Found problems: 913
2005 Today's Calculation Of Integral, 18
Calculate the following indefinite integrals.
[1] $\int (\sin x+\cos x)^4 dx$
[2] $\int \frac{e^{2x}}{e^x+1}dx$
[3] $\int \sin ^ 4 xdx$
[4] $\int \sin 6x\cos 2xdx$
[5] $\int \frac{x^2}{\sqrt{(x+1)^3}}dx$
2011 Pre-Preparation Course Examination, 6
We call a subset $S$ of vertices of graph $G$, $2$-dominating, if and only if for every vertex $v\notin S,v\in G$, $v$ has at least two neighbors in $S$. prove that every $r$-regular $(r\ge3)$ graph has a $2$-dominating set with size at most $\frac{n(1+\ln(r))}{r}$.(15 points)
time of this exam was 3 hours
2009 Today's Calculation Of Integral, 450
Let $ a,\ b$ be postive real numbers. Find $ \lim_{n\to\infty} \sum_{k\equal{}1}^n \frac{n}{(k\plus{}an)(k\plus{}bn)}.$
1983 IMO Longlists, 32
Let $a, b, c$ be positive real numbers and let $[x]$ denote the greatest integer that does not exceed the real number $x$. Suppose that $f$ is a function defined on the set of non-negative integers $n$ and taking real values such that $f(0) = 0$ and
\[f(n) \leq an + f([bn]) + f([cn]), \qquad \text{ for all } n \geq 1.\]
Prove that if $b + c < 1$, there is a real number $k$ such that
\[f(n) \leq kn \qquad \text{ for all } n \qquad (1)\]
while if $b + c = 1$, there is a real number $K$ such that $f(n) \leq K n \log_2 n$ for all $n \geq 2$. Show that if $b + c = 1$, there may not be a real number $k$ that satisfies $(1).$
2004 AMC 12/AHSME, 16
The set of all real numbers $ x$ for which
\[ \log_{2004}(\log_{2003}(\log_{2002}(\log_{2001}{x})))
\]is defined is $ \{x|x > c\}$. What is the value of $ c$?
$ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 2001^{2002} \qquad \textbf{(C)}\ 2002^{2003} \qquad \textbf{(D)}\ 2003^{2004} \qquad \textbf{(E)}\ 2001^{2002^{2003}}$
1992 IMO Longlists, 74
Let $S = \{\frac{\pi^n}{1992^m} | m,n \in \mathbb Z \}.$ Show that every real number $x \geq 0$ is an accumulation point of $S.$
2009 Vietnam National Olympiad, 4
Let $ a$, $ b$, $ c$ be three real numbers. For each positive integer number $ n$, $ a^n \plus{} b^n \plus{} c^n$ is an integer number. Prove that there exist three integers $ p$, $ q$, $ r$ such that $ a$, $ b$, $ c$ are the roots of the equation $ x^3 \plus{} px^2 \plus{} qx \plus{} r \equal{} 0$.
1949-56 Chisinau City MO, 39
Solve the equation: $\log_{x} 2 \cdot \log_{2x} 2 = \log_{4x} 2$.
2010 IMC, 2
Compute the sum of the series
$\sum_{k=0}^{\infty} \frac{1}{(4k+1)(4k+2)(4k+3)(4k+4)} = \frac{1}{1\cdot2\cdot3\cdot4} + \frac{1}{5\cdot6\cdot7\cdot8} + ...$
2005 Today's Calculation Of Integral, 7
Calculate the following indefinite integrals.
[1] $\int \sqrt{x}(\sqrt{x}+1)^2 dx$
[2] $\int (e^x+2e^{x+1}-3e^{x+2})dx$
[3] $\int (\sin ^2 x+\cos x)\sin x dx$
[4] $\int x\sqrt{2-x} dx$
[5] $\int x\ln x dx$
2008 Putnam, A6
Prove that there exists a constant $ c>0$ such that in every nontrivial finite group $ G$ there exists a sequence of length at most $ c\ln |G|$ with the property that each element of $ G$ equals the product of some subsequence. (The elements of $ G$ in the sequence are not required to be distinct. A [i]subsequence[/i] of a sequence is obtained by selecting some of the terms, not necessarily consecutive, without reordering them; for example, $ 4,4,2$ is a subesequence of $ 2,4,6,4,2,$ but $ 2,2,4$ is not.)
2013 Today's Calculation Of Integral, 860
For a function $f(x)\ (x\geq 1)$ satisfying $f(x)=(\log_e x)^2-\int_1^e \frac{f(t)}{t}dt$, answer the questions as below.
