Found problems: 2215
2005 Today's Calculation Of Integral, 34
Let $p$ be a constant number such that $0<p<1$.
Evaluate
\[\sum_{k=0}^{2004} \frac{p^k (1-p)^{2004-k}}{\displaystyle \int_0^1 x^k (1-x)^{2004-k} dx}\]
2010 Today's Calculation Of Integral, 535
Let $ C$ be the parameterized curve for a given positive number $ r$ and $ 0\leq t\leq \pi$,
$ C: \left\{\begin{array}{ll} x \equal{} 2r(t \minus{} \sin t\cos t) & \quad \\
y \equal{} 2r\sin ^ 2 t & \quad \end{array} \right.$
When the point $ P$ moves on the curve $ C$,
(1) Find the magnitude of acceleralation of the point $ P$ at time $ t$.
(2) Find the length of the locus by which the point $ P$ sweeps for $ 0\leq t\leq \pi$.
(3) Find the volume of the solid by rotation of the region bounded by the curve $ C$ and the $ x$-axis about the $ x$-axis.
Edited.
2007 Princeton University Math Competition, 8
What is the area of the region defined by $x^2+3y^2 \le 4$ and $y^2+3x^2 \le 4$?
2024 CMIMC Integration Bee, 14
\[\int_1^\infty \frac{\lfloor x \rfloor}{9x^3-x}\mathrm dx\]
[i]Proposed by Robert Trosten and Connor Gordon[/i]
2021 JHMT HS, 7
In three-dimensional space, let $\mathcal{S}$ be the surface consisting of all points $(x, y, 0)$ satisfying $x^2 + 1 \leq y \leq 2,$ and let $A$ be the point $(0, 0, 900).$ Compute the volume of the solid obtained by taking the union of all line segments with endpoints in $\mathcal{S} \cup \{A\}.$
2009 VTRMC, Problem 3
Define $f(x)=\int^x_0\int^x_0e^{u^2v^2}dudv$. Calculate $2f''(2)+f'(2)$.
2014 NIMO Problems, 6
Let $N=10^6$. For which integer $a$ with $0 \leq a \leq N-1$ is the value of \[\binom{N}{a+1}-\binom{N}{a}\] maximized?
[i]Proposed by Lewis Chen[/i]
2005 Today's Calculation Of Integral, 58
Let $f(x)=\frac{e^x}{e^x+1}$
Prove the following equation.
\[\int_a^b f(x)dx+\int_{f(a)}^{f(b)} f^{-1}(x)dx=bf(b)-af(a)\]
1982 IMO Shortlist, 15
Show that
\[ \frac{1 - s^a}{1 - s} \leq (1 + s)^{a-1}\]
holds for every $1 \neq s > 0$ real and $0 < a \leq 1$ rational.
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 $\lim_{n\to\infty} n\left\{\int_0^1 f(x)\ dx-\frac{1}{n}\sum_{k=0}^{n-1} f\left(\frac{k}{n}\right)\right\}.$
2009 ISI B.Stat Entrance Exam, 4
A sequence is called an [i]arithmetic progression of the first order[/i] if the differences of the successive terms are constant. It is called an [i]arithmetic progression of the second order[/i] if the differences of the successive terms form an arithmetic progression of the first order. In general, for $k\geq 2$, a sequence is called an [i]arithmetic progression of the $k$-th order[/i] if the differences of the successive terms form an arithmetic progression of the $(k-1)$-th order.
The numbers
\[4,6,13,27,50,84\]
are the first six terms of an arithmetic progression of some order. What is its least possible order? Find a formula for the $n$-th term of this progression.
2011 China Team Selection Test, 2
Let $\ell$ be a positive integer, and let $m,n$ be positive integers with $m\geq n$, such that $A_1,A_2,\cdots,A_m,B_1,\cdots,B_m$ are $m+n$ pairwise distinct subsets of the set $\{1,2,\cdots,\ell\}$. It is known that $A_i\Delta B_j$ are pairwise distinct, $1\leq i\leq m, 1\leq j\leq n$, and runs over all nonempty subsets of $\{1,2,\cdots,\ell\}$. Find all possible values of $m,n$.
2008 Iran MO (3rd Round), 1
Prove that for $ n > 0$ and $ a\neq0$ the polynomial $ p(z) \equal{} az^{2n \plus{} 1} \plus{} bz^{2n} \plus{} \bar bz \plus{} \bar a$ has a root on unit circle
2009 German National Olympiad, 6
Let a sequences: $ x_0\in [0;1],x_{n\plus{}1}\equal{}\frac56\minus{}\frac43 \Big|x_n\minus{}\frac12\Big|$. Find the "best" $ |a;b|$ so that for all $ x_0$ we have $ x_{2009}\in [a;b]$
2010 Today's Calculation Of Integral, 528
Consider a function $ f(x)\equal{}xe^{\minus{}x^3}$ defined on any real numbers.
