2006 District Olympiad, 1
Let $f_1,f_2,\ldots,f_n : [0,1]\to (0,\infty)$ be $n$ continuous functions, $n\geq 1$, and let $\sigma$ be a permutation of the set $\{1,2,\ldots, n\}$. Prove that \[ \prod^n_{i=1} \int^1_0 \frac{ f_i^2(x) }{ f_{\sigma(i)}(x) } dx \geq \prod^n_{i=1} \int^1_0 f_i(x) dx. \]
2007 Today's Calculation Of Integral, 222
Find $ \lim_{a\rightarrow\infty}\int_{a}^{a\plus{}1}\frac{x}{x\plus{}\ln x}\ dx$.
2023 CMIMC Integration Bee, 7
\[\int_0^{\frac \pi 2} \left(\frac{1}{1-\cos(x)}-\frac{2}{x^2}\right)\,\mathrm dx\]
[i]Proposed by Vlad Oleksenko[/i]
2011 SEEMOUS, Problem 4
Let $f:[0,1]\to\mathbb R$ be a twice continuously differentiable increasing function. Define the sequences given by $L_n=\frac1n\sum_{k=0}^{n-1}f\left(\frac kn\right)$ and $U_n=\frac1n\sum_{k=0}^nf\left(\frac kn\right)$, $n\ge1$. 1. The interval $[L_n,U_n]$ is divided into three equal segments. Prove that, for large enough $n$, the number $I=\int^1_0f(x)\text dx$ belongs to the middle one of these three segments.
1989 Putnam, B6
Let $(x_1,x_2,\ldots,x_n)$ be a point chosen at random in the $n$-dimensional region defined by $0<x_1<x_2<\ldots<x_n<1$, denoting $x_0=0$ and $x_{n+1}=1$. Let $f$ be a continuous function on $[0,1]$ with $f(1)=0$. Show that the expected value of the sum
$$\sum_{i=0}^n(x_{i+1}-x_i)f(x_{i+1})$$is $\int^1_0f(t)P(t)dt$., where $P$ is a polynomial of degree $n$, independent of $f$, with $0\le P(t)\le1$ for $0\le t\le1$.
2009 ISI B.Stat Entrance Exam, 2
Let $f(x)$ be a continuous function, whose first and second derivatives are continuous on $[0,2\pi]$ and $f''(x) \geq 0$ for all $x$ in $[0,2\pi]$. Show that
\[\int_{0}^{2\pi} f(x)\cos x dx \geq 0\]
1949 Miklós Schweitzer, 2
Compute
$ \lim_{n\rightarrow \infty} \int_{0}^{\pi} \frac {\sin{x}}{1 \plus{} \cos^2 nx}dx$ .
1994 Cono Sur Olympiad, 2
Solve the following equation in integers with gcd (x, y) = 1
$x^2 + y^2 = 2 z^2$
2012 USAMO, 6
For integer $n\geq2$, let $x_1, x_2, \ldots, x_n$ be real numbers satisfying \[x_1+x_2+\ldots+x_n=0, \qquad \text{and}\qquad x_1^2+x_2^2+\ldots+x_n^2=1.\]For each subset $A\subseteq\{1, 2, \ldots, n\}$, define\[S_A=\sum_{i\in A}x_i.\](If $A$ is the empty set, then $S_A=0$.)
Prove that for any positive number $\lambda$, the number of sets $A$ satisfying $S_A\geq\lambda$ is at most $2^{n-3}/\lambda^2$. For which choices of $x_1, x_2, \ldots, x_n, \lambda$ does equality hold?
2008 All-Russian Olympiad, 5
The numbers from $ 51$ to $ 150$ are arranged in a $ 10\times 10$ array. Can this be done in such a way that, for any two horizontally or vertically adjacent numbers $ a$ and $ b$, at least one of the equations $ x^2 \minus{} ax \plus{} b \equal{} 0$ and $ x^2 \minus{} bx \plus{} a \equal{} 0$ has two integral roots?
2021 CMIMC Integration Bee, 15
$$\int_{-\infty}^\infty \frac{\sin(\pi x)}{x(1+x^2)}\,dx$$
[i]Proposed by Vlad Oleksenko[/i]
2024 CMIMC Integration Bee, 12
\[\int_1^\infty \frac{\sec^{-1}(x^{2})-\sec^{-1}(\sqrt x)}{x\log(x)}\mathrm dx\]
[i]Proposed by Connor Gordon[/i]
2004 China Team Selection Test, 3
Given arbitrary positive integer $ a$ larger than $ 1$, show that for any positive integer $ n$, there always exists a n-degree integral coefficient polynomial $ p(x)$, such that $ p(0)$, $ p(1)$, $ \cdots$, $ p(n)$ are pairwise distinct positive integers, and all have the form of $ 2a^k\plus{}3$, where $ k$ is also an integer.
2003 Alexandru Myller, 4
[b]a)[/b] Prove that the function $ 1\le t\mapsto\int_{1}^t\frac{\sin x}{x^n} dx $ has an horizontal asymptote, for any natural number $ n. $
[b]b)[/b] Calculate $ \lim_{n\to\infty }\lim_{t\to\infty }\int_{1}^t\frac{\sin x}{x^n} . $
[i]Mihai Piticari[/i]
2021 CMIMC Integration Bee, 12
$$\int_1^\infty \frac{1 + 2x \ln 2}{x\sqrt{x 4^x - 1}}\,dx$$
[i]Proposed by Vlad Oleksenko[/i]
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 $0\le x<\infty$).