Found problems: 1687
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]
1986 AMC 12/AHSME, 29
Two of the altitudes of the scalene triangle $ABC$ have length $4$ and $12$. If the length of the third altitude is also an integer, what is the biggest it can be?
$ \textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ \text{none of these} $
2010 Harvard-MIT Mathematics Tournament, 8
Let $f(n)=\displaystyle\sum_{k=2}^\infty \dfrac{1}{k^n\cdot k!}.$ Calculate $\displaystyle\sum_{n=2}^\infty f(n)$.
2009 Today's Calculation Of Integral, 511
Suppose that $ f(x),\ g(x)$ are differential fuctions and their derivatives are continuous.
Find $ f(x),\ g(x)$ such that $ f(x)\equal{}\frac 12\minus{}\int_0^x \{f'(t)\plus{}g(t)\}\ dt\ \ g(x)\equal{}\sin x\minus{}\int_0^{\pi} \{f(t)\minus{}g'(t)\}\ dt$.
2003 Romania National Olympiad, 3
Let be a continuous function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that has the property that
$$ xf(x)\ge \int_0^x f(t)dt , $$
for all real numbers $ x. $ Prove that
[b]a)[/b] the mapping $ x\mapsto \frac{1}{x}\int_0^x f(t) dt $ is nondecreasing on the restrictions $ \mathbb{R}_{<0 } $ and $ \mathbb{R}_{>0 } . $
[b]b)[/b] if $ \int_x^{x+1} f(t)dt=\int_{x-1}^x f(t)dt , $ for any real number $ x, $ then $ f $ is constant.
[i]Mihai Piticari[/i]
2005 Today's Calculation Of Integral, 29
Let $a$ be a real number.
Evaluate
\[\int _{-\pi+a}^{3\pi+a} |x-a-\pi|\sin \left(\frac{x}{2}\right)dx\]
2010 Today's Calculation Of Integral, 635
Suppose that a function $f(x)$ defined in $-1<x<1$ satisfies the following properties (i) , (ii), (iii).
(i) $f'(x)$ is continuous.
(ii) When $-1<x<0,\ f'(x)<0,\ f'(0)=0$, when $0<x<1,\ f'(x)>0$.
(iii) $f(0)=-1$
Let $F(x)=\int_0^x \sqrt{1+\{f'(t)\}^2}dt\ (-1<x<1)$. If $F(\sin \theta)=c\theta\ (c :\text{constant})$ holds for $-\frac{\pi}{2}<\theta <\frac{\pi}{2}$, then find $f(x)$.
[i]1975 Waseda University entrance exam/Science and Technology[/i]
2005 Today's Calculation Of Integral, 26
Evaluate
\[{{\int_{e^{e^{e}}}^{e^{e^{e^{e}}}}} \frac{dx}{x\ln x\cdot \ln (\ln x)\cdot \ln \{\ln (\ln x)\}}}\]
2010 Today's Calculation Of Integral, 642
Evaluate
\[\int_0^{\frac{\pi}{6}} \frac{(\tan ^ 2 2x)\sqrt{\cos 2x}+2}{(\cos ^ 2 x)\sqrt{\cos 2x}}dx.\]
Own
2011 Today's Calculation Of Integral, 766
Let $f(x)$ be a continuous function defined on $0\leq x\leq \pi$ and satisfies $f(0)=1$ and
\[\left\{\int_0^{\pi} (\sin x+\cos x)f(x)dx\right\}^2=\pi \int_0^{\pi}\{f(x)\}^2dx.\]
Evaluate $\int_0^{\pi} \{f(x)\}^3dx.$
2012 Uzbekistan National Olympiad, 2
For any positive integers $n$ and $m$ satisfying the equation $n^3+(n+1)^3+(n+2)^3=m^3$, prove that $4\mid n+1$.
2005 Today's Calculation Of Integral, 73
Find the minimum value of $\int_0^{\pi} (a\sin x+b\sin 2x+c\sin 3x-x)^2\ dx$
2009 Today's Calculation Of Integral, 477
Suppose that $ P_1(x)\equal{}\frac{d}{dx}(x^2\minus{}1),\ P_2(x)\equal{}\frac{d^2}{dx^2}(x^2\minus{}1)^2,\ P_3(x)\equal{}\frac{d^3}{dx^3}(x^2\minus{}1)^3$.
Find all possible values for which $ \int_{\minus{}1}^1 P_k(x)P_l(x)\ dx\ (k\equal{}1,\ 2,\ 3,\ l\equal{}1,\ 2,\ 3)$ can be valued.
2011 Today's Calculation Of Integral, 694
Prove the following inequality:
\[\int_1^e \frac{(\ln x)^{2009}}{x^2}dx>\frac{1}{2010\cdot 2011\cdot2012}\]
created by kunny
1952 Miklós Schweitzer, 9
Let $ C$ denote the set of functions $ f(x)$, integrable (according to either Riemann or Lebesgue) on $ (a,b)$, with $ 0\le f(x)\le1$. An element $ \phi(x)\in C$ is said to be an "extreme point" of $ C$ if it can not be represented as the arithmetical mean of two different elements of $ C$. Find the extreme points of $ C$ and the functions $ f(x)\in C$ which can be obtained as "weak limits" of extreme points $ \phi_n(x)$ of $ C$.
