Found problems: 913
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
Find the positive constant $c_0$ such that the series \[ \displaystyle\sum_{n = 0}^{\infty} \dfrac {n!}{(cn)^n} \] converges for $c>c_0$ and diverges for $0<c<c_0$.
Examine the behavior of the expression
$ \sum_{\nu\equal{}1}^{n\minus{}1}\frac{\log(n\minus{}\nu)}{\nu}\minus{}\log^2 n$
as $ n\rightarrow \infty$
If $a>1$, $b>1$, and $p=\frac{\log_b(\log_ba)}{\log_ba}$, then $a^n$ equals
$\textbf {(A) } 1 \qquad \textbf {(B) } b \qquad \textbf {(C) } \log_ab \qquad \textbf {(D) } \log_ba \qquad \textbf {(E) } a^{\log_ba}$
For a function $f(x)=\ln (1+\sqrt{1-x^2})-\sqrt{1-x^2}-\ln x\ (0<x<1)$, answer the following questions:
(1) Find $f'(x)$.
(2) Sketch the graph of $y=f(x)$.
(3) Let $P$ be a mobile point on the curve $y=f(x)$ and $Q$ be a point which is on the tangent at $P$ on the curve $y=f(x)$ and such that $PQ=1$. Note that the $x$-coordinate of $Q$ is les than that of $P$. Find the locus of $Q$.
Let $n>2,n \in \mathbb{N}$ and $a>0,a \in \mathbb{R}$ such that $2^a + \log_2 a = n^2$. Prove that: \[ 2 \cdot \log_2 n>a>2 \cdot \log_2 n -\frac{1}{n} . \]
[i]Radu Gologan[/i]
Find the positive integer $n$ such that $32$ is the product of the real number solutions of $x^{\log_2(x^3)-n} = 13$
Let be two real numbers $ a,b, $ that satisfy $ 3^a+13^b=17^a $ and $ 5^a+7^b=11^b. $
Show that $ a<b. $
There is a unique positive real number $x$ such that the three numbers $\log_8(2x),\log_4x,$ and $\log_2x,$ in that order, form a geometric progression with positive common ratio. The number $x$ can be written as $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Evaluate $\displaystyle\lim_{x\to 1}x^{\dfrac{x}{\sin(1-x)}}$.
Let $L$ be a positive constant. For a point $P(t,\ 0)$ on the positive part of the $x$ axis on the coordinate plane, denote $Q(u(t),\ v(t))$ the point at which the point reach starting from $P$ proceeds by distance $L$ in counter-clockwise on the perimeter of a circle passing the point $P$ with center $O$.
(1) Find $u(t),\ v(t)$.
(2) For real number $a$ with $0<a<1$, find $f(a)=\int_a^1 \sqrt{\{u'(t)\}^2+\{v'(t)\}^2}\ dt$.
(3) Find $\lim_{a\rightarrow +0} \frac{f(a)}{\ln a}$.
[i]2011 Tokyo University entrance exam/Science, Problem 3[/i]
Find the infinite integrals as follows.
(1) 2013 Hiroshima City University entrance exam/Informatic Science
$\int \frac{x^2}{2-x^2}dx$
(2) 2013 Kanseigakuin University entrance exam/Science and Technology
$\int x^4\ln x\ dx$
(3) 2013 Shinsyu University entrance exam/Textile Science and Technology, Second-exam
$\int \frac{\cos ^ 3 x}{\sin ^ 2 x}\ dx$
Let $ \alpha,\beta$ be real numbers satisfying $ 1 < \alpha < \beta.$ Find the greatest positive integer $ r$ having the following property: each of positive integers is colored by one of $ r$ colors arbitrarily, there always exist two integers $ x,y$ having the same color such that $ \alpha\le \frac {x}{y}\le\beta.$
Find the limit
\[\lim_{n\to\infty} \frac{1}{n\ln n}\int_{\pi}^{(n+1)\pi} (\sin ^ 2 t)(\ln t)\ dt.\]
Let Q+ and Z denote the set of positive rationals and the set of inte-
gers, respectively. Find all functions f : Q+ → Z satisfying the following
conditions:
(i) f(1999) = 1;
(ii) f(ab) = f(a) + f(b) for all a, b ∈ Q+;
(iii) f(a + b) ≥ min{f(a), f(b)} for all a, b ∈ Q+.
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 $KL$.
Suppose that $a^{\log_{b}c}+b^{\log_{c}a}=m$.
Find the value of $c^{\log_{b}a}+a^{\log_{c}b}$ .
if a,b∈R+,$a^{\log b}=2$,$a^{\log a}b^{\log b}=5$,find out $(ab)^{\log ab}$
For the function $f(x)=\int_0^x \frac{dt}{1+t^2}$, answer the questions as follows.
Note : Please solve the problems without using directly the formula $\int \frac{1}{1+x^2}\ dx=\tan^{-1}x +C$ for Japanese High School students those who don't study arc sin x, arc cos x, arc tanx.
(1) Find $f(\sqrt{3})$
(2) Find $\int_0^{\sqrt{3}} xf(x)\ dx$
(3) Prove that for $x>0$. $f(x)+f\left(\frac{1}{x}\right)$ is constant, then find the value.
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 $m+n$?
$\textbf{(A) }19\qquad
\textbf{(B) }31\qquad
\textbf{(C) }271\qquad
\textbf{(D) }319\qquad
\textbf{(E) }511\qquad$
Suppose that $p, q \in \mathbb{N}$ satisfy the inequality \[\exp(1)\cdot( \sqrt{p+q}-\sqrt{q})^{2}<1.\] Show that $\ln \left(1+\frac{p}{q}\right)$ is irrational.
What is more $\frac{2}{201}$ or $\ln\frac{101}{100}$? (No differential calculus allowed).
Let be three real numbers $ a,b,c, $ all in the interval $ (0,\infty ) $ or all in the interval $ (0,1). $ Prove the following inequality:
$$ \sum_{\text{cyc}}\log_a bc\ge 4\cdot\sum_{\text{cyc}} \log_{ab} c . $$
For each positive integer $ n,$ let $ f(n)$ be the number of ways to make $ n!$ cents using an unordered collection of coins, each worth $ k!$ cents for some $ k,\ 1\le k\le n.$ Prove that for some constant $ C,$ independent of $ n,$
\[ n^{n^2/2\minus{}Cn}e^{\minus{}n^2/4}\le f(n)\le n^{n^2/2\plus{}Cn}e^{\minus{}n^2/4}.\]
Prove that $2^{n - 1}$ divides $n!$ if and only if $n = 2^{k - 1}$ for some positive integer $k$.