Found problems: 913
Evaluate $ \int_0^1 \frac{(1\minus{}x\plus{}x^2)\cos \ln (x\plus{}\sqrt{1\plus{}x^2})\minus{}\sqrt{1\plus{}x^2}\sin \ln (x\plus{}\sqrt{1\plus{}x^2})}{(1\plus{}x^2)^{\frac{3}{2}}}\ dx$.
Evaluate
\[\int_0^{\frac{\pi}{2}} \cos ^ {2004}x\cos 2004x\ dx\]
A system of n numbers $x_1, x_2, \ldots, x_n$ is given such that
\[x_1 = \log_{x_{n-1}} x_n, x_2 = \log_{x_{n}} x_1, \ldots, x_n = \log_{x_{n-2}} x_{n-1}.\]
Prove that $\prod_{k=1}^n x_k =1.$
Given $a_0 > 1$, the sequence $a_0, a_1, a_2, ...$ is such that for all $k > 0$, $a_k$ is the smallest integer greater than $a_{k-1}$ which is relatively prime to all the earlier terms in the sequence.
Find all $a_0$ for which all terms of the sequence are primes or prime powers.
Compare the values of the following definite integrals.
\[\int_{0}^{\infty}\ln \left(x+\frac{1}{x}\right)\frac{dx}{1+x^{2}},\ \ \int_{0}^{\frac{\pi}{2}}\left(\frac{\theta}{\sin \theta}\right)^{2}d\theta\]
Throw $ n$ balls in to $ 2n$ boxes. Suppose each ball comes into each box with equal probability of entering in any boxes.
Let $ p_n$ be the probability such that any box has ball less than or equal to one. Find the limit $ \lim_{n\to\infty} \frac{\ln p_n}{n}$
(a) A polynomial $P(x)$ with integer coefficients takes the value $-2$ for at least seven distinct integers $x$. Prove that it cannot take the value $1996$.
(b) Prove that there are irrational numbers $x,y$ such that $x^y$ is rational.
Suppose that $p(n)$ is the number of partitions of a natural number $n$. Prove that there exists $c>0$ such that $P(n)\ge n^{c \cdot \log n}$.
[i]proposed by Mohammad Mansouri[/i]
Suppose that for every $n$ the number $m(n)$ is chosen such that $m(n)\ln(m(n))=n-\frac 12$. Show that $b_n$ is asymptotic to the following expression where $b_n$ is the $n-$th Bell number, that is the number of ways to partition $\{1,2,\ldots,n\}$:
\[ \frac{m(n)^ne^{m(n)-n-\frac 12}}{\sqrt{\ln n}}. \]
Two functions $f(n)$ and $g(n)$ are asymptotic to each other if $\lim_{n\rightarrow \infty}\frac{f(n)}{g(n)}=1$.
Show that there exists a positive real number $x\neq 2$ such that $\log_2x=\frac{x}{2}$. Hence obtain the set of real numbers $c$ such that
\[\frac{\log_2x}{x}=c\]
has only one real solution.
Evaluate
\[\int_0^1 \frac{(x^2+x+1)^3\{\ln (x^2+x+1)+2\}}{(x^2+x+1)^3}(2x+1)e^{x^2+x+1}dx.\]
Let \[F=\log\dfrac{1+x}{1-x}.\] Find a new function $G$ by replacing each $x$ in $F$ by \[\dfrac{3x+x^3}{1+3x^2},\] and simplify. The simplified expression $G$ is equal to:
$\textbf{(A)}\ -F \qquad
\textbf{(B)}\ F\qquad
\textbf{(C)}\ 3F \qquad
\textbf{(D)}\ F^3 \qquad
\textbf{(E)}\ F^3-F$
Given $\frac{\log{a}}{p}=\frac{\log{b}}{q}=\frac{\log{c}}{r}=\log{x}$, all logarithms to the same base and $x \not= 1$. If $\frac{b^2}{ac}=x^y$, then $y$ is:
$ \text{(A)}\ \frac{q^2}{p+r}\qquad\text{(B)}\ \frac{p+r}{2q}\qquad\text{(C)}\ 2q-p-r\qquad\text{(D)}\ 2q-pr\qquad\text{(E)}\ q^2-pr$
Find the values of $a$ such that $\log (ax+1) = \log (x-a) + \log (2-x)$ has a unique real solution.
