Found problems: 894
2014 Contests, 3
Let \[ A = \frac{1}{6}((\log_2(3))^3-(\log_2(6))^3-(\log_2(12))^3+(\log_2(24))^3) \].
Compute $2^A$.
2007 Moldova National Olympiad, 11.2
Define $a_{n}$ as satisfying: $\left(1+\frac{1}{n}\right)^{n+a_{n}}=e$.
Find $\lim_{n\rightarrow\infty}a_{n}$.
1985 AMC 12/AHSME, 24
A non-zero digit is chosen in such a way that the probability of choosing digit $ d$ is $ \log_{10}(d\plus{}1) \minus{} \log_{10} d$. The probability that the digit $ 2$ is chosen is exactly $ \frac12$ the probability that the digit chosen is in the set
$ \textbf{(A)}\ \{2,3\} \qquad \textbf{(B)}\ \{3,4\} \qquad \textbf{(C)}\ \{4,5,6,7,8\} \qquad \textbf{(D)}\ \{5,6,7,8,9\} \qquad \textbf{(E)}\ \{4,5,6,7,8,9\}$
2012 China Western Mathematical Olympiad, 2
Define a sequence $\{a_n\}$ by\[a_0=\frac{1}{2},\ a_{n+1}=a_{n}+\frac{a_{n}^2}{2012}, (n=0,\ 1,\ 2,\ \cdots),\] find integer $k$ such that $a_{k}<1<a_{k+1}.$
(September 29, 2012, Hohhot)
2013 ELMO Shortlist, 5
Let $a,b,c$ be positive reals satisfying $a+b+c = \sqrt[7]{a} + \sqrt[7]{b} + \sqrt[7]{c}$. Prove that $a^a b^b c^c \ge 1$.
[i]Proposed by Evan Chen[/i]
1990 AMC 12/AHSME, 23
If $x,y>0$, $\log_yx+\log_xy=\frac{10}{3}$ and $xy=144$, then $\frac{x+y}{2}=$
$ \textbf{(A)}\ 12\sqrt{2} \qquad\textbf{(B)}\ 13\sqrt{3} \qquad\textbf{(C)}\ 24 \qquad\textbf{(D)}\ 30 \qquad\textbf{(E)}\ 36 $
2012 Postal Coaching, 3
Given an integer $n\ge 2$, prove that
\[\lfloor \sqrt n \rfloor + \lfloor \sqrt[3]n\rfloor + \cdots +\lfloor \sqrt[n]n\rfloor = \lfloor \log_2n\rfloor + \lfloor \log_3n\rfloor + \cdots +\lfloor \log_nn\rfloor\].
[hide="Edit"] Thanks to shivangjindal for pointing out the mistake (and sorry for the late edit)[/hide]
1956 AMC 12/AHSME, 18
If $ 10^{2y} \equal{} 25$, then $ 10^{ \minus{} y}$ equals:
$ \textbf{(A)}\ \minus{} \frac {1}{5} \qquad\textbf{(B)}\ \frac {1}{625} \qquad\textbf{(C)}\ \frac {1}{50} \qquad\textbf{(D)}\ \frac {1}{25} \qquad\textbf{(E)}\ \frac {1}{5}$
2012 ELMO Shortlist, 2
Determine whether it's possible to cover a $K_{2012}$ with
a) 1000 $K_{1006}$'s;
b) 1000 $K_{1006,1006}$'s.
[i]David Yang.[/i]
2005 Today's Calculation Of Integral, 3
Calculate the following indefinite integrals.
[1] $\int \sin x\sin 2x dx$
[2] $\int \frac{e^{2x}}{e^x-1}dx$
[3] $\int \frac{\tan ^2 x}{\cos ^2 x}dx$
[4] $\int \frac{e^x+e^{-x}}{e^x-e^{-x}}dx$
[5] $\int \frac{e^x}{e^x+1}dx$
2012 Romania Team Selection Test, 1
Prove that for any positive integer $n\geq 2$ we have that \[\sum_{k=2}^n \lfloor \sqrt[k]{n}\rfloor=\sum_{k=2}^n\lfloor\log_{k}n\rfloor.\]
2012 ELMO Shortlist, 6
Consider a directed graph $G$ with $n$ vertices, where $1$-cycles and $2$-cycles are permitted. For any set $S$ of vertices, let $N^{+}(S)$ denote the out-neighborhood of $S$ (i.e. set of successors of $S$), and define $(N^{+})^k(S)=N^{+}((N^{+})^{k-1}(S))$ for $k\ge2$.
For fixed $n$, let $f(n)$ denote the maximum possible number of distinct sets of vertices in $\{(N^{+})^k(X)\}_{k=1}^{\infty}$, where $X$ is some subset of $V(G)$. Show that there exists $n>2012$ such that $f(n)<1.0001^n$.
