Found problems: 913
Evaluate $\int_0^1 \frac{x^2+5}{(x+1)^2(x-2)}dx.$
[i]2011 Doshisya University entrance exam/Science and Technology[/i]
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}$
Compute $e^A$ where $A$ is defined as \[\int_{3/4}^{4/3}\dfrac{2x^2+x+1}{x^3+x^2+x+1}dx.\]
Evaluate $ \int_1^e \{(1 \plus{} x)e^x \plus{} (1 \minus{} x)e^{ \minus{} x}\}\ln x\ dx$.
For a positive integer $n \ge 2 $, define set $ T = \{ (i,j) | 1 \le i < j \le n , i | j \} $. For nonnegative real numbers $ x_1 , x_2 , \cdots , x_n $ with $ x_1 + x_2 + \cdots + x_n = 1 $, find the maximum value of
\[ \sum_{(i,j) \in T} x_i x_j \]
in terms of $n$.
Let $ a$, $ b$, $ c$ be three real numbers. For each positive integer number $ n$, $ a^n \plus{} b^n \plus{} c^n$ is an integer number. Prove that there exist three integers $ p$, $ q$, $ r$ such that $ a$, $ b$, $ c$ are the roots of the equation $ x^3 \plus{} px^2 \plus{} qx \plus{} r \equal{} 0$.
Let $ f$ be a real function defined on the positive half-axis for which $ f(xy)\equal{}xf(y)\plus{}yf(x)$ and $ f(x\plus{}1) \leq f(x)$ hold for every positive $ x$ and $ y$. Show that if $ f(1/2)\equal{}1/2$, then \[ f(x)\plus{}f(1\minus{}x) \geq \minus{}x \log_2 x \minus{}(1\minus{}x) \log_2 (1\minus{}x)\] for every $ x\in (0,1)$.
[i]Z. Daroczy, Gy. Maksa[/i]
If $\log_2(\log_3(\log_4 x))=\log_3(\log_4(\log_2 y))=\log_4(\log_2(\log_3 z))=0$, then the sum $x+y+z$ is equal to
$\textbf{(A) }50\qquad\textbf{(B) }58\qquad\textbf{(C) }89\qquad\textbf{(D) }111\qquad \textbf{(E) }1296$
Let \[ A = \frac{1}{6}((\log_2(3))^3-(\log_2(6))^3-(\log_2(12))^3+(\log_2(24))^3) \].
Compute $2^A$.
Prove that for any positive integer $n\geq 2$ we have that \[\sum_{k=2}^n \lfloor \sqrt[k]{n}\rfloor=\sum_{k=2}^n\lfloor\log_{k}n\rfloor.\]
Evaluate $\int_0^{\frac{\pi}{2}} \frac{\sin x-\cos x}{1+\cos x}\ dx.$
Given sequences $a_n=\frac{1}{n}{\sqrt[n] {_{2n}P_n}},\ b_n=\frac{1}{n^2}{\sqrt[n] {_{4n}P_{2n}}}$ and $c_n=\sqrt[n]{\frac{_{8n}P_{4n}}{_{6n}P_{4n}}}$, find $\lim_{n\to\infty} a_n,\ \lim_{n\to\infty} b_n$and $\lim_{n\to\infty} c_n.$
Let $a, b, c$ be strictly positive real numbers such that $1 < b \le c^2 \le a^{10}$, and
\[\log_ab + 2\log_bc + 5\log_ca = 12.\]
Prove that
\[2\log_ac + 5\log_cb + 10\log_ba \ge 21.\]
For $0\leq a\leq 2$, find the minimum value of $\int_0^2 \left|\frac{1}{1+e^x}-\frac{1}{1+e^a}\right|\ dx.$
[i]2010 Kyoto Institute of Technology entrance exam/Textile e.t.c.[/i]
Find all differentiable functions $f: (0,\infty)\mapsto (0,\infty)$ for which there is a positive real number $a$ such that
\[ f'\left(\frac ax\right)=\frac x{f(x)} \]
for all $x>0.$
Let $a\neq 1$ be a positive real number. Find all real solutions to the equation $a^x=x^x+\log_a(\log_a(x)).$
[i]Mihai Opincariu[/i]
Let $f:\mathbb{R}\to\mathbb{R}^+$ be a continuous and periodic function. Prove that for all $\alpha\in\mathbb{R}$ the following inequality holds:
$\int_0^T\frac{f(x)}{f(x+\alpha)}dx\ge T$,
where $T$ is the period of $f(x)$.
Let $a$ and $b$ be real numbers greater than $1$ for which there exists a positive real number $c$, different from $1$, such that \[2(\log_ac+\log_bc)=9\log_{ab}c.\] Find the largest possible value of $\log_ab$.
$\textbf{(A) }\sqrt2\qquad\textbf{(B) }\sqrt3\qquad\textbf{(C) }2\qquad\textbf{(D) }\sqrt6\qquad\textbf{(E) }3$
Evaluate $\int_0^{\frac{\pi}{4}} \frac{\sin \theta (\sin \theta \cos \theta +2)}{\cos ^ 4 \theta}\ d\theta$.
Let $ K$ be the figure bounded by the graph of function $ y \equal{} \frac {x}{\sqrt {1 \minus{} x^2}}$, $ x$ axis and the line $ x \equal{} \frac {1}{2}$.
(1) Find the volume $ V_1$ of the solid generated by rotation of $ K$ around $ x$ axis.
(2) Find the volume $ V_2$ of the solid generated by rotation of $ K$ around $ y$ axis.
Please solve question (2) without using the shell method for Japanese High School Students those who don't learn it.
Evaluate $\int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{2+\sin x}{1+\cos x}\ dx.$
Let $a$ be a constant number such that $0<a<1$ and $V(a)$ be the volume formed by the revolution of the figure
which is enclosed by the curve $y=\ln (x-a)$, the $x$-axis and two lines $x=1,x=3$ about the $x$-axis.
If $a$ varies in the range of $0<a<1$, find the minimum value of $V(a)$.
Let $ f(x)\equal{}\left(1\plus{}\frac{1}{x}\right)^{x}\ (x>0)$.
Find $ \lim_{n\to\infty}\left\{f\left(\frac{1}{n}\right)f\left(\frac{2}{n}\right)f\left(\frac{3}{n}\right)\cdots\cdots f\left(\frac{n}{n}\right)\right\}^{\frac{1}{n}}$.
Prove that there exists a real constant $c$ such that for any pair $(x,y)$ of real numbers, there exist relatively prime integers $m$ and $n$ satisfying the relation
\[
\sqrt{(x-m)^2 + (y-n)^2} < c\log (x^2 + y^2 + 2).
\]
Let $X$ be a non empty subset of $\mathbb{N} = \{1,2,\ldots \}$. Suppose that for all $x \in X$, $4x \in X$ and $\lfloor \sqrt{x} \rfloor \in X$. Prove that $X=\mathbb{N}$.