Found problems: 1687
2016 District Olympiad, 2
Let $ f:\mathbb{R}\longrightarrow (0,\infty ) $ be a continuous and periodic function having a period of $ 2, $ and such that the integral $ \int_0^2 \frac{f(1+x)}{f(x)} dx $ exists. Show that
$$ \int_0^2 \frac{f(1+x)}{f(x)} dx\ge 2, $$
with equality if and only if $ 1 $ is also a period of $ f. $
2005 VTRMC, Problem 6
Compute $\int^1_0\left((e-1)\sqrt{\ln(1+ex-x)}+e^{x^2}\right)dx$.
1987 Vietnam National Olympiad, 2
Let $ f : [0, \plus{}\infty) \to \mathbb R$ be a differentiable function. Suppose that $ \left|f(x)\right| \le 5$ and $ f(x)f'(x) \ge \sin x$ for all $ x \ge 0$. Prove that there exists $ \lim_{x\to\plus{}\infty}f(x)$.
2025 Romania EGMO TST, P1
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]
2007 Today's Calculation Of Integral, 204
Evaluate
\[\int_{0}^{1}\frac{x\ dx}{(x^{2}+x+1)^{\frac{3}{2}}}\]
2017 Hong Kong TST, 1
Given that $\{a_n\}$ is a sequence of integers satisfying the following condition for all positive integral values of $n$: $a_n+a_{n+1}=2a_{n+2}a_{n+3}+2016$. Find all possible values of $a_1$ and $a_2$
2007 Today's Calculation Of Integral, 203
Let $\alpha ,\ \beta$ be the distinct positive roots of the equation of $2x=\tan x$.
Evaluate the following definite integral.
\[\int_{0}^{1}\sin \alpha x\sin \beta x\ dx \]
2001 Vietnam Team Selection Test, 1
Let a sequence of integers $\{a_n\}$, $n \in \mathbb{N}$ be given, defined by
\[a_0 = 1, a_n= a_{n-1} + a_{[n/3]}\]
for all $n \in \mathbb{N}^{*}$.
Show that for all primes $p \leq 13$, there are infinitely many integer numbers $k$ such that $a_k$ is divided by $p$.
(Here $[x]$ denotes the integral part of real number $x$).
2012 Today's Calculation Of Integral, 821
Prove that : $\ln \frac{11}{27}<\int_{\frac 14}^{\frac 34} \frac{1}{\ln (1-x)}\ dx<\ln \frac{7}{15}.$
2005 Romania National Olympiad, 3
Let $f:[0,\infty)\to(0,\infty)$ a continous function such that $\lim_{n\to\infty} \int^x_0 f(t)dt$ exists and it is finite. Prove that
\[ \lim_{x\to\infty} \frac 1{\sqrt x} \int^x_0 \sqrt {f(t)} dt = 0. \]
[i]Radu Miculescu[/i]
2006 ISI B.Math Entrance Exam, 5
A domino is a $2$ by $1$ rectangle . For what integers $m$ and $n$ can we cover an $m*n$ rectangle with non-overlapping dominoes???
2005 Today's Calculation Of Integral, 49
For $x\geq 0$, Prove that $\int_0^x (t-t^2)\sin ^{2002} t \,dt<\frac{1}{2004\cdot 2005}$
2014 VJIMC, Problem 4
Let $0<a<b$ and let $f:[a,b]\to\mathbb R$ be a continuous function with $\int^b_af(t)dt=0$. Show that
$$\int^b_a\int^b_af(x)f(y)\ln(x+y)dxdy\le0.$$
1996 Putnam, 2
Prove the inequality for all positive integer $n$ :
\[ \left(\frac{2n-1}{e}\right)^{\frac{2n-1}{2}}<1\cdot 3\cdot 5\cdots (2n-1)<\left(\frac{2n+1}{e}\right)^{\frac{2n+1}{2}} \]
2007 ITest, 40
Let $S$ be the sum of all $x$ such that $1\leq x\leq 99$ and \[\{x^2\}=\{x\}^2.\] Compute $\lfloor S\rfloor$.
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}$.
2023 OMpD, 4
Let $n \geq 0$ be an integer and $f: [0, 1] \rightarrow \mathbb{R}$ an integrable function such that: $$\int^1_0f(x)dx = \int^1_0xf(x)dx = \int^1_0x^2f(x)dx = \ldots = \int^1_0x^nf(x)dx = 1$$ Prove that: $$\int_0^1f(x)^2dx \geq (n+1)^2$$
2014 PUMaC Number Theory A, 7
Find the number of positive integers $n \le 2014$ such that there exists integer $x$ that satisfies the condition that $\frac{x+n}{x-n}$ is an odd perfect square.
2009 Today's Calculation Of Integral, 409
Evaluate $ \int_0^1 \sqrt{\frac{x\plus{}\sqrt{x^2\plus{}1}}{x^2\plus{}1}}\ dx$.
2012 Today's Calculation Of Integral, 828
Find a function $f(x)$, which is differentiable and $f'(x) $ is continuous, such that $\int_0^x f(t)\cos (x-t)\ dt=xe^{2x}.$
2009 Today's Calculation Of Integral, 404
Evaluate $ \int_{ \minus{} \pi}^{\pi} \frac {\sin nx}{(1 \plus{} 2009^x)\sin x}\ dx\ (n\equal{}0,\ 1,\ 2,\ \cdots)$.
Today's calculation of integrals, 897
Find the volume $V$ of the solid formed by a rotation of the region enclosed by the curve $y=2^{x}-1$ and two lines $x=0,\ y=1$ around the $y$ axis.
2012 Today's Calculation Of Integral, 772
Given are three points $A(2,\ 0,\ 2),\ B(1,\ 1,\ 0),\ C(0,\ 0,\ 3)$ in the coordinate space. Find the volume of the solid of a triangle $ABC$ generated by a rotation about $z$-axis.
2007 AIME Problems, 12
The increasing geometric sequence $x_{0},x_{1},x_{2},\ldots$ consists entirely of integral powers of $3.$ Given that \[\sum_{n=0}^{7}\log_{3}(x_{n}) = 308\qquad\text{and}\qquad 56 \leq \log_{3}\left ( \sum_{n=0}^{7}x_{n}\right ) \leq 57,\] find $\log_{3}(x_{14}).$
1995 Flanders Math Olympiad, 2
How many values of $x\in\left[ 1,3 \right]$ are there, for which $x^2$ has the same decimal part as $x$?