Found problems: 913
Let $a$ and $b$ be positive integers with $a>1$ and $b>2$. Prove that $a^b+1\ge b(a+1)$ and determine when there is inequality.
Solve in the real numbers the equation:
$$ \cos^2 \frac{(x-2)\pi }{4} +\cos\frac{(x-2)\pi }{3} =\log_3 (x^2-4x+6) $$
[i]Gheorghe Mihai[/i]
Determine whether $\sqrt{1001^2+1}+\sqrt{1002^2+1}+ \cdots + \sqrt{2000^2+1}$ be a rational number or not?
Let $f:\mathbb N\rightarrow\mathbb N$ be a non-decreasing function and let $n$ be an arbitrary natural number. Suppose that there are prime numbers $p_1,p_2,\dots,p_n$ and natural numbers $s_1,s_2,\dots,s_n$ such that for each $1\leq i\leq n$ the set $\{f(p_ir+s_i)|r=1,2,\dots\}$ is an infinite arithmetic progression. Prove that there is a natural number $a$ such that
\[f(a+1), f(a+2), \dots, f(a+n)\]
form an arithmetic progression.
Which is greater: $ (3^5)^{(5^3)}$ or $ (5^3)^{(3^5)}$?
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$
For real numbers $x$ let $$f(x)=\frac{4^x}{25^{x+1}}+\frac{5^x}{2^{x+1}}.$$ Then $f\left(\frac{1}{1-\log_{10}4}\right)=\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
If $\log_{b^2}x+\log_{x^2}b=1, b>0, b \neq 1, x \neq 1$, then $x$ equals:
$ \textbf{(A)}\ 1/b^2 \qquad\textbf{(B)}\ 1/b \qquad\textbf{(C)}\ b^2 \qquad\textbf{(D)}\ b \qquad\textbf{(E)}\ \sqrt{b} $
Let $f\left(x\right)$ be a non-constant polynomial with integer coefficients, and let $u$ be an arbitrary positive integer. Prove that there is an integer $n$ such that $f\left(n\right)$ has at least $u$ distinct prime factors and $f\left(n\right) \neq 0$.
Consider a positive real number $a$ and a positive integer $m$. The sequence $(x_k)_{k\in \mathbb{Z}^{+}}$ is defined as:
$x_1=1$, $x_2=a$, $x_{n+2}=\sqrt[m+1]{x_{n+1}^mx_n}$.
$a)$ Prove that the sequence is converging.
$b)$ Find $\lim_{n\rightarrow \infty}{x_n}$.
Of the following
(1) $ a(x\minus{}y)\equal{}ax\minus{}ay$
(2) $ a^{x\minus{}y}\equal{}a^x\minus{}a^y$
(3) $ \log (x\minus{}y)\equal{}\log x\minus{}\log y$
(4) $ \frac {\log x}{\log y}\equal{} \log{x}\minus{} \log{y}$
(5) $ a(xy)\equal{}ax\times ay$
$\textbf{(A)}\ \text{Only 1 and 4 are true} \qquad\\
\textbf{(B)}\ \text{Only 1 and 5 are true} \qquad\\
\textbf{(C)}\ \text{Only 1 and 3 are true} \qquad\\
\textbf{(D)}\ \text{Only 1 and 2 are true} \qquad\\
\textbf{(E)}\ \text{Only 1 is true}$
Show that positive integers $n_i,m_i$ $(i=1,2,3, \cdots )$ can be found such that $ \mathop{\lim }\limits_{i \to \infty } \frac{2^{n_i}}{3^{m_i }} = 1$
A sequence $\{a_n\}_{n\geq 1}$ is defined by a recurrence relation
$$a_1 = 1,\quad a_{n+1} = \log \frac{e^{a_n}-1}{a_n}$$
And a sequence $\{b_n\}_{n\geq 1}$ is defined as $b_n = \prod\limits_{i=1}^n a_i$. Evaluate an infinite series $\sum\limits_{n=1}^\infty b_n$.
Evaluate
\[\int_{\frac{\pi}{2}}^{\frac{2\pi}{3}} \frac{1}{\sin x \sqrt{1-\cos x}}dx\]
The sum of all real $x$ such that $(2^{x} - 4)^{3} + (4^{x} - 2)^{3} = (4^{x} + 2^{x} - 6)^{3}$ is
$ \textbf{(A)}\ 3/2\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 5/2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 7/2 $
Prove that\[(1+\frac{1}{3})(1+\frac{1}{3^2})\cdots(1+\frac{1}{3^n})< 2.\]
If $ x$ is greater than zero, then the correct relationship is:
$ \textbf{(A)}\ \log (1\plus{}x) \equal{} \frac{x}{1\plus{}x} \qquad\textbf{(B)}\ \log (1\plus{}x) < \frac{x}{1\plus{}x} \\
\textbf{(C)}\ \log(1\plus{}x) > x \qquad\textbf{(D)}\ \log (1\plus{}x) < x \qquad\textbf{(E)}\ \text{none of these}$
For a function $ f(x)\equal{}\int_x^{\frac{\pi}{4}\minus{}x} \log_4 (1\plus{}\tan t)dt\ \left(0\leq x\leq \frac{\pi}{8}\right)$, answer the following questions.
(1) Find $ f'(x)$.
(2) Find the $ n$ th term of the sequence $ a_n$ such that $ a_1\equal{}f(0),\ a_{n\plus{}1}\equal{}f(a_n)\ (n\equal{}1,\ 2,\ 3,\ \cdots)$.
If $a=\log_8225$ and $b=\log_215$, then
$\textbf{(A) }a=\frac{1}{2}b\qquad\textbf{(B) }a=\frac{2b}{3}\qquad\textbf{(C) }a=b\qquad\textbf{(D) }b=\frac{1}{2}a\qquad \textbf{(E) }a=\frac{3b}{2}$
The number \[ \frac 2{\log_4{2000^6}}+\frac 3{\log_5{2000^6}} \] can be written as $\frac mn$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
Let $ f(n)$ be that largest integer $ k$ such that $ n^k$ divides $ n!$, and let $ F(n)\equal{} \max_{2 \leq m \leq n} f(m)$. Show that \[ \lim_{n\rightarrow \infty} \frac{F(n) \log n}{n \log \log n}\equal{}1.\]
[i]P. Erdos[/i]
Evaluate
\[{{\int_{e^{e^{e}}}^{e^{e^{e^{e}}}}} \frac{dx}{x\ln x\cdot \ln (\ln x)\cdot \ln \{\ln (\ln x)\}}}\]
It is known that $ k$ and $ n$ are positive integers and \[ k \plus{} 1\leq\sqrt {\frac {n \plus{} 1}{\ln(n \plus{} 1)}}.\] Prove that there exists a polynomial $ P(x)$ of degree $ n$ with coefficients in the set $ \{0,1, \minus{} 1\}$ such that $ (x \minus{} 1)^{k}$ divides $ P(x)$.
Evaluate $\int_0^{\infty} \frac{\ln (1+e^{4x})}{e^x}dx.$
Find the volume $V$ of the solid formed by a rotation of the region enclosed by the curve $y=2^{x}-1$ and two lines $x=0,\ y=1$ around the $y$ axis.