Found problems: 894
2007 Today's Calculation Of Integral, 249
Determine the sign of $ \int_{\frac{1}{2}}^2 \frac{\ln t}{1\plus{}t^n}\ dt\ (n\equal{}1, 2, \cdots)$.
2017 Bosnia And Herzegovina - Regional Olympiad, 1
In terms of real parameter $a$ solve inequality:
$\log _{a} {x} + \mid a+\log _{a} {x} \mid \cdot \log _{\sqrt{x}} {a} \geq a\log _{x} {a}$
in set of real numbers
2018 Moscow Mathematical Olympiad, 7
$x^3+(\log_2{5}+\log_3{2}+\log_5{3})x=(\log_2{3}+\log_3{5}+\log_5{2})x^2+1$
2005 Croatia National Olympiad, 3
If $a, b, c$ are real numbers greater than $1$, prove that for any real number $r$
\[(\log_{a}bc)^{r}+(\log_{b}ca)^{r}+(\log_{c}ab)^{r}\geq 3 \cdot 2^{r}. \]
2007 Junior Tuymaada Olympiad, 5
What minimum number of colours is sufficient to colour all positive real numbers so that every two numbers whose ratio is 4 or 8 have different colours?
2014 Online Math Open Problems, 30
For a positive integer $n$, an [i]$n$-branch[/i] $B$ is an ordered tuple $(S_1, S_2, \dots, S_m)$ of nonempty sets (where $m$ is any positive integer) satisfying $S_1 \subset S_2 \subset \dots \subset S_m \subseteq \{1,2,\dots,n\}$. An integer $x$ is said to [i]appear[/i] in $B$ if it is an element of the last set $S_m$. Define an [i]$n$-plant[/i] to be an (unordered) set of $n$-branches $\{ B_1, B_2, \dots, B_k\}$, and call it [i]perfect[/i] if each of $1$, $2$, \dots, $n$ appears in exactly one of its branches.
Let $T_n$ be the number of distinct perfect $n$-plants (where $T_0=1$), and suppose that for some positive real number $x$ we have the convergence \[ \ln \left( \sum_{n \ge 0} T_n \cdot \frac{\left( \ln x \right)^n}{n!} \right) = \frac{6}{29}. \] If $x = \tfrac mn$ for relatively prime positive integers $m$ and $n$, compute $m+n$.
[i]Proposed by Yang Liu[/i]
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
2005 AIME Problems, 5
Determine the number of ordered pairs $(a,b)$ of integers such that $\log_a b + 6\log_b a=5$, $2 \leq a \leq 2005$, and $2 \leq b \leq 2005$.
1969 AMC 12/AHSME, 17
The equation $2^{2x}-8\cdot 2^x+12=0$ is satisfied by:
$\textbf{(A) }\log3\qquad
\textbf{(B) }\tfrac12\log6\qquad
\textbf{(C) }1+\log\tfrac34\qquad$
$\textbf{(D) }1+\tfrac{\log3}{\log2}\qquad
\textbf{(E) }\text{none of these}$
2014 District Olympiad, 2
Solve in real numbers the equation
\[ x+\log_{2}\left( 1+\sqrt{\frac{5^{x}}{3^{x}+4^{x}}}\right) =4+\log_{1/2}\left(1+\sqrt{\frac{25^{x}}{7^{x}+24^{x}}}\right) \]
2016 CMIMC, 2
Suppose that some real number $x$ satisfies
\[\log_2 x + \log_8 x + \log_{64} x = \log_x 2 + \log_x 16 + \log_x 128.\] Given that the value of $\log_2 x + \log_x 2$ can be expressed as $\tfrac{a\sqrt{b}}{c}$, where $a$ and $c$ are coprime positive integers and $b$ is squarefree, compute $abc$.
2013 Waseda University Entrance Examination, 3
Let $f(x)=\frac 12e^{2x}+2e^x+x$. Answer the following questions.
(1) For a real number $t$, set $g(x)=tx-f(x).$ When $x$ moves in the range of all real numbers, find the range of $t$ for which $g(x)$ has maximum value, then for the range of $t$, find the maximum value of $g(x)$ and the value of $x$ which gives the maximum value.
(2) Denote by $m(t)$ the maximum value found in $(1)$. Let $a$ be a constant, consider a function of $t$, $h(t)=at-m(t)$. When $t$ moves in the range of $t$ found in $(1)$, find the maximum value of $h(t)$.
