Found problems: 1687
2009 Today's Calculation Of Integral, 489
Find the following limit.
$ \lim_{n\to\infty} \int_{\minus{}1}^1 |x|\left(1\plus{}x\plus{}\frac{x^2}{2}\plus{}\frac{x^3}{3}\plus{}\cdots \plus{}\frac{x^{2n}}{2n}\right)\ dx$.
2019 Korea USCM, 3
Two vector fields $\mathbf{F},\mathbf{G}$ are defined on a three dimensional region $W=\{(x,y,z)\in\mathbb{R}^3 : x^2+y^2\leq 1, |z|\leq 1\}$.
$$\mathbf{F}(x,y,z) = (\sin xy, \sin yz, 0),\quad \mathbf{G} (x,y,z) = (e^{x^2+y^2+z^2}, \cos xz, 0)$$
Evaluate the following integral.
\[\iiint_{W} (\mathbf{G}\cdot \text{curl}(\mathbf{F}) - \mathbf{F}\cdot \text{curl}(\mathbf{G})) dV\]
Today's calculation of integrals, 855
Let $f(x)$ be a function which is differentiable twice and $f''(x)>0$ on $[0,\ 1]$.
For a positive integer $n$, find $\lim_{n\to\infty} n\left\{\int_0^1 f(x)\ dx-\frac{1}{n}\sum_{k=0}^{n-1} f\left(\frac{k}{n}\right)\right\}.$
2014 Online Math Open Problems, 16
Let $OABC$ be a tetrahedron such that $\angle AOB = \angle BOC = \angle COA = 90^\circ$ and its faces have integral surface areas. If $[OAB] = 20$ and $[OBC] = 14$, find the sum of all possible values of $[OCA][ABC]$. (Here $[\triangle]$ denotes the area of $\triangle$.)
[i]Proposed by Robin Park[/i]
2006 District Olympiad, 4
Let $\mathcal F = \{ f: [0,1] \to [0,\infty) \mid f$ continuous $\}$ and $n$ an integer, $n\geq 2$. Find the smallest real constant $c$ such that for any $f\in \mathcal F$ the following inequality takes place \[ \int^1_0 f \left( \sqrt [n] x \right) dx \leq c \int^1_0 f(x) dx. \]
2001 District Olympiad, 3
Consider a continuous function $f:[0,1]\rightarrow \mathbb{R}$ such that for any third degree polynomial function $P:[0,1]\to [0,1]$, we have
\[\int_0^1f(P(x))dx=0\]
Prove that $f(x)=0,\ (\forall)x\in [0,1]$.
[i]Mihai Piticari[/i]
Today's calculation of integrals, 890
A function $f_n(x)\ (n=1,\ 2,\ \cdots)$ is defined by $f_1(x)=x$ and
\[f_n(x)=x+\frac{e}{2}\int_0^1 f_{n-1}(t)e^{x-t}dt\ (n=2,\ 3,\ \cdots)\].
Find $f_n(x)$.
1987 Swedish Mathematical Competition, 4
A differentiable function $f$ with $f(0) = f(1) = 0$ is defined on the interval $[0,1]$.
Prove that there exists a point $y \in [0,1]$ such that $| f' (y)| = 4 \int _0^1 | f(x)|dx$.
2006 Czech-Polish-Slovak Match, 5
Find the number of sequences $(a_n)_{n=1}^\infty$ of integers satisfying $a_n \ne -1$ and
\[a_{n+2} =\frac{a_n + 2006}{a_{n+1} + 1}\]
for each $n \in \mathbb{N}$.
2002 Romania National Olympiad, 3
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous and bounded function such that
\[x\int_{x}^{x+1}f(t)\, \text{d}t=\int_{0}^{x}f(t)\, \text{d}t,\quad\text{for any}\ x\in\mathbb{R}.\]
Prove that $f$ is a constant function.
2010 N.N. Mihăileanu Individual, 2
Let be a continuous function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ having the property that there exists a continuous and bounded function $ g:\mathbb{R}\longrightarrow\mathbb{R} $ that verifies the equality
$$ f(x)=\int_0^x f(\xi )g(\xi )d\xi , $$
for any real number $ x. $ Prove that $ f=0. $
[i]Nelu Chichirim[/i]
2007 Today's Calculation Of Integral, 234
For $ x\geq 0,$ define a function $ f(x)\equal{}\sin \left(\frac{n\pi}{4}\right)\sin x\ (n\pi \leq x<(n\plus{}1)\pi )\ (n\equal{}0,\ 1,\ 2,\ \cdots)$.