(1) Examine the variation and convexity of $ f(x)$ to draw the garph of $ f(x)$.
(2) For a positive number $ C$, let $ D_1$ be the region bounded by $ y\equal{}f(x)$, the $ x$-axis and $ x\equal{}C$. Denote $ V_1(C)$ the volume obtained by rotation of $ D_1$ about the $ x$-axis. Find $ \lim_{C\rightarrow \infty} V_1(C)$.
(3) Let $ M$ be the maximum value of $ y\equal{}f(x)$ for $ x\geq 0$. Denote $ D_2$ the region bounded by $ y\equal{}f(x)$, the $ y$-axis and $ y\equal{}M$.
Find the volume $ V_2$ obtained by rotation of $ D_2$ about the $ y$-axis.
MIPT Undergraduate Contest 2019, 2.2
Petya and Vasya are playing the following game. Petya chooses a non-negative random value $\xi$ with expectation $\mathbb{E} [\xi ] = 1$, after which Vasya chooses his own value $\eta$ with expectation $\mathbb{E} [\eta ] = 1$ without reference to the value of $\xi$. For which maximal value $p$ can Petya choose a value $\xi$ in such a way that for any choice of Vasya's $\eta$, the inequality $\mathbb{P}[\eta \geq \xi ] \leq p$ holds?
2012 Today's Calculation Of Integral, 825
Answer the following questions.
(1) For $x\geq 0$, show that $x-\frac{x^3}{6}\leq \sin x\leq x.$
(2) For $x\geq 0$, show that $\frac{x^3}{3}-\frac{x^5}{30}\leq \int_0^x t\sin t\ dt\leq \frac{x^3}{3}.$
(3) Find the limit
\[\lim_{x\rightarrow 0} \frac{\sin x-x\cos x}{x^3}.\]
2009 Today's Calculation Of Integral, 452
Let $ a,\ b$ are postive constant numbers.
(1) Differentiate $ \ln (x\plus{}\sqrt{x^2\plus{}a})\ (x>0).$
(2) For $ a\equal{}\frac{4b^2}{(e\minus{}e^{\minus{}1})^2}$, evaluate $ \int_0^b \frac{1}{\sqrt{x^2\plus{}a}}\ dx.$
2009 Today's Calculation Of Integral, 438
Evaluate $ \int_{\sqrt{2}\minus{}1}^{\sqrt{2}\plus{}1} \frac{x^4\plus{}x^2\plus{}2}{(x^2\plus{}1)^2}\ dx.$
2010 Contests, 522
Find $ \lim_{a\rightarrow{\infty}} \frac{1}{a^2}\int_0^a \ln (1\plus{}e^x)dx$.
2018 VJIMC, 4
Compute the integral
\[\iint_{\mathbb{R}^2} \left(\frac{1-e^{-xy}}{xy}\right)^2 e^{-x^2-y^2} dx dy.\]
1971 Canada National Olympiad, 5
Let \[ p(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x+a_0, \] where the coefficients $a_i$ are integers. If $p(0)$ and $p(1)$ are both odd, show that $p(x)$ has no integral roots.
2004 Harvard-MIT Mathematics Tournament, 6
For $x>0$, let $f(x)=x^x$. Find all values of $x$ for which $f(x)=f'(x)$.
2020 Jozsef Wildt International Math Competition, W23
Prove that
$$\int^{\frac\pi3}_{\frac\pi6}\frac u{\sin u}du=\frac83\sum_{k=0}^\infty\frac{(-1)^k}{3^k(2k+1)^2}+\frac{\pi\ln3}{3\sqrt3}-\frac{4C}3+\frac\pi6\ln\left(2+\sqrt3\right)-\operatorname{Im}\left(\frac2{\sqrt3}\operatorname{Li}_2\left(\frac{1-i\sqrt3}2\right)-\frac2{\sqrt3}\operatorname{Li}_2\left(\frac{\sqrt3-i}{2\sqrt3}\right)\right)$$
where as usual
$$\operatorname{Li}_2(z)=-\int^z_0\frac{\ln(1-t)}tdt,z\in\mathbb C\setminus[1,\infty)$$
and $C=\sum_{k=0}^\infty\frac{(-1)^k}{(2k+1)^2}$ is the Catalan constant.
[i]Proposed by Paolo Perfetti[/i]
1989 IMO Shortlist, 3
Ali Barber, the carpet merchant, has a rectangular piece of carpet whose dimensions are unknown. Unfortunately, his tape measure is broken and he has no other measuring instruments. However, he finds that if he lays it flat on the floor of either of his storerooms, then each corner of the carpet touches a different wall of that room. He knows that the sides of the carpet are integral numbers of feet and that his two storerooms have the same (unknown) length, but widths of 38 feet and 50 feet respectively. What are the carpet dimensions?