1978 Putnam, A3
Find the value of $ k\ (0<k<5)$ such that $ \int_0^{\infty} \frac{x^k}{2\plus{}4x\plus{}3x^2\plus{}5x^3\plus{}3x^4\plus{}4x^5\plus{}2x^6}\ dx$ is minimal.
2005 Harvard-MIT Mathematics Tournament, 2
A plane curve is parameterized by $x(t)=\displaystyle\int_{t}^{\infty} \dfrac {\cos u}{u} \, \mathrm{d}u $ and $ y(t) = \displaystyle\int_{t}^{\infty} \dfrac {\sin u}{u} \, \mathrm{d}u $ for $ 1 \le t \le 2 $. What is the length of the curve?
2011 Today's Calculation Of Integral, 691
Let $a$ be a constant. In the $xy$ palne, the curve $C_1:y=\frac{\ln x}{x}$ touches $C_2:y=ax^2$.
Find the volume of the solid generated by a rotation of the part enclosed by $C_1,\ C_2$ and the $x$ axis about the $x$ axis.
[i]2011 Yokohama National Universty entrance exam/Engineering[/i]
2010 Polish MO Finals, 3
Real number $C > 1$ is given. Sequence of positive real numbers $a_1, a_2, a_3, \ldots$, in which $a_1=1$ and $a_2=2$, satisfy the conditions
\[a_{mn}=a_ma_n, \] \[a_{m+n} \leq C(a_m + a_n),\]
for $m, n = 1, 2, 3, \ldots$. Prove that $a_n = n$ for $n=1, 2, 3, \ldots$.
2013 ELMO Shortlist, 2
Let $n$ be a fixed positive integer. Initially, $n$ 1's are written on a blackboard. Every minute, David picks two numbers $x$ and $y$ written on the blackboard, erases them, and writes the number $(x+y)^4$ on the blackboard. Show that after $n-1$ minutes, the number written on the blackboard is at least $2^{\frac{4n^2-4}{3}}$.
[i]Proposed by Calvin Deng[/i]
2012 Today's Calculation Of Integral, 832
Find the limit
\[\lim_{n\to\infty} \frac{1}{n\ln n}\int_{\pi}^{(n+1)\pi} (\sin ^ 2 t)(\ln t)\ dt.\]
2013 ELMO Problems, 2
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]
2014 AMC 12/AHSME, 18
The domain of the function $f(x)=\log_{\frac12}(\log_4(\log_{\frac14}(\log_{16}(\log_{\frac1{16}}x))))$ is an interval of length $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
$\textbf{(A) }19\qquad
\textbf{(B) }31\qquad
\textbf{(C) }271\qquad
\textbf{(D) }319\qquad
\textbf{(E) }511\qquad$
2022 IMC, 4
Let $n > 3$ be an integer. Let $\Omega$ be the set of all triples of distinct elements of
$\{1, 2, \ldots , n\}$. Let $m$ denote the minimal number of colours which suffice to colour $\Omega$ so that whenever
$1\leq a<b<c<d \leq n$, the triples $\{a,b,c\}$ and $\{b,c,d\}$ have different colours. Prove that $\frac{1}{100}\log\log n \leq m \leq100\log \log n$.
1991 USAMO, 2
For any nonempty set $\,S\,$ of numbers, let $\,\sigma(S)\,$ and $\,\pi(S)\,$ denote the sum and product, respectively, of the elements of $\,S\,$. Prove that
\[ \sum \frac{\sigma(S)}{\pi(S)} = (n^2 + 2n) - \left(1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} \right) (n+1), \]
where ``$\Sigma$'' denotes a sum involving all nonempty subsets $S$ of $\{1,2,3, \ldots,n\}$.
2012 China Team Selection Test, 1
Given an integer $n\ge 4$. $S=\{1,2,\ldots,n\}$. $A,B$ are two subsets of $S$ such that for every pair of $(a,b),a\in A,b\in B, ab+1$ is a perfect square. Prove that
\[\min \{|A|,|B|\}\le\log _2n.\]
2006 Pre-Preparation Course Examination, 5
Powers of $2$ in base $10$ start with $3$ or $4$ more frequently? What is their state in base $3$? First write down an exact form of the question.
2009 Princeton University Math Competition, 3
It is known that a certain mechanical balance can measure any object of integer mass anywhere between 1 and 2009 (both included). This balance has $k$ weights of integral values. What is the minimum $k$ for which there exist weights that satisfy this condition?