(The latter means that
$ \lim_{n\to \infty}\int_a^b \phi_n(x)h(x)\,dx\equal{}\int_a^bf(x)h(x)\,dx$
holds for every integrable function $ h(x)$.)
2009 Today's Calculation Of Integral, 431
Consider the function $ f(\theta) \equal{} \int_0^1 |\sqrt {1 \minus{} x^2} \minus{} \sin \theta|dx$ in the interval of $ 0\leq \theta \leq \frac {\pi}{2}$.
(1) Find the maximum and minimum values of $ f(\theta)$.
(2) Evaluate $ \int_0^{\frac {\pi}{2}} f(\theta)\ d\theta$.
1980 Putnam, A5
Let $P(t)$ be a nonconstant polynomial with real coefficients. Prove that the system of simultaneous equations
$$ \int_{0}^{x} P(t)\sin t \, dt =0, \;\;\;\; \int_{0}^{x} P(t) \cos t \, dt =0 $$
has only finitely many solutions $x.$
2003 China Western Mathematical Olympiad, 1
The sequence $ \{a_n\}$ satisfies $ a_0 \equal{} 0, a_{n \plus{} 1} \equal{} ka_n \plus{} \sqrt {(k^2 \minus{} 1)a_n^2 \plus{} 1}, n \equal{} 0, 1, 2, \ldots$, where $ k$ is a fixed positive integer. Prove that all the terms of the sequence are integral and that $ 2k$ divides $ a_{2n}, n \equal{} 0, 1, 2, \ldots$.
2012 Today's Calculation Of Integral, 824
In the $xy$-plane, for $a>1$ denote by $S(a)$ the area of the figure bounded by the curve $y=(a-x)\ln x$ and the $x$-axis.
Find the value of integer $n$ for which $\lim_{a\rightarrow \infty} \frac{S(a)}{a^n\ln a}$ is non-zero real number.
1991 Putnam, A1
The rectangle with vertices $(0,0)$, $(0,3)$, $(2,0)$ and $(2,3)$ is rotated clockwise through a right angle about the point $(2,0)$, then about $(5,0)$, then about $(7,0$), and finally about $(10,0)$. The net effect is to translate it a distance $10$ along the $x$-axis. The point initially at $(1,1)$ traces out a curve. Find the area under this curve (in other words, the area of the region bounded by the curve, the $x$-axis and the lines parallel to the $y$-axis through $(1,0)$ and $(11,0)$).
2011 Today's Calculation Of Integral, 743
Evaluate $\int_0^{\frac{\pi}{2}} \ln (1+\sqrt[3]{\sin \theta})\cos \theta\ d\theta.$
2005 Putnam, A6
Let $n$ be given, $n\ge 4,$ and suppose that $P_1,P_2,\dots,P_n$ are $n$ randomly, independently and uniformly, chosen points on a circle. Consider the convex $n$-gon whose vertices are the $P_i.$ What is the probability that at least one of the vertex angles of this polygon is acute.?
2005 Today's Calculation Of Integral, 21
[1] Tokyo Univ. of Science: $\int \frac{\ln x}{(x+1)^2}dx$
[2] Saitama Univ.: $\int \frac{5}{3\sin x+4\cos x}dx$
[3] Yokohama City Univ.: $\int_1^{\sqrt{3}} \frac{1}{\sqrt{x^2+1}}dx$
[4] Daido Institute of Technology: $\int_0^{\frac{\pi}{2}} \frac{\sin ^ 3 x}{\sin x +\cos x}dx$
[5] Gunma Univ.: $\int_0^{\frac{3\pi}{4}} \{(1+x)\sin x+(1-x)\cos x\}dx$
2012 Today's Calculation Of Integral, 801
Answer the following questions:
(1) Let $f(x)$ be a function such that $f''(x)$ is continuous and $f'(a)=f'(b)=0$ for some $a<b$.
Prove that $f(b)-f(a)=\int_a^b \left(\frac{a+b}{2}-x\right)f''(x)dx$.
(2) Consider the running a car on straight road. After a car which is at standstill at a traffic light started at time 0, it stopped again at the next traffic light apart a distance $L$ at time $T$. During the period, prove that there is an instant for which the absolute value of the acceleration of the car is more than or equal to $\frac{4L}{T^2}.$
2011 Today's Calculation Of Integral, 737
Let $a,\ b$ real numbers such that $a>1,\ b>1.$
Prove the following inequality.
\[\int_{-1}^1 \left(\frac{1+b^{|x|}}{1+a^{x}}+\frac{1+a^{|x|}}{1+b^{x}}\right)\ dx<a+b+2\]