Find the maximum possible value of $8\cdot 27^{\log_6 x}+27\cdot 8^{\log_6 x}-x^3$ as $x$ varies over the positive real numbers.
To the nearest thousandth, $\log_{10}2$ is $.301$ and $\log_{10}3$ is $.477$. Which of the following is the best approximation of $\log_5 10$?
$\textbf{(A) }\frac{8}{7}\qquad\textbf{(B) }\frac{9}{7}\qquad\textbf{(C) }\frac{10}{7}\qquad\textbf{(D) }\frac{11}{7}\qquad\textbf{(E) }\frac{12}{7}$
Evaluate $ \int_{ \minus{} 1}^ {a^2} \frac {1}{x^2 \plus{} a^2}\ dx\ (a > 0).$
You may not use $ \tan ^{ \minus{} 1} x$ or Complex Integral here.
Let $n$ be a positive integer. Daniel and Merlijn are playing a game. Daniel
has $k$ sheets of paper lying next to each other on a table, where $k$ is a
positive integer. On each of the sheets, he writes some of the numbers
from $1$ up to $n$ (he is allowed to write no number at all, or all numbers).
On the back of each of the sheets, he writes down the remaining numbers.
Once Daniel is finished, Merlijn can flip some of the sheets of paper (he is
allowed to flip no sheet at all, or all sheets). If Merlijn succeeds in making
all of the numbers from $1$ up to n visible at least once, then he wins.
Determine the smallest $k$ for which Merlijn can always win, regardless of
Daniel’s actions.
([b]7[/b]) Evaluate the limit $ \lim_{n\rightarrow\infty}
n^{\minus{}\frac{1}{2}\left(1\plus{}\frac{1}{n}\right)}
\left(1^1\cdot2^2\cdot\cdots\cdot n^n\right)^{\frac{1}{n^2}}$.
Let $<a_n>$ be a sequence of non-negative real numbers such that $a_{m+n} \le a_m +a_n$ for all $m,n \in \mathbb{N}$.
Prove that
\[\sum_{k=1}^{N} \frac{a_k}{k^2}\ge \frac{a_N}{4N}\ln N\]
for any $N \in \mathbb{N}$, where $\ln$ denotes the natural logarithm.
The set of solutions of the equation $\log_{10}\left( a^2-15a\right)=2$ consists of
$ \textbf{(A)}\ \text{two integers } \qquad\textbf{(B)}\ \text{one integer and one fraction}\qquad$ $\textbf{(C)}\ \text{two irrational numbers }\qquad\textbf{(D)}\ \text{two non-real numbers} \qquad\textbf{(E)}\ \text{no numbers, that is, the empty set} $
Let $a$ and $b$ be positive real numbers such that $a+b=1$. Prove that $a^ab^b+a^bb^a\le 1$.
Determine whether it's possible to cover a $K_{2012}$ with
a) 1000 $K_{1006}$'s;
b) 1000 $K_{1006,1006}$'s.
[i]David Yang.[/i]
Let $ f$ be a function given by $ f(x) = \lg(x+1)-\frac{1}{2}\cdot\log_{3}x$.
a) Solve the equation $ f(x) = 0$.
b) Find the number of the subsets of the set \[ \{n | f(n^{2}-214n-1998) \geq 0,\ n \in\mathbb{Z}\}.\]
How many ordered triples of integers $ (a,b,c)$, with $ a \ge 2$, $ b\ge 1$, and $ c \ge 0$, satisfy both $ \log_a b \equal{} c^{2005}$ and $ a \plus{} b \plus{} c \equal{} 2005$?
$ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$