Found problems: 913
2008 Harvard-MIT Mathematics Tournament, 4
([b]4[/b]) Let $ a$, $ b$ be constants such that $ \lim_{x\rightarrow1}\frac {(\ln(2 \minus{} x))^2}{x^2 \plus{} ax \plus{} b} \equal{} 1$. Determine the pair $ (a,b)$.
2005 AMC 10, 17
Suppose that $ 4^a \equal{} 5$, $ 5^b \equal{} 6$, $ 6^c \equal{} 7$, and $ 7^d \equal{} 8$. What is $ a\cdot b\cdot c\cdot d$?
$ \textbf{(A)}\ 1\qquad
\textbf{(B)}\ \frac{3}{2}\qquad
\textbf{(C)}\ 2\qquad
\textbf{(D)}\ \frac{5}{2}\qquad
\textbf{(E)}\ 3$
1958 AMC 12/AHSME, 17
If $ x$ is positive and $ \log{x} \ge \log{2} \plus{} \frac{1}{2}\log{x}$, then:
$ \textbf{(A)}\ {x}\text{ has no minimum or maximum value}\qquad \\
\textbf{(B)}\ \text{the maximum value of }{x}\text{ is }{1}\qquad \\
\textbf{(C)}\ \text{the minimum value of }{x}\text{ is }{1}\qquad \\
\textbf{(D)}\ \text{the maximum value of }{x}\text{ is }{4}\qquad \\
\textbf{(E)}\ \text{the minimum value of }{x}\text{ is }{4}$
2007 Stanford Mathematics Tournament, 15
Evaluate $\int_{0}^{\infty}\frac{\tan^{-1}(\pi x)-\tan^{-1}x}{x}dx$
2007 Today's Calculation Of Integral, 207
Evaluate the following definite integral.
\[\int_{e^{e}}^{e^{e+1}}\left\{\frac{1}{\ln x \cdot\ln (\ln x)}+\ln (\ln (\ln x))\right\}dx\]
Today's calculation of integrals, 882
Find $\lim_{n\to\infty} \sum_{k=1}^n \frac{1}{n+k}(\ln (n+k)-\ln\ n)$.
2011 Today's Calculation Of Integral, 726
Let $P(x,\ y)\ (x>0,\ y>0)$ be a point on the curve $C: x^2-y^2=1$. If $x=\frac{e^u+e^{-u}}{2}\ (u\geq 0)$, then find the area bounded by the line $OP$, the $x$ axis and the curve $C$ in terms of $u$.
1973 Miklós Schweitzer, 3
Find a constant $ c > 1$ with the property that, for arbitrary positive integers $ n$ and $ k$ such that $ n>c^k$, the number of distinct prime factors of $ \binom{n}{k}$ is at least $ k$.
[i]P. Erdos[/i]
1954 AMC 12/AHSME, 15
$ \log 125$ equals:
$ \textbf{(A)}\ 100 \log 1.25 \qquad \textbf{(B)}\ 5 \log 3 \qquad \textbf{(C)}\ 3 \log 25$
$ \textbf{(D)}\ 3 \minus{} 3\log 2 \qquad \textbf{(E)}\ (\log 25)(\log 5)$
1971 IMO Longlists, 44
Let $m$ and $n$ denote integers greater than $1$, and let $\nu (n)$ be the number of primes less than or equal to $n$. Show that if the equation $\frac{n}{\nu(n)}=m$ has a solution, then so does the equation $\frac{n}{\nu(n)}=m-1$.
2011 ELMO Shortlist, 5
Prove there exists a constant $c$ (independent of $n$) such that for any graph $G$ with $n>2$ vertices, we can split $G$ into a forest and at most $cf(n)$ disjoint cycles, where
a) $f(n)=n\ln{n}$;
b) $f(n)=n$.
[i]David Yang.[/i]
2007 Princeton University Math Competition, 1
Take the square with vertices $(0,0)$, $(1,0)$, $(0,1)$, and $(1,1)$. Choose a random point in this square and draw the line segment from it to $(0,0)$. Choose a second random point in this square and draw the line segment from it to $(1,0)$. What is the probability that the two line segments intersect?
