Found problems: 894
Today's calculation of integrals, 897
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.
1999 Brazil Team Selection Test, Problem 4
Let Q+ and Z denote the set of positive rationals and the set of inte-
gers, respectively. Find all functions f : Q+ → Z satisfying the following
conditions:
(i) f(1999) = 1;
(ii) f(ab) = f(a) + f(b) for all a, b ∈ Q+;
(iii) f(a + b) ≥ min{f(a), f(b)} for all a, b ∈ Q+.
2019 AIME Problems, 6
In a Martian civilization, all logarithms whose bases are not specified are assumed to be base $b$, for some fixed $b \geq 2$. A Martian student writes down
\begin{align*}3 \log(\sqrt{x}\log x) &= 56\\\log_{\log (x)}(x) &= 54
\end{align*}
and finds that this system of equations has a single real number solution $x > 1$. Find $b$.
1998 AMC 8, 17
Problems 15, 16, and 17 all refer to the following:
In the very center of the Irenic Sea lie the beautiful Nisos Isles. In 1998 the number of people on these islands is only 200, but the population triples every 25 years. Queen Irene has decreed that there must be at least 1.5 square miles for every person living in the Isles. The total area of the Nisos Isles is 24,900 square miles.
17. In how many years, approximately, from 1998 will the population of Nisos be as much as Queen Irene has proclaimed that the islands can support?
$ \text{(A)}\ 50\text{ yrs.}\qquad\text{(B)}\ 75\text{ yrs.}\qquad\text{(C)}\ 100\text{ yrs.}\qquad\text{(D)}\ 125\text{ yrs.}\qquad\text{(E)}\ 150\text{ yrs.} $
2010 Today's Calculation Of Integral, 522
Find $ \lim_{a\rightarrow{\infty}} \frac{1}{a^2}\int_0^a \ln (1\plus{}e^x)dx$.
1997 IMC, 3
Show that $\sum^{\infty}_{n=1}\frac{(-1)^{n-1}\sin(\log n)}{n^\alpha}$ converges iff $\alpha>0$.
2013 Today's Calculation Of Integral, 896
Given sequences $a_n=\frac{1}{n}{\sqrt[n] {_{2n}P_n}},\ b_n=\frac{1}{n^2}{\sqrt[n] {_{4n}P_{2n}}}$ and $c_n=\sqrt[n]{\frac{_{8n}P_{4n}}{_{6n}P_{4n}}}$, find $\lim_{n\to\infty} a_n,\ \lim_{n\to\infty} b_n$and $\lim_{n\to\infty} c_n.$
2002 AMC 12/AHSME, 14
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$
2010 Today's Calculation Of Integral, 538
Evaluate $ \int_1^{\sqrt{2}} \frac{x^2\plus{}1}{x\sqrt{x^4\plus{}1}}\ dx$.
1966 IMO Longlists, 30
Let $n$ be a positive integer, prove that :
[b](a)[/b] $\log_{10}(n + 1) > \frac{3}{10n} +\log_{10}n ;$
[b](b)[/b] $ \log n! > \frac{3n}{10}\left( \frac 12+\frac 13 +\cdots +\frac 1n -1\right).$
2009 Today's Calculation Of Integral, 399
Evaluate $ \int_0^{\sqrt{2}\minus{}1} \frac{1\plus{}x^2}{1\minus{}x^2}\ln \left(\frac{1\plus{}x}{1\minus{}x}\right)\ dx$.
2011 Iran MO (3rd Round), 3
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]
2011 Today's Calculation Of Integral, 730
Let $a_n$ be the local maximum of $f_n(x)=\frac{x^ne^{-x+n\pi}}{n!}\ (n=1,\ 2,\ \cdots)$ for $x>0$.
Find $\lim_{n\to\infty} \ln \left(\frac{a_{2n}}{a_n}\right)^{\frac{1}{n}}$.
2010 Today's Calculation Of Integral, 666
Let $f(x)$ be a function defined in $0<x<\frac{\pi}{2}$ satisfying:
(i) $f\left(\frac{\pi}{6}\right)=0$
(ii) $f'(x)\tan x=\int_{\frac{\pi}{6}}^x \frac{2\cos t}{\sin t}dt$.
Find $f(x)$.
[i]1987 Sapporo Medical University entrance exam[/i]
2010 Today's Calculation Of Integral, 613
Find the area of the part, in the $x$-$y$ plane, enclosed by the curve $|ye^{2x}-6e^{x}-8|=-(e^{x}-2)(e^{x}-4).$
[i]2010 Tokyo University of Agriculture and Technology entrance exam[/i]
2009 Unirea, 4
Evaluate the limit:
\[ \lim_{n \to \infty}{n \cdot \sin{1} \cdot \sin{2} \cdot \dots \cdot \sin{n}}.\]
Proposed to "Unirea" Intercounty contest, grade 11, Romania
1986 India National Olympiad, 2
Solve
\[ \left\{ \begin{array}{l}
\log_2 x\plus{}\log_4 y\plus{}\log_4 z\equal{}2 \\
\log_3 y\plus{}\log_9 z\plus{}\log_9 x\equal{}2 \\
\log_4 z\plus{}\log_{16} x\plus{}\log_{16} y\equal{}2 \\
\end{array} \right.\]
1989 AMC 12/AHSME, 15
Hi guys,
I was just reading over old posts that I made last year ( :P ) and saw how much the level of Getting Started became harder. To encourage more people from posting, I decided to start a Problem of the Day. This is how I'll conduct this:
1. In each post (not including this one since it has rules, etc) everyday, I'll post the problem. I may post another thread after it to give hints though.
