Found problems: 913
1973 AMC 12/AHSME, 28
If $ a$, $ b$, and $ c$ are in geometric progression (G.P.) with $ 1 < a < b < c$ and $ n > 1$ is an integer, then $ \log_an$, $ \log_b n$, $ \log_c n$ form a sequence
$ \textbf{(A)}\ \text{which is a G.P} \qquad$
$ \textbf{(B)}\ \text{whichi is an arithmetic progression (A.P)} \qquad$
$ \textbf{(C)}\ \text{in which the reciprocals of the terms form an A.P} \qquad$
$ \textbf{(D)}\ \text{in which the second and third terms are the }n\text{th powers of the first and second respectively} \qquad$
$ \textbf{(E)}\ \text{none of these}$
2010 Today's Calculation Of Integral, 661
Consider a sequence $1^{0.01},\ 2^{0.02},\ 2^{0.02},\ 3^{0.03},\ 3^{0.03},\ 3^{0.03},\ 4^{0.04},\ 4^{0.04},\ 4^{0.04},\ 4^{0.04},\ \cdots$.
(1) Find the 36th term.
(2) Find $\int x^2\ln x\ dx$.
(3) Let $A$ be the product of from the first term to the 36th term. How many digits does $A$ have integer part?
If necessary, you may use the fact $2.0<\ln 8<2.1,\ 2.1<\ln 9<2.2,\ 2.30<\ln 10<2.31$.
[i]2010 National Defense Medical College Entrance Exam, Problem 4[/i]
2014 Junior Balkan MO, 4
For a positive integer $n$, two payers $A$ and $B$ play the following game: Given a pile of $s$ stones, the players take turn alternatively with $A$ going first. On each turn the player is allowed to take either one stone, or a prime number of stones, or a positive multiple of $n$ stones. The winner is the one who takes the last stone. Assuming both $A$ and $B$ play perfectly, for how many values of $s$ the player $A$ cannot win?
1963 AMC 12/AHSME, 30
Let \[F=\log\dfrac{1+x}{1-x}.\] Find a new function $G$ by replacing each $x$ in $F$ by \[\dfrac{3x+x^3}{1+3x^2},\] and simplify. The simplified expression $G$ is equal to:
$\textbf{(A)}\ -F \qquad
\textbf{(B)}\ F\qquad
\textbf{(C)}\ 3F \qquad
\textbf{(D)}\ F^3 \qquad
\textbf{(E)}\ F^3-F$
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
\[ a^{(\log_3 7)^2} \plus{} b^{(\log_7 11)^2} \plus{} c^{(\log_{11} 25)^2}.
\]
2007 Stanford Mathematics Tournament, 15
Evaluate $\int_{0}^{\infty}\frac{\tan^{-1}(\pi x)-\tan^{-1}x}{x}dx$
2010 Today's Calculation Of Integral, 659
Evaluate $\int_0^1 \frac{\ln (x+2)}{x+1}dx.$
2009 Today's Calculation Of Integral, 466
For $ n \equal{} 1,\ 2,\ 3,\ \cdots$, let $ (p_n,\ q_n)\ (p_n > 0,\ q_n > 0)$ be the point of intersection of $ y \equal{} \ln (nx)$ and $ \left(x \minus{} \frac {1}{n}\right)^2 \plus{} y^2 \equal{} 1$.
(1) Show that $ 1 \minus{} q_n^2\leq \frac {(e \minus{} 1)^2}{n^2}$ to find $ \lim_{n\to\infty} q_n$.
(2) Find $ \lim_{n\to\infty} n\int_{\frac {1}{n}}^{p_n} \ln (nx)\ dx$.
2013 Harvard-MIT Mathematics Tournament, 35
Let $P$ be the number of ways to partition $2013$ into an ordered tuple of prime numbers. What is $\log_2 (P)$? If your answer is $A$ and the correct answer is $C$, then your score on this problem will be $\left\lfloor\frac{125}2\left(\min\left(\frac CA,\frac AC\right)-\frac35\right)\right\rfloor$ or zero, whichever is larger.
PEN O Problems, 40
Let $X$ be a non-empty set of positive integers which satisfies the following: [list] [*] if $x \in X$, then $4x \in X$, [*] if $x \in X$, then $\lfloor \sqrt{x}\rfloor \in X$. [/list] Prove that $X=\mathbb{N}$.