[i]Linus Hamilton.[/i]
2009 Harvard-MIT Mathematics Tournament, 2
Let $S$ be the sum of all the real coefficients of the expansion of $(1+ix)^{2009}$. What is $\log_2(S)$?
2012 ELMO Shortlist, 2
Determine whether it's possible to cover a $K_{2012}$ with
a) 1000 $K_{1006}$'s;
b) 1000 $K_{1006,1006}$'s.
[i]David Yang.[/i]
2010 Today's Calculation Of Integral, 638
Let $(a,\ b)$ be a point on the curve $y=\frac{x}{1+x}\ (x\geq 0).$ Denote $U$ the volume of the figure enclosed by the curve , the $x$ axis and the line $x=a$, revolved around the the $x$ axis and denote $V$ the volume of the figure enclosed by the curve , the $y$ axis and th line $y=b$, revolved around the $y$ axis. What's the relation of $U$ and $V?$
1978 Chuo university entrance exam/Science and Technology
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.
2002 Moldova National Olympiad, 2
Let $ a,b,c\in \mathbb R$ such that $ a\ge b\ge c > 1$. Prove the inequality:
$ \log_c\log_c b \plus{} \log_b\log_b a \plus{} \log_a\log_a c\geq 0$
1991 Arnold's Trivium, 5
Calculate the $100$th derivative of the function
\[\frac{1}{x^2+3x+2}\]
at $x=0$ with $10\%$ relative error.
Today's calculation of integrals, 848
Evaluate $\int_0^{\frac {\pi}{4}} \frac {\sin \theta -2\ln \frac{1-\sin \theta}{\cos \theta}}{(1+\cos 2\theta)\sqrt{\ln \frac{1+\sin \theta}{\cos \theta}}}d\theta .$
2009 AIME Problems, 2
Suppose that $ a$, $ b$, and $ c$ are positive real numbers such that $ a^{\log_3 7} \equal{} 27$, $ b^{\log_7 11} \equal{} 49$, and $ c^{\log_{11} 25} \equal{} \sqrt {11}$. Find
\[ a^{(\log_3 7)^2} \plus{} b^{(\log_7 11)^2} \plus{} c^{(\log_{11} 25)^2}.
\]
2010 Today's Calculation Of Integral, 634
Prove that :
\[\int_1^{\sqrt{e}} (\ln x)^n dx=(-1)^{n-1}n!+\sqrt{e}\sum_{m=0}^{n} (-1)^{n-m}\frac{n!}{m!}\left(\frac 12\right)^m\ (n=1,\ 2,\ \cdots)\]
[i]2010 Miyazaki University entrance exam/Medicine[/i]
2003 Brazil National Olympiad, 2
Let $S$ be a set with $n$ elements. Take a positive integer $k$. Let $A_1, A_2, \ldots, A_k$ be any distinct subsets of $S$. For each $i$ take $B_i = A_i$ or $B_i = S - A_i$. Find the smallest $k$ such that we can always choose $B_i$ so that $\bigcup_{i=1}^k B_i = S$, no matter what the subsets $A_i$ are.
2010 District Olympiad, 1
Prove the following equalities of sets:
\[ \text{i)} \{x\in \mathbb{R}\ |\ \log_2 \lfloor x \rfloor \equal{} \lfloor \log_2 x\rfloor \} \equal{} \bigcup_{m\in \mathbb{N}} \left[2^m,2^m \plus{} 1\right)\]
\[ \text{ii)} \{x\in \mathbb{R}\ |\ 2^{\lfloor x\rfloor} \equal{} \left\lfloor 2^x\right\rfloor \} \equal{} \bigcup_{m\in \mathbb{N}} \left[m, \log_2 (2^m \plus{} 1) \right)\]
2024 AMC 12/AHSME, 8
What value of $x$ satisfies \[\frac{\log_2x\cdot\log_3x}{\log_2x+\log_3x}=2?\]
$
\textbf{(A) }25\qquad
\textbf{(B) }32\qquad
\textbf{(C) }36\qquad
\textbf{(D) }42\qquad
\textbf{(E) }48\qquad
$
1976 AMC 12/AHSME, 20
Let $a,~b,$ and $x$ be positive real numbers distinct from one. Then \[4(\log_ax)^2+3(\log_bx)^2=8(\log_ax)(\log_bx)\]
$\textbf{(A) }\text{for all values of }a,~b,\text{ and }x\qquad$
$\textbf{(B) }\text{if and only if }a=b^2\qquad$
$\textbf{(C) }\text{if and only if }b=a^2\qquad$
$\textbf{(D) }\text{if and only if }x=ab\qquad$
$ \textbf{(E) }\text{for none of these}$