Evaluate $ \int_0^{100\pi } f(x)\ dx.$
Today's calculation of integrals, 882
Find $\lim_{n\to\infty} \sum_{k=1}^n \frac{1}{n+k}(\ln (n+k)-\ln\ n)$.
2000 USA Team Selection Test, 4
Let $n$ be a positive integer. Prove that
\[ \binom{n}{0}^{-1} + \binom{n}{1}^{-1} + \cdots + \binom{n}{n}^{-1} = \frac{n+1}{2^{n+1}} \left( \frac{2}{1} + \frac{2^2}{2} + \cdots + \frac{2^{n+1}}{n+1} \right). \]
2009 Today's Calculation Of Integral, 462
Evaluate $ \int_0^1 \frac{(1\minus{}x\plus{}x^2)\cos \ln (x\plus{}\sqrt{1\plus{}x^2})\minus{}\sqrt{1\plus{}x^2}\sin \ln (x\plus{}\sqrt{1\plus{}x^2})}{(1\plus{}x^2)^{\frac{3}{2}}}\ dx$.
2000 IMC, 6
Let $f: \mathbb{R}\rightarrow ]0,+\infty[$ be an increasing differentiable function with $\lim_{x\rightarrow+\infty}f(x)=+\infty$ and $f'$ is bounded, and let $F(x)=\int^x_0 f(t) dt$.
Define the sequence $(a_n)$ recursively by $a_0=1,a_{n+1}=a_n+\frac1{f(a_n)}$
Define the sequence $(b_n)$ by $b_n=F^{-1}(n)$.
Prove that $\lim_{x\rightarrow+\infty}(a_n-b_n)=0$.
2010 Today's Calculation Of Integral, 551
In the coordinate plane, find the area of the region bounded by the curve $ C: y\equal{}\frac{x\plus{}1}{x^2\plus{}1}$ and the line $ L: y\equal{}1$.
2014 Dutch IMO TST, 5
Let $P(x)$ be a polynomial of degree $n \le 10$ with integral coefficients such that for every $k \in \{1, 2, \dots, 10\}$ there is an integer $m$ with $P(m) = k$. Furthermore, it is given that $|P(10) - P(0)| < 1000$. Prove that for every integer $k$ there is an integer $m$ such that $P(m) = k.$
2009 Olympic Revenge, 2
Prove that $\int_{0}^{\frac{\pi}{2}} arctg (1 - \sin^2x\cos^2x)dx = \frac{\pi^2}{4} - \pi arctg\sqrt{\frac{\sqrt{2}-1}{2}}$
2010 Today's Calculation Of Integral, 556
Prove the following inequality.
\[ \sqrt[3]{\int_0^{\frac {\pi}{4}} \frac {x}{\cos ^ 2 x\cos ^ 2 (\tan x)\cos ^ 2(\tan (\tan x))\cos ^ 2(\tan (\tan (\tan x)))}dx}<\frac{4}{\pi}\]
Last Edited.
Sorry, I have changed the problem.
kunny
2013 Romania National Olympiad, 1
Determine continuous functions $f:\mathbb{R}\to \mathbb{R}$ such that $\left( {{a}^{2}}+ab+{{b}^{2}} \right)\int\limits_{a}^{b}{f\left( x \right)dx=3\int\limits_{a}^{b}{{{x}^{2}}f\left( x \right)dx,}}$ for every $a,b\in \mathbb{R}$ .
1998 Vietnam Team Selection Test, 1
Find all integer polynomials $P(x)$, the highest coefficent is 1 such that: there exist infinitely irrational numbers $a$ such that $p(a)$ is a positive integer.
2005 Today's Calculation Of Integral, 15
Calculate the following indefinite integrals.
[1] $\int \frac{(x^2-1)^2}{x^4}dx$
[2] $\int \frac{e^{3x}}{\sqrt{e^x+1}}dx$
[3] $\int \sin 2x\cos 3xdx$
[4] $\int x\ln (x+1)dx$
[5] $\int \frac{x}{(x+3)^2}dx$
2005 Today's Calculation Of Integral, 41
Evaluate
\[\int_0^a \sqrt{2ax-x^2}\ dx \ (a>0)\]
2007 District Olympiad, 3
Find all continuous functions $f : \mathbb R \to \mathbb R$ such that:
(a) $\lim_{x \to \infty}f(x)$ exists;
(b) $f(x) = \int_{x+1}^{x+2}f(t) \, dt$, for all $x \in \mathbb R$.