1986 Vietnam National Olympiad, 3
Suppose $ M(y)$ is a polynomial of degree $ n$ such that $ M(y) \equal{} 2^y$ for $ y \equal{} 1, 2, \ldots, n \plus{} 1$. Compute $ M(n \plus{} 2)$.
2000 Stanford Mathematics Tournament, 15
Which is greater: $ (3^5)^{(5^3)}$ or $ (5^3)^{(3^5)}$?
1970 Czech and Slovak Olympiad III A, 6
Determine all real $x$ such that \[\sqrt{\tan(x)-1}\,\Bigl(\log_{\tan(x)}\bigl(2+4\cos^2(x)-2\bigr)\Bigr)\ge0.\]
2011 Kosovo National Mathematical Olympiad, 3
Prove that the following inequality holds:
\[ \left( \log_{24}48 \right)^2+ \left( \log_{12}54 \right)^2>4\]
2006 AIME Problems, 2
The lengths of the sides of a triangle with positive area are $\log_{10} 12$, $\log_{10} 75$, and $\log_{10} n$, where $n$ is a positive integer. Find the number of possible values for $n$.
2014 Indonesia MO, 3
Let $ABCD$ be a trapezoid (quadrilateral with one pair of parallel sides) such that $AB < CD$. Suppose that $AC$ and $BD$ meet at $E$ and $AD$ and $BC$ meet at $F$. Construct the parallelograms $AEDK$ and $BECL$. Prove that $EF$ passes through the midpoint of the segment $KL$.
PEN G Problems, 15
Prove that for any $ p, q\in\mathbb{N}$ with $ q > 1$ the following inequality holds:
\[ \left\vert\pi\minus{}\frac{p}{q}\right\vert\ge q^{\minus{}42}.\]
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\}$
2007 Princeton University Math Competition, 5
Find the values of $a$ such that $\log (ax+1) = \log (x-a) + \log (2-x)$ has a unique real solution.
2010 Today's Calculation Of Integral, 562
(1) Show the following inequality for every natural number $ k$.
\[ \frac {1}{2(k \plus{} 1)} < \int_0^1 \frac {1 \minus{} x}{k \plus{} x}dx < \frac {1}{2k}\]
(2) Show the following inequality for every natural number $ m,\ n$ such that $ m > n$.
\[ \frac {m \minus{} n}{2(m \plus{} 1)(n \plus{} 1)} < \log \frac {m}{n} \minus{} \sum_{k \equal{} n \plus{} 1}^{m} \frac {1}{k} < \frac {m \minus{} n}{2mn}\]
2007 Princeton University Math Competition, 6
Take the square with vertices $(0,0)$, $(1,0)$, $(0,1)$, and $(1,1)$. Choose a random point in this square and draw the line segment from it to $(0,0)$. Choose a second random point in this square and draw the line segment from it to $(1,0)$. What is the probability that the two line segments intersect?
2006 China Second Round Olympiad, 5
Suppose $f(x) = x^3 + \log_2(x + \sqrt{x^2+1})$. For any $a,b \in \mathbb{R}$, to satisfy $f(a) + f(b) \ge 0$, the condition $a + b \ge 0$ is
$ \textbf{(A)}\ \text{necessary and sufficient}\qquad\textbf{(B)}\ \text{not necessary but sufficient}\qquad\textbf{(C)}\ \text{necessary but not sufficient}\qquad$
$\textbf{(D)}\ \text{neither necessary nor sufficient}\qquad$
2024 AMC 12/AHSME, 8
How many angles $\theta$ with $0\le\theta\le2\pi$ satisfy $\log(\sin(3\theta))+\log(\cos(2\theta))=0$?
$
\textbf{(A) }0 \qquad
\textbf{(B) }1 \qquad
\textbf{(C) }2 \qquad
\textbf{(D) }3 \qquad
\textbf{(E) }4 \qquad
$