(a) Find $f(x)$ and the $y$-coordinate of the inflection point of the curve $y=f(x)$.
(b) Find the area of the figure bounded by the tangent line of $y=f(x)$ at the point $(e,\ f(e))$, the curve $y=f(x)$ and the line $x=1$.
1995 Canada National Olympiad, 2
Let $\{a,b,c\}\in \mathbb{R}^{+}$. Prove that $a^a b^b c^c \ge (abc)^{\frac{a+b+c}{3}}$.
2005 Today's Calculation Of Integral, 38
Let $a$ be a constant number such that $0<a<1$ and $V(a)$ be the volume formed by the revolution of the figure
which is enclosed by the curve $y=\ln (x-a)$, the $x$-axis and two lines $x=1,x=3$ about the $x$-axis.
If $a$ varies in the range of $0<a<1$, find the minimum value of $V(a)$.
2019 Jozsef Wildt International Math Competition, W. 6
Compute$$\int \limits_{\frac{\pi}{6}}^{\frac{\pi}{4}}\frac{(1+\ln x)\cos x+x\sin x\ln x}{\cos^2 x + x^2 \ln^2 x}dx$$
2010 Today's Calculation Of Integral, 630
Evaluate $\int_0^{\infty} \frac{\ln (1+e^{4x})}{e^x}dx.$
2014 NIMO Summer Contest, 6
Suppose $x$ is a random real number between $1$ and $4$, and $y$ is a random real number between $1$ and $9$. If the expected value of \[ \left\lceil \log_2 x \right\rceil - \left\lfloor \log_3 y \right\rfloor \] can be expressed as $\frac mn$ where $m$ and $n$ are relatively prime positive integers, compute $100m + n$.
[i]Proposed by Lewis Chen[/i]
1951 AMC 12/AHSME, 45
If you are given $ \log 8 \approx .9031$ and $ \log 9 \approx .9542$, then the only logarithm that cannot be found without the use of tables is:
$ \textbf{(A)}\ \log 17 \qquad\textbf{(B)}\ \log \frac {5}{4} \qquad\textbf{(C)}\ \log 15 \qquad\textbf{(D)}\ \log 600 \qquad\textbf{(E)}\ \log .4$
2012 Today's Calculation Of Integral, 848
Evaluate $\int_0^{\frac {\pi}{4}} \frac {\sin \theta -2\ln \frac{1-\sin \theta}{\cos \theta}}{(1+\cos 2\theta)\sqrt{\ln \frac{1+\sin \theta}{\cos \theta}}}d\theta .$
2011 Math Prize For Girls Problems, 4
If $x > 10$, what is the greatest possible value of the expression
\[
{( \log x )}^{\log \log \log x} - {(\log \log x)}^{\log \log x} ?
\]
All the logarithms are base 10.
2014 District Olympiad, 3
Let $p$ and $n$ be positive integers, with $p\geq2$, and let $a$ be a real number such that $1\leq a<a+n\leq p$. Prove that the set
\[ \mathcal {S}=\left\{\left\lfloor \log_{2}x\right\rfloor +\left\lfloor \log_{3}x\right\rfloor +\cdots+\left\lfloor \log_{p}x\right\rfloor\mid x\in\mathbb{R},a\leq x\leq a+n\right\} \]
has exactly $n+1$ elements.
2003 CentroAmerican, 3
Let $a$ and $b$ be positive integers with $a>1$ and $b>2$. Prove that $a^b+1\ge b(a+1)$ and determine when there is inequality.
2014 Harvard-MIT Mathematics Tournament, 3
Let \[ A = \frac{1}{6}((\log_2(3))^3-(\log_2(6))^3-(\log_2(12))^3+(\log_2(24))^3) \].
Compute $2^A$.
2005 USAMO, 6
For $m$ a positive integer, let $s(m)$ be the sum of the digits of $m$. For $n\ge 2$, let $f(n)$ be the minimal $k$ for which there exists a set $S$ of $n$ positive integers such that $s\left(\sum_{x\in X} x\right)=k$ for any nonempty subset $X\subset S$. Prove that there are constants $0<C_1<C_2$ with
\[C_1 \log_{10} n \le f(n) \le C_2 \log_{10} n.\]
2010 Today's Calculation Of Integral, 578
Find the range of $ k$ for which the following inequality holds for $ 0\leq x\leq 1$.
\[ \int_0^x \frac {dt}{\sqrt {(3 \plus{} t^2)^3}}\geq k\int _0^x \frac {dt}{\sqrt {3 \plus{} t^2}}\]
If necessary, you may use $ \ln 3 \equal{} 1.10$.