Found problems: 913
2004 Romania Team Selection Test, 4
Let $D$ be a closed disc in the complex plane. Prove that for all positive integers $n$, and for all complex numbers $z_1,z_2,\ldots,z_n\in D$ there exists a $z\in D$ such that $z^n = z_1\cdot z_2\cdots z_n$.
2006 MOP Homework, 1
Determine all positive real numbers $a$ such that there exists a positive integer $n$ and partition $A_1$, $A_2$, ..., $A_n$ of infinity sets of the set of the integers satisfying the following condition: for every set $A_i$, the positive difference of any pair of elements in $A_i$ is at least $a^i$.
1996 Canadian Open Math Challenge, 9
If $\log_{2n} 1994 = \log_n \left(486 \sqrt{2}\right)$, compute $n^6$.
2007 Harvard-MIT Mathematics Tournament, 33
Compute \[\int_1^2\dfrac{9x+4}{x^5+3x^2+x}dx.\]
(No, your TI-89 doesn’t know how to do this one. Yes, the end is near.)
2009 Harvard-MIT Mathematics Tournament, 3
Compute $e^A$ where $A$ is defined as \[\int_{3/4}^{4/3}\dfrac{2x^2+x+1}{x^3+x^2+x+1}dx.\]
1976 Euclid, 5
Source: 1976 Euclid Part A Problem 5
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If $\log_8 m+\log_8 \frac{1}{6}=\frac{2}{3}$, then $m$ equals
$\textbf{(A) } \frac{1}{2} \qquad \textbf{(B) } \frac{2}{3} \qquad \textbf{(C) } \frac{23}{6} \qquad \textbf{(D) } 4 \qquad \textbf{(E) } 24$
2009 Today's Calculation Of Integral, 451
Find $ \lim_{n\to\infty} \sum_{k\equal{}1}^n \ln \left(1\plus{}\frac{k^a}{n^{a\plus{}1}}\right).$
2005 Brazil Undergrad MO, 4
Let $a_{n+1} = a_n + \frac{1}{{a_n}^{2005}}$ and $a_1=1$. Show that $\sum^{\infty}_{n=1}{\frac{1}{n a_n}}$ converge.
2010 Today's Calculation Of Integral, 581
For real numer $ c$ for which $ cx^2\geq \ln (1\plus{}x^2)$ for all real numbers $ x$, find the value of $ c$ such that the area of the figure bounded by two curves $ y\equal{}cx^2$ and $ y\equal{}\ln (1\plus{}x^2)$ and two lines $ x\equal{}1,\ x\equal{}\minus{}1$ is 4.
1950 Miklós Schweitzer, 7
Examine the behavior of the expression
$ \sum_{\nu\equal{}1}^{n\minus{}1}\frac{\log(n\minus{}\nu)}{\nu}\minus{}\log^2 n$
as $ n\rightarrow \infty$
1976 Miklós Schweitzer, 3
Let $ H$ denote the set of those natural numbers for which $ \tau(n)$ divides $ n$, where $ \tau(n)$ is the number of divisors of $ n$. Show that
a) $ n! \in H$ for all sufficiently large $ n$,
b)$ H$ has density $ 0$.
[i]P. Erdos[/i]
2005 Putnam, A5
Evaluate $\int_0^1\frac{\ln(x+1)}{x^2+1}\,dx.$
1998 National Olympiad First Round, 2
Let $ A$, $ B$ be the number of digits of $ 2^{1998}$ and $ 5^{1998}$ in decimal system. $ A \plus{} B \equal{} ?$
$\textbf{(A)}\ 1998 \qquad\textbf{(B)}\ 1999 \qquad\textbf{(C)}\ 2000 \qquad\textbf{(D)}\ 3996 \qquad\textbf{(E)}\ 3998$
1962 AMC 12/AHSME, 17
If $ a \equal{} \log_8 225$ and $ b \equal{} \log_2 15,$ then $ a$, in terms of $ b,$ is:
$ \textbf{(A)}\ \frac{b}{2} \qquad
\textbf{(B)}\ \frac{2b}{3}\qquad
\textbf{(C)}\ b \qquad
\textbf{(D)}\ \frac{3b}{2} \qquad
\textbf{(E)}\ 2b$
2023 China Second Round, 2
if a,b∈R+,$a^{\log b}=2$,$a^{\log a}b^{\log b}=5$,find out $(ab)^{\log ab}$
2011 Today's Calculation Of Integral, 711
Evaluate $\int_e^{e^2} \frac{4(\ln x)^2+1}{(\ln x)^{\frac 32}}\ dx.$
2011 Today's Calculation Of Integral, 757
Evaluate
\[\int_0^1 \frac{(x^2+x+1)^3\{\ln (x^2+x+1)+2\}}{(x^2+x+1)^3}(2x+1)e^{x^2+x+1}dx.\]
1995 Irish Math Olympiad, 1
Prove that for every positive integer $ n$,
$ n^n \le (n!)^2 \le \left( \frac{(n\plus{}1)(n\plus{}2)}{6} \right) ^n.$
2014 Paenza, 1
Let $\{a_n\}_{n\geq 1}$ be a sequence of real numbers which satisfies the following relation:
\[a_{n+1}=10^n a_n^2\]
(a) Prove that if $a_1$ is small enough, then $\displaystyle\lim_{n\to\infty} a_n =0$.
(b) Find all possible values of $a_1\in \mathbb{R}$, $a_1\geq 0$, such that $\displaystyle\lim_{n\to\infty} a_n =0$.
2010 Today's Calculation Of Integral, 537
Evaluate $ \int_0^{\frac{\pi}{6}} \frac{\sqrt{1\plus{}\sin x}}{\cos x}\ dx$.
2010 Today's Calculation Of Integral, 575
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)$.
2009 ISI B.Stat Entrance Exam, 6
Let $f(x)$ be a function satisfying
\[xf(x)=\ln x \ \ \ \ \ \ \ \ \text{for} \ \ x>0\]
Show that $f^{(n)}(1)=(-1)^{n+1}n!\left(1+\frac{1}{2}+\cdots+\frac{1}{n}\right)$ where $f^{(n)}(x)$ denotes the $n$-th derivative evaluated at $x$.
1985 AMC 12/AHSME, 24
A non-zero digit is chosen in such a way that the probability of choosing digit $ d$ is $ \log_{10}(d\plus{}1) \minus{} \log_{10} d$. The probability that the digit $ 2$ is chosen is exactly $ \frac12$ the probability that the digit chosen is in the set
$ \textbf{(A)}\ \{2,3\} \qquad \textbf{(B)}\ \{3,4\} \qquad \textbf{(C)}\ \{4,5,6,7,8\} \qquad \textbf{(D)}\ \{5,6,7,8,9\} \qquad \textbf{(E)}\ \{4,5,6,7,8,9\}$
2012 Today's Calculation Of Integral, 773
For $x\geq 0$ find the value of $x$ by which $f(x)=\int_0^x 3^t(3^t-4)(x-t)dt$ is minimized.
1984 AMC 12/AHSME, 9
The number of digits in $4^{16} 5^{25}$ (when written in the usual base 10 form) is
A. 31
B. 30
C. 29
D. 28
E. 27