2007 AMC 12/AHSME, 23
Square $ ABCD$ has area $ 36,$ and $ \overline{AB}$ is parallel to the x-axis. Vertices $ A,$ $ B,$ and $ C$ are on the graphs of $ y \equal{} \log_{a}x,$ $ y \equal{} 2\log_{a}x,$ and $ y \equal{} 3\log_{a}x,$ respectively. What is $ a?$
$ \textbf{(A)}\ \sqrt [6]{3}\qquad \textbf{(B)}\ \sqrt {3}\qquad \textbf{(C)}\ \sqrt [3]{6}\qquad \textbf{(D)}\ \sqrt {6}\qquad \textbf{(E)}\ 6$
2012 Today's Calculation Of Integral, 829
Let $a$ be a positive constant. Find the value of $\ln a$ such that
\[\frac{\int_1^e \ln (ax)\ dx}{\int_1^e x\ dx}=\int_1^e \frac{\ln (ax)}{x}\ dx.\]
2009 Today's Calculation Of Integral, 483
Let $ n\geq 2$ be natural number. Answer the following questions.
(1) Evaluate the definite integral $ \int_1^n x\ln x\ dx.$
(2) Prove the following inequality.
$ \frac 12n^2\ln n \minus{} \frac 14(n^2 \minus{} 1) < \sum_{k \equal{} 1}^n k\ln k < \frac 12n^2\ln n \minus{} \frac 14 (n^2 \minus{} 1) \plus{} n\ln n.$
(3) Find $ \lim_{n\to\infty} (1^1\cdot 2^2\cdot 3^3\cdots\cdots n^n)^{\frac {1}{n^2 \ln n}}.$
2011 Bogdan Stan, 1
If $ a,b,c $ are all in the interval $ (0,1) $ or all in the interval $ \left( 1,\infty \right), $ then
$$ 1\le\sum_{\text{cyc}} \frac{\log_a^7 b\cdot \log_b^3c}{\log_c a +2\log_a b} . $$
[i]Gheorghe Duță[/i]
2010 Today's Calculation Of Integral, 611
Let $g(t)$ be the minimum value of $f(x)=x2^{-x}$ in $t\leq x\leq t+1$.
Evaluate $\int_0^2 g(t)dt$.
[i]2010 Kumamoto University entrance exam/Science[/i]
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
$
1964 AMC 12/AHSME, 11
Given $2^x=8^{y+1}$ and $9^y=3^{x-9}$, find the value of $x+y$.
${{ \textbf{(A)}\ 18 \qquad\textbf{(B)}\ 21 \qquad\textbf{(C)}\ 24 \qquad\textbf{(D)}\ 27 }\qquad\textbf{(E)}\ 30 } $
2009 Today's Calculation Of Integral, 490
For a positive real number $ a > 1$, prove the following inequality.
$ \frac {1}{a \minus{} 1}\left(1 \minus{} \frac {\ln a}{a\minus{}1}\right) < \int_0^1 \frac {x}{a^x}\ dx < \frac {1}{\ln a}\left\{1 \minus{} \frac {\ln (\ln a \plus{} 1)}{\ln a}\right\}$
1994 AIME Problems, 4
Find the positive integer $n$ for which \[ \lfloor \log_2{1}\rfloor+\lfloor\log_2{2}\rfloor+\lfloor\log_2{3}\rfloor+\cdots+\lfloor\log_2{n}\rfloor=1994. \] (For real $x$, $\lfloor x\rfloor$ is the greatest integer $\le x.$)
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
\[C_1 \log_{10} n \le f(n) \le C_2 \log_{10} n.\]
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?
2012 Today's Calculation Of Integral, 804
For $a>0$, find the minimum value of $I(a)=\int_1^e |\ln ax|\ dx.$
2002 AMC 12/AHSME, 14
For all positive integers $ n$, let $ f(n) \equal{} \log_{2002} n^2$. Let
\[ N \equal{} f(11) \plus{} f(13) \plus{} f(14)
\]
Which of the following relations is true?
$ \textbf{(A)}\ N < 1 \qquad \textbf{(B)}\ N \equal{} 1 \qquad \textbf{(C)}\ 1 < N < 2 \qquad \textbf{(D)}\ N \equal{} 2 \qquad \textbf{(E)}\ N > 2$
2009 Today's Calculation Of Integral, 488
For $ 0\leq x <\frac{\pi}{2}$, prove the following inequality.
$ x\plus{}\ln (\cos x)\plus{}\int_0^1 \frac{t}{1\plus{}t^2}\ dt\leq \frac{\pi}{4}$
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] $\int \frac{x^2}{\sqrt{(x+1)^3}}dx$