Found problems: 913
Prove that
\[\frac{1995}{2}-\frac{1994}{3}+\frac{1993}{4}-\ldots -\frac{2}{1995}+\frac{1}{1996}=\frac{1}{999}+\frac{3}{1000}+\ldots +\frac{1995}{1996}\]
If $ \log_{k}{x}\cdot \log_{5}{k} \equal{} 3$, then $ x$ equals:
$ \textbf{(A)}\ k^6\qquad
\textbf{(B)}\ 5k^3\qquad
\textbf{(C)}\ k^3\qquad
\textbf{(D)}\ 243\qquad
\textbf{(E)}\ 125$
Let $a$ be a positive real number, $n$ a positive integer, and define the [i]power tower[/i] $a\uparrow n$ recursively with $a\uparrow 1=a$, and $a\uparrow(i+1)=a^{a\uparrow i}$ for $i=1,2,3,\ldots$. For example, we have $4\uparrow 3=4^{(4^4)}=4^{256}$, a number which has $155$ digits. For each positive integer $k$, let $x_k$ denote the unique positive real number solution of the equation $x\uparrow k=10\uparrow (k+1)$. Which is larger: $x_{42}$ or $x_{43}$?
If $ \log x\minus{}5 \log 3\equal{}\minus{}2$, then $ x$ equals:
$ \textbf{(A)}\ 1.25 \qquad
\textbf{(B)}\ 0.81 \qquad
\textbf{(C)}\ 2.43 \qquad
\textbf{(D)}\ 0.8 \qquad
\textbf{(E)}\ \text{either 0.8 or 1.25}$
Let $ a,b,c$ be positive numbers.Find $ k$ such that:
$ (k \plus{} \frac {a}{b \plus{} c})(k \plus{} \frac {b}{c \plus{} a})(k \plus{} \frac {c}{a \plus{} b}) \ge (k \plus{} \frac {1}{2})^3$
Let $n$ be a fixed natural number.
[b]a)[/b] Find all solutions to the following equation :
\[ \sum_{k=1}^n [\frac x{2^k}]=x-1 \]
[b]b)[/b] Find the number of solutions to the following equation ($m$ is a fixed natural) :
\[ \sum_{k=1}^n [\frac x{2^k}]=x-m \]
How many real numbers $x$ satisfy the equation $\frac{1}{5}\log_2 x = \sin (5\pi x)$?
Prove the following equalities of sets:
\[ \text{i)} \{x\in \mathbb{R}\ |\ \log_2 \lfloor x \rfloor \equal{} \lfloor \log_2 x\rfloor \} \equal{} \bigcup_{m\in \mathbb{N}} \left[2^m,2^m \plus{} 1\right)\]
\[ \text{ii)} \{x\in \mathbb{R}\ |\ 2^{\lfloor x\rfloor} \equal{} \left\lfloor 2^x\right\rfloor \} \equal{} \bigcup_{m\in \mathbb{N}} \left[m, \log_2 (2^m \plus{} 1) \right)\]
Find all increasing functions $f$ from the nonnegative integers to the integers satisfying $f(2)=7$ and \[ f(mn) = f(m) + f(n) + f(m)f(n) \] for all nonnegative integers $m$ and $n$.
Evaluate $\sum_{n=0}^{\infty}\frac{1}{(2n+1)2^{2n+1}}.$
Let $a, b, c$ be positive real numbers and let $[x]$ denote the greatest integer that does not exceed the real number $x$. Suppose that $f$ is a function defined on the set of non-negative integers $n$ and taking real values such that $f(0) = 0$ and
\[f(n) \leq an + f([bn]) + f([cn]), \qquad \text{ for all } n \geq 1.\]
Prove that if $b + c < 1$, there is a real number $k$ such that
\[f(n) \leq kn \qquad \text{ for all } n \qquad (1)\]
while if $b + c = 1$, there is a real number $K$ such that $f(n) \leq K n \log_2 n$ for all $n \geq 2$. Show that if $b + c = 1$, there may not be a real number $k$ that satisfies $(1).$
Prove that : $\ln \frac{11}{27}<\int_{\frac 14}^{\frac 34} \frac{1}{\ln (1-x)}\ dx<\ln \frac{7}{15}.$
Let $0<t<\frac{\pi}{2}$.
Evaluate
\[\lim_{t\rightarrow \frac{\pi}{2}} \int_0^t \tan \theta \sqrt{\cos \theta}\ln (\cos \theta)d\theta\]
Compute $\displaystyle\int_0^1\dfrac{dx}{\sqrt{x}+\sqrt[3]{x}}$.
Prove that the equation $4^x +6^x =9^x$ has no rational solutions.
Find the extremum of $f(t)=\int_1^t \frac{\ln x}{x+t}dx\ (t>0)$.
If $x = cos 1^o cos 2^o cos 3^o...cos 89^o$ and $y = cos 2^o cos 6^o cos 10^o...cos 86^o$, then what is the integer nearest to $\frac27 \log_2 \frac{y}{x}$ ?
Let $\otimes$ be a binary operation that takes two positive real numbers and returns a positive real number. Suppose further that $\otimes$ is continuous, commutative $(a\otimes b=b\otimes a)$, distributive across multiplication $(a\otimes(bc)=(a\otimes b)(a\otimes c))$, and that $2\otimes 2=4$. Solve the equation $x\otimes y=x$ for $y$ in terms of $x$ for $x>1$.
For what value of $ x$ does
\[ \log_{\sqrt{2}} \sqrt{x} \plus{} \log_2 x \plus{} \log_4 (x^2) \plus{} \log_8 (x^3) \plus{} \log_{16} (x^4) \equal{} 40?\]
$ \textbf{(A)}\ 8 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\ 256 \qquad \textbf{(E)}\ 1024$
Calculate the following indefinite integrals.
[1] $\int \frac{\sin x\cos x}{1+\sin ^ 2 x}dx$
[2] $\int x\log_{10} x dx$
[3] $\int \frac{x}{\sqrt{2x-1}}dx$
[4] $\int (x^2+1)\ln x dx$
[5] $\int e^x\cos x dx$
Determine the leftmost three digits of the number
\[1^1+2^2+3^3+...+999^{999}+1000^{1000}.\]
Show that \[\log_{a}bc+\log_bca+\log_cab \ge 4(\log_{ab}c+\log_{bc}a+\log_{ca}b)\] for all $a,b,c$ greater than 1.
Through the use of theorems on logarithms
\[ \log{\frac{a}{b}} \plus{} \log{\frac{b}{c}} \plus{} \log{\frac{c}{d}} \minus{} \log{\frac{ay}{dx}}\]
can be reduced to:
$ \textbf{(A)}\ \log{\frac{y}{x}}\qquad
\textbf{(B)}\ \log{\frac{x}{y}}\qquad
\textbf{(C)}\ 1\qquad
\textbf{(D)}\ 0\qquad
\textbf{(E)}\ \log{\frac{a^2y}{d^2x}}$
There is a unique positive integer $n$ such that \[\log_2{(\log_{16}{n})} = \log_4{(\log_4{n})}.\]
What is the sum of the digits of $n?$
$\textbf{(A) } 4 \qquad \textbf{(B) } 7 \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 11 \qquad \textbf{(E) } 13$
The value of $ 10^{\log_{10}7}$ is:
$ \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 10 \qquad\textbf{(D)}\ \log_{10} 7 \qquad\textbf{(E)}\ \log_7 10$