Found problems: 913
2008 persons take part in a programming contest. In one round, the 2008 programmers are divided into two groups. Find the minimum number of groups such that every two programmers ever be in the same group.
The solution of the equation $ 7^{x\plus{}7}\equal{}8^x$ can be expressed in the form $ x\equal{}\log_b 7^7$. What is $ b$?
$ \textbf{(A)}\ \frac{7}{15} \qquad
\textbf{(B)}\ \frac{7}{8} \qquad
\textbf{(C)}\ \frac{8}{7} \qquad
\textbf{(D)}\ \frac{15}{8} \qquad
\textbf{(E)}\ \frac{15}{7}$
Prove the following identity for every $ n\in N$:
$ \sum_{j\plus{}h\equal{}n,j\geq h}\frac{(\minus{}1)^h2^{j\minus{}h}\binom{j}{h}}{j}\equal{}\frac{2}{n}$
Square $ ABCD$ has area $ 36,$ and $ \overline{AB}$ is parallel to the x-axis. Vertices $ A,$ $ B,$ and $ C$ are on the graphs of $ y \equal{} \log_{a}x,$ $ y \equal{} 2\log_{a}x,$ and $ y \equal{} 3\log_{a}x,$ respectively. What is $ a?$
$ \textbf{(A)}\ \sqrt [6]{3}\qquad \textbf{(B)}\ \sqrt {3}\qquad \textbf{(C)}\ \sqrt [3]{6}\qquad \textbf{(D)}\ \sqrt {6}\qquad \textbf{(E)}\ 6$
Let $a_n\ (n\geq 1)$ be the value for which $\int_x^{2x} e^{-t^n}dt\ (x\geq 0)$ is maximal. Find $\lim_{n\to\infty} \ln a_n.$
Solve in the set of real numbers the equation $ a^{[ x ]} +\log_a\{ x \} =x , $ where $ a $ is a real number from the interval $ (0,1). $
$ [] $ and $ \{\} $ [i]denote the floor, respectively, the fractional part.[/i]
$p,q$ satisfies $px+q\geq \ln x$ at $a\leq x\leq b\ (0<a<b)$.
Find the value of $p,q$ for which the following definite integral is minimized and then the minimum value.
\[\int_a^b (px+q-\ln x)dx\]
Find at least one root of the equation$$\sin(2 \log_2 x) + tg(3\log_2 x) = \sin6+tg9$$less than $0.01$.
$(a)$ Compute -
\begin{align*}
\frac{\mathrm{d}}{\mathrm{d}x} \bigg[ \int_{0}^{e^x} \log ( t ) \cos^4 ( t ) \mathrm{d}t \bigg]
\end{align*}
$(b)$ For $x > 0 $ define $F ( x ) = \int_{1}^{x} t \log ( t ) \mathrm{d}t . $\\
\\$1.$ Determine the open interval(s) (if any) where $F ( x )$ is decreasing and all the open interval(s) (if any) where $F ( x )$ is increasing.\\
\\$2.$ Determine all the local minima of $F ( x )$ (if any) and all the local maxima of $F ( x )$ (if any) $.$
For every positive integer $n$, form the number $n/s(n)$, where $s(n)$ is the sum of digits of $n$ in base 10. Determine the minimum value of $n/s(n)$ in each of the following cases:
(i) $10 \leq n \leq 99$
(ii) $100 \leq n \leq 999$
(iii) $1000 \leq n \leq 9999$
(iv) $10000 \leq n \leq 99999$
Evaluate $\lim_{x\rightarrow 0^+}\int^{2x}_x\frac{\sin^m(t)}{t^n}dt$. ($m,n\in\mathbb{N}$)
Let $x_1, x_2 , x_3, .... x_n$ be $n$ real numbers such that $0 < x_j < \frac{1}{2}$. Prove that \[ \frac{ \prod\limits_{j=1}^{n} x_j } { \left( \sum\limits_{j=1}^{n} x_j \right)^n} \leq \frac{ \prod\limits_{j=1}^{n} (1-x_j) } { \left( \sum\limits_{j=1}^{n} (1 - x_j) \right)^n} \]
If you are given $ \log 8 \approx .9031$ and $ \log 9 \approx .9542$, then the only logarithm that cannot be found without the use of tables is:
$ \textbf{(A)}\ \log 17 \qquad\textbf{(B)}\ \log \frac {5}{4} \qquad\textbf{(C)}\ \log 15 \qquad\textbf{(D)}\ \log 600 \qquad\textbf{(E)}\ \log .4$
Simplify
\[\frac{1}{\log_a(abc)}+\frac{1}{\log_b(abc)}+\frac{1}{\log_c(abc)},\]
where $a, b, c$ are positive real numbers.
If $x=(\log_82)^{(\log_28)}$, then $\log_3x$ equals:
$\text{(A)} \ -3 \qquad \text{(B)} \ -\frac13 \qquad \text{(C)} \ \frac13 \qquad \text{(D)} \ 3 \qquad \text{(E)} \ 9$
Evaluate $ \int_0^{\frac{\pi}{4}} \frac{\tan x}{1\plus{}\sin x}\ dx$.
Take the square with vertices $(0,0)$, $(1,0)$, $(0,1)$, and $(1,1)$. Choose a random point in this square and draw the line segment from it to $(0,0)$. Choose a second random point in this square and draw the line segment from it to $(1,0)$. What is the probability that the two line segments intersect?
The equation $2^{2x}-8\cdot 2^x+12=0$ is satisfied by:
$\textbf{(A) }\log3\qquad
\textbf{(B) }\tfrac12\log6\qquad
\textbf{(C) }1+\log\tfrac34\qquad$
$\textbf{(D) }1+\tfrac{\log3}{\log2}\qquad
\textbf{(E) }\text{none of these}$
If $\log_7 \Big(\log_3 (\log_2 x) \Big) = 0$, then $x^{-1/2}$ equals
$\displaystyle \text{(A)} \ \frac{1}{3} \qquad \text{(B)} \ \frac{1}{2 \sqrt 3} \qquad \text{(C)} \ \frac{1}{3 \sqrt 3} \qquad \text{(D)} \ \frac{1}{\sqrt{42}} \qquad \text{(E)} \ \text{none of these}$
Prove that for all natural numbers $n$ greater than $1$ :
$$[\sqrt{n}] + [\sqrt[3]{n}] +...+[ \sqrt[n]{n}] = [\log_2 n] + [\log_3 n] + ... + [\log_n n]$$
(VV Kisil)
Solve in the interval $ (2,\infty ) $ the following equation:
$$ 1=\cos\left( \pi\log_3 (x+6)\right)\cdot\cos\left( \pi\log_3 (x-2)\right) . $$
Prove that
\[ \sum_{n=1}^\infty {1\over n^n} = \int_0^1 x^{-x}\,dx. \]
For $ 0\leq x <\frac{\pi}{2}$, prove the following inequality.
$ x\plus{}\ln (\cos x)\plus{}\int_0^1 \frac{t}{1\plus{}t^2}\ dt\leq \frac{\pi}{4}$
Evaluate $\int_2^a \frac{x^a-1-xa^x\ln a}{(x^a-1)^2}dx.$
proposed by kunny
Determine all functions $f:[0,\infty)\rightarrow\mathbb{R}$ such that $f(0)=0$ and \[f(x)=1+5f\left(\left\lfloor{\frac{x}{2}\right\rfloor}\right)-6f\left(\left\lfloor{\frac{x}{4}\right\rfloor}\right)\] for all $x>0$.