2. Level of problem.. This is VERY important. All problems in this thread will be all AHSME or problems similar to this level. No AIME. Some AHSME problems, however, that involve tough insight or skills will not be posted. The chosen problems will be usually ones that everyone can solve after working. Calculators are allowed when you solve problems but it is NOT necessary.
3. Response.. All you have to do is simply solve the problem and post the solution. There is no credit given or taken away if you get the problem wrong. This isn't like other threads where the number of problems you get right or not matters. As for posting, post your solutions here in this thread. Do NOT PM me. Also, here are some more restrictions when posting solutions:
A. No single answer post. It doesn't matter if you put hide and say "Answer is ###..." If you don't put explanation, it simply means you cheated off from some other people. I've seen several posts that went like "I know the answer" and simply post the letter. What is the purpose of even posting then? Huh?
B. Do NOT go back to the previous problem(s). This causes too much confusion.
C. You're FREE to give hints and post different idea, way or answer in some cases in problems. If you see someone did wrong or you don't understand what they did, post here. That's what this thread is for.
4. Main purpose.. This is for anyone who visits this forum to enjoy math. I rememeber when I first came into this forum, I was poor at math compared to other people. But I kindly got help from many people such as JBL, joml88, tokenadult, and many other people that would take too much time to type. Perhaps without them, I wouldn't be even a moderator in this forum now. This site clearly made me to enjoy math more and more and I'd like to do the same thing. That's about the rule.. Have fun problem solving!
Next post will contain the Day 1 Problem. You can post the solutions until I post one. :D
2013 Today's Calculation Of Integral, 885
Find the infinite integrals as follows.
(1) 2013 Hiroshima City University entrance exam/Informatic Science
$\int \frac{x^2}{2-x^2}dx$
(2) 2013 Kanseigakuin University entrance exam/Science and Technology
$\int x^4\ln x\ dx$
(3) 2013 Shinsyu University entrance exam/Textile Science and Technology, Second-exam
$\int \frac{\cos ^ 3 x}{\sin ^ 2 x}\ dx$
Today's calculation of integrals, 896
Given sequences $a_n=\frac{1}{n}{\sqrt[n] {_{2n}P_n}},\ b_n=\frac{1}{n^2}{\sqrt[n] {_{4n}P_{2n}}}$ and $c_n=\sqrt[n]{\frac{_{8n}P_{4n}}{_{6n}P_{4n}}}$, find $\lim_{n\to\infty} a_n,\ \lim_{n\to\infty} b_n$and $\lim_{n\to\infty} c_n.$
1985 Canada National Olympiad, 4
Prove that $2^{n - 1}$ divides $n!$ if and only if $n = 2^{k - 1}$ for some positive integer $k$.
2010 Putnam, A4
Prove that for each positive integer $n,$ the number $10^{10^{10^n}}+10^{10^n}+10^n-1$ is not prime.
2014 IPhOO, 3
Consider a charged capacitor made with two square plates of side length $L$, uniformly charged, and separated by a very small distance $d$. The EMF across the capacitor is $\xi$. One of the plates is now rotated by a very small angle $\theta$ to the original axis of the capacitor. Find an expression for the difference in charge between the two plates of the capacitor, in terms of (if necessary) $d$, $\theta$, $\xi$, and $L$.
Also, approximate your expression by transforming it to algebraic form: i.e. without any non-algebraic functions. For example, logarithms and trigonometric functions are considered non-algebraic. Assume $ d << L $ and $ \theta \approx 0 $.
$\emph{Hint}$: You may assume that $ \frac {\theta L}{d} $ is also very small.
[i]Problem proposed by Trung Phan[/i]
[hide="Clarification"]
There are two possible ways to rotate the capacitor. Both were equally scored but this is what was meant: [asy]size(6cm);
real h = 7;
real w = 2;
draw((-w,0)--(-w,h));
draw((0,0)--(0,h), dashed);
draw((0,0)--h*dir(64));
draw(arc((0,0),2,64,90));
label("$\theta$", 2*dir(77), dir(77));
[/asy]
[/hide]
2023 AIME, 2
If $\sqrt{\log_bn}=\log_b\sqrt n$ and $b\log_bn=\log_bbn,$ then the value of $n$ is equal to $\frac jk,$ where $j$ and $k$ are relatively prime. What is $j+k$?
2010 Today's Calculation Of Integral, 562
(1) Show the following inequality for every natural number $ k$.
\[ \frac {1}{2(k \plus{} 1)} < \int_0^1 \frac {1 \minus{} x}{k \plus{} x}dx < \frac {1}{2k}\]
(2) Show the following inequality for every natural number $ m,\ n$ such that $ m > n$.
\[ \frac {m \minus{} n}{2(m \plus{} 1)(n \plus{} 1)} < \log \frac {m}{n} \minus{} \sum_{k \equal{} n \plus{} 1}^{m} \frac {1}{k} < \frac {m \minus{} n}{2mn}\]