Found problems: 1687
2009 Bulgaria National Olympiad, 6
Prove that if $ a_{1},a_{2},\ldots,a_{n}$, $ b_{1},b_{2},\ldots,b_{n}$ are arbitrary taken real numbers and $ c_{1},c_{2},\ldots,c_{n}$
are positive real numbers, than
$ \left(\sum_{i,j \equal{} 1}^{n}\frac {a_{i}a_{j}}{c_{i} \plus{} c_{j}}\right)\left(\sum_{i,j \equal{} 1}^{n}\frac {b_{i}b_{j}}{c_{i} \plus{} c_{j}}\right)\ge \left(\sum_{i,j \equal{} 1}^{n}\frac {a_{i}b_{j}}{c_{i} \plus{} c_{j}}\right)^{2}$.
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]
2003 Baltic Way, 10
A [i]lattice point[/i] in the plane is a point with integral coordinates. The[i] centroid[/i] of four points $(x_i,y_i )$, $i = 1, 2, 3, 4$, is the point $\left(\frac{x_1 +x_2 +x_3 +x_4}{4},\frac{y_1 +y_2 +y_3 +y_4 }{4}\right)$.
Let $n$ be the largest natural number for which there are $n$ distinct lattice points in the plane such that the centroid of any four of them is not a lattice point. Prove that $n = 12$.
2013 Stanford Mathematics Tournament, 7
The function $f(x)$ has the property that, for some real positive constant $C$, the expression \[\frac{f^{(n)}(x)}{n+x+C}\] is independent of $n$ for all nonnegative integers $n$, provided that $n+x+C\neq 0$. Given that $f'(0)=1$ and $\int_{0}^{1}f(x) \, dx = C+(e-2)$, determine the value of $C$.
Note: $f^{(n)}(x)$ is the $n$-th derivative of $f(x)$, and $f^{(0)}(x)$ is defined to be $f(x)$.
2010 Albania National Olympiad, 3
[b](a)[/b]Prove that every pentagon with integral coordinates has at least two vertices , whose respective coordinates have the same parity.
[b](b)[/b]What is the smallest area possible of pentagons with integral coordinates.
Albanian National Mathematical Olympiad 2010---12 GRADE Question 3.
2018 Ramnicean Hope, 1
Let be two nonzero real numbers $ a,b $ such that $ |a|\neq |b| $ and let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a function satisfying the functional relation
$$ af(x)+bf(-x)=(x^3+x)^5+\sin^5 x . $$
Calculate $ \int_{-2019}^{2019}f(x)dx . $
[i]Constantin Rusu[/i]
2011 Today's Calculation Of Integral, 758
Find the slope of a line passing through the point $(0,\ 1)$ with which the area of the part bounded by the line and the parabola $y=x^2$ is $\frac{5\sqrt{5}}{6}.$
2009 Today's Calculation Of Integral, 408
Evaluate $ \int_1^e \{(1 \plus{} x)e^x \plus{} (1 \minus{} x)e^{ \minus{} x}\}\ln x\ dx$.
2010 Today's Calculation Of Integral, 585
Evaluate $ \int_0^{\ln 2} (x\minus{}\ln 2)e^{\minus{}2\ln (1\plus{}e^x)\plus{}x\plus{}\ln 2}dx$.
2020 Jozsef Wildt International Math Competition, W3
Let $n \geq 2$ be an integer. Calculate$$\int \limits_{0}^{\frac{\pi}{2}}\frac{\sin x}{\sin^{2n-1}x+\cos^{2n-1}x}dx$$
2023 CMIMC Integration Bee, 11
\[\int_{-1}^1 \frac{1}{(8 + x^2)\sqrt{1-x^2}} \,\mathrm dx\]
[i]Proposed by Vlad Oleksenko[/i]
1995 AIME Problems, 10
What is the largest positive integer that is not the sum of a positive integral multiple of 42 and a positive composite integer?
