Found problems: 913
$ \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)$
For a positive real number $x$, find the minimum value of $f(x)=\int_x^{2x} (t\ln t-t)dt.$
Answer the following questions.
(1) By setting $ x\plus{}\sqrt{x^2\minus{}1}\equal{}t$, find the indefinite integral $ \int \sqrt{x^2\minus{}1}\ dx$.
(2) Given two points $ P(p,\ q)\ (p>1,\ q>0)$ and $ A(1,\ 0)$ on the curve $ x^2\minus{}y^2\equal{}1$. Find the area $ S$ of the figure bounded by two lines $ OA,\ OP$ and the curve in terms of $ p$.
(3) Let $ S\equal{}\frac{\theta}{2}$. Express $ p,\ q$ in terms of $ \theta$.
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] $\int x\ln x dx$
Let $f(x)$ be a function defined in $0<x<\frac{\pi}{2}$ satisfying:
(i) $f\left(\frac{\pi}{6}\right)=0$
(ii) $f'(x)\tan x=\int_{\frac{\pi}{6}}^x \frac{2\cos t}{\sin t}dt$.
Find $f(x)$.
[i]1987 Sapporo Medical University entrance exam[/i]
The equation \[2^{333x-2}+2^{111x+2}=2^{222x+1}+1\] has three real roots. Given that their sum is $m/n$ where $m$ and $n$ are relatively prime positive integers, find $m+n$.
What is the value of $[\log_{10}(5\log_{10}100)]^2$?
${{ \textbf{(A)}\ \log_{10}50 \qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 10 \qquad\textbf{(D)}\ 2}\qquad\textbf{(E)}\ 1 } $
If $ a$, $ b$ and $ c$ are positive reals prove inequality:
\[ \left(1\plus{}\frac{4a}{b\plus{}c}\right)\left(1\plus{}\frac{4b}{a\plus{}c}\right)\left(1\plus{}\frac{4c}{a\plus{}b}\right) > 25.\]
Define $ \{ a_n \}_{n\equal{}1}$ as follows: $ a_1 \equal{} 1989^{1989}; \ a_n, n > 1,$ is the sum of the digits of $ a_{n\minus{}1}$. What is the value of $ a_5$?
i) If $x = \left(1+\frac{1}{n}\right)^{n}$ and $y=\left(1+\frac{1}{n}\right)^{n+1}$, show that $y^{x}= x^{y}$.
ii) Show that, for all positive integers $n$, \[1^{2}-2^{2}+3^{2}-4^{2}+\cdots+(-1)^{n}(n-1)^{2}+(-1)^{n+1}n^{2}= (-1)^{n+1}(1+2+\cdots+n).\]
Professor Gamble buys a lottery ticket, which requires that he pick six different integers from $ 1$ through $ 46$, inclusive. He chooses his numbers so that the sum of the base-ten logarithms of his six numbers is an integer. It so happens that the integers on the winning ticket have the same property--- the sum of the base-ten logarithms is an integer. What is the probability that Professor Gamble holds the winning ticket?
$ \textbf{(A)}\ 1/5 \qquad \textbf{(B)}\ 1/4 \qquad \textbf{(C)}\ 1/3 \qquad \textbf{(D)}\ 1/2 \qquad \textbf{(E)}\ 1$
Consider the number $\alpha$ obtained by writing one after another the decimal representations of $1, 1987, 1987^2, \dots$ to the right the decimal point. Show that $\alpha$ is irrational.
Determine the real numbers $a$ such that the equation $a 3^x + 3^{-x} = 3$ has exactly one solution $x.$
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 $
Given $ m$ and $ n$ as relatively prime positive integers greater than one, show that
\[ \frac{\log_{10} m}{\log_{10} n}\]
is not a rational number.
Evaluate $\int_1^e \frac{\{1-(x-1)e^{x}\}\ln x}{(1+e^x)^2}dx.$
For each $m\in \mathbb N$ we define $rad\ (m)=\prod p_i$, where $m=\prod p_i^{\alpha_i}$.
[b]abc Conjecture[/b]
Suppose $\epsilon >0$ is an arbitrary number, then there exist $K$ depinding on $\epsilon$ that for each 3 numbers $a,b,c\in\mathbb Z$ that $gcd (a,b)=1$ and $a+b=c$ then: \[ max\{|a|,|b|,|c|\}\leq K(rad\ (abc))^{1+\epsilon} \]
Now prove each of the following statements by using the $abc$ conjecture :
a) Fermat's last theorem for $n>N$ where $N$ is some natural number.
b) We call $n=\prod p_i^{\alpha_i}$ strong if and only $\alpha_i\geq 2$.
c) Prove that there are finitely many $n$ such that $n,\ n+1,\ n+2$ are strong.
d) Prove that there are finitely many rational numbers $\frac pq$ such that: \[ \Big| \sqrt[3]{2}-\frac pq \Big|<\frac{2^ {1384}}{q^3} \]
Prove the following inequality.
\[\frac{e-1}{n+1}\leqq\int^e_1(\log x)^n dx\leqq\frac{(n+1)e+1}{(n+1)(n+2)}\ (n=1,2,\cdot\cdot\cdot) \]
1994 Kyoto University entrance exam/Science
Solve the equation $\sqrt{x} = \log_2 x$.
If $\frac{4^x}{2^{x+y}}=8$ and $\frac{9^{x+y}}{3^{5y}}=243$, $x$ and $y$ are real numbers, then $xy$ equals:
$\text{(A)} \ \frac{12}{5} \qquad \text{(B)} \ 4 \qquad \text{(C)} \ 6 \qquad \text{(D)} \ 12 \qquad \text{(E)} \ -4$
At a competition with $N$ players, the number of players given elite status is equal to \[2^{1+\lfloor\log_2{(N-1)}\rfloor} - N. \] Suppose that $19$ players are given elite status. What is the sum of the two smallest possible values of $N$?
$ \textbf{(A)}\ 38\qquad
\textbf{(B)}\ 90 \qquad
\textbf{(C)}\ 154 \qquad
\textbf{(D)}\ 406 \qquad
\textbf{(E)}\ 1024$
For $n=1,2,\dots$ let
\[S_n=\log\left(\sqrt[n^2]{1^1 \cdot 2^2 \cdot \dotsc \cdot n^n}\right)-\log(\sqrt{n}),\]
where $\log$ denotes the natural logarithm. Find $\lim_{n \to \infty} S_n$.
Real numbers $x$ and $y$ with $x,y>1$ satisfy $\log_x(y^x)=\log_y(x^{4y})=10.$ What is the value of $xy$?
Prove that $ \frac {( \minus{} 1)^n}{n!}\int_1^2 (\ln x)^n\ dx \equal{} 2\sum_{k \equal{} 1}^n \frac {( \minus{} \ln 2)^k}{k!} \plus{} 1$.
Evaluate the following definite integral.
\[\int_{e^{e}}^{e^{e+1}}\left\{\frac{1}{\ln x \cdot\ln (\ln x)}+\ln (\ln (\ln x))\right\}dx\]