2005 Putnam, B6
Let $S_n$ denote the set of all permutations of the numbers $1,2,\dots,n.$ For $\pi\in S_n,$ let $\sigma(\pi)=1$ if $\pi$ is an even permutation and $\sigma(\pi)=-1$ if $\pi$ is an odd permutation. Also, let $v(\pi)$ denote the number of fixed points of $\pi.$ Show that
\[ \sum_{\pi\in S_n}\frac{\sigma(\pi)}{v(\pi)+1}=(-1)^{n+1}\frac{n}{n+1}. \]
2005 Today's Calculation Of Integral, 64
Let $f(t)$ be the cubic polynomial for $t$ such that $\cos 3x=f(\cos x)$ holds for all real number $x$.
Evaluate
\[\int_0^1 \{f(t)\}^2 \sqrt{1-t^2}dt\]
2007 Today's Calculation Of Integral, 185
Evaluate the following integrals.
(1) $\int_{0}^{\frac{\pi}{4}}\frac{dx}{1+\sin x}.$
(2) $\int_{\frac{4}{3}}^{2}\frac{dx}{x^{2}\sqrt{x-1}}.$
Today's calculation of integrals, 894
Let $a$ be non zero real number. Find the area of the figure enclosed by the line $y=ax$, the curve $y=x\ln (x+1).$
1998 IberoAmerican Olympiad For University Students, 6
Take the following differential equation:
\[3(3+x^2)\frac{dx}{dt}=2(1+x^2)^2e^{-t^2}\]
If $x(0)\leq 1$, prove that there exists $M>0$ such that $|x(t)|<M$ for all $t\geq 0$.
2011 Today's Calculation Of Integral, 715
Find the differentiable function $f(x)$ with $f(0)\neq 0$ satisfying $f(x+y)=f(x)f'(y)+f'(x)f(y)$ for all real numbers $x,\ y$.
2022 IMC, 1
Let $f: [0,1] \to (0, \infty)$ be an integrable function such that $f(x)f(1-x) = 1$ for all $x\in [0,1]$. Prove that $\int_0^1f(x)dx \geq 1$.
Today's calculation of integrals, 873
Let $a,\ b$ be positive real numbers. Consider the circle $C_1: (x-a)^2+y^2=a^2$ and the ellipse $C_2: x^2+\frac{y^2}{b^2}=1.$
(1) Find the condition for which $C_1$ is inscribed in $C_2$.
(2) Suppose $b=\frac{1}{\sqrt{3}}$ and $C_1$ is inscribed in $C_2$. Find the coordinate $(p,\ q)$ of the point of tangency in the first quadrant for $C_1$ and $C_2$.
(3) Under the condition in (1), find the area of the part enclosed by $C_1,\ C_2$ for $x\geq p$.
60 point
2006 VJIMC, Problem 4
Let $f:[0,\infty)\to\mathbb R$ ba a strictly convex continuous function such that
$$\lim_{x\to+\infty}\frac{f(x)}x=+\infty.$$Prove that the improper integral $\int^{+\infty}_0\sin(f(x))\text dx$ is convergent but not absolutely convergent.
2012 Gulf Math Olympiad, 4
Fawzi cuts a spherical cheese completely into (at least three) slices of equal thickness. He starts at one end, making successive parallel cuts, working through the cheese until the slicing is complete. The discs exposed by the first two cuts have integral areas.
[list](i) Prove that all the discs that he cuts have integral areas.
(ii) Prove that the original sphere had integral surface area if, and only if, the area of the second disc that he exposes is even.[/list]
2012 Today's Calculation Of Integral, 813
Let $a$ be a real number. Find the minimum value of $\int_0^1 |ax-x^3|dx$.
How many solutions (including University Mathematics )are there for the problem?
Any advice would be appreciated.
2010 Putnam, B5
Is there a strictly increasing function $f:\mathbb{R}\to\mathbb{R}$ such that $f'(x)=f(f(x))$ for all $x?$
2011 Today's Calculation Of Integral, 689
Let $C: y=x^2+ax+b$ be a parabola passing through the point $(1,\ -1)$. Find the minimum volume of the figure enclosed by $C$ and the $x$ axis by a rotation about the $x$ axis.
Proposed by kunny