Found problems: 913
2009 Today's Calculation Of Integral, 464
Evaluate $ \int_1^e \frac {(1 \plus{} 2x^2)\ln x}{\sqrt {1 \plus{} x^2}}\ dx$.
2009 Today's Calculation Of Integral, 434
Evaluate $ \int_0^1 \frac{x\minus{}e^{2x}}{x^2\minus{}e^{2x}}dx$.
2002 APMO, 1
Let $a_1,a_2,a_3,\ldots,a_n$ be a sequence of non-negative integers, where $n$ is a positive integer. Let
\[ A_n={a_1+a_2+\cdots+a_n\over n}\ . \]
Prove that
\[ a_1!a_2!\ldots a_n!\ge\left(\lfloor A_n\rfloor !\right)^n \]
where $\lfloor A_n\rfloor$ is the greatest integer less than or equal to $A_n$, and $a!=1\times 2\times\cdots\times a$ for $a\ge 1$(and $0!=1$). When does equality hold?
PEN D Problems, 22
Prove that $1980^{1981^{1982}} + 1982^{1981^{1980}}$ is divisible by $1981^{1981}$.
1948 Moscow Mathematical Olympiad, 145
Without tables and such, prove that $\frac{1}{\log_2 \pi}+\frac{1}{\log_5 \pi} >2$
1986 AIME Problems, 8
Let $S$ be the sum of the base 10 logarithms of all the proper divisors of 1000000. What is the integer nearest to $S$?
1969 AMC 12/AHSME, 25
If it is known that $\log_2a+\log_2b\geq 6$, then the least value that can be taken on by $a+b$ is:
$\textbf{(A) }2\sqrt6\qquad
\textbf{(B) }6\qquad
\textbf{(C) }8\sqrt2\qquad
\textbf{(D) }16\qquad
\textbf{(E) }\text{none of these.}$
2012 Putnam, 1
Let $S$ be a class of functions from $[0,\infty)$ to $[0,\infty)$ that satisfies:
(i) The functions $f_1(x)=e^x-1$ and $f_2(x)=\ln(x+1)$ are in $S;$
(ii) If $f(x)$ and $g(x)$ are in $S,$ the functions $f(x)+g(x)$ and $f(g(x))$ are in $S;$
(iii) If $f(x)$ and $g(x)$ are in $S$ and $f(x)\ge g(x)$ for all $x\ge 0,$ then the function $f(x)-g(x)$ is in $S.$
Prove that if $f(x)$ and $g(x)$ are in $S,$ then the function $f(x)g(x)$ is also in $S.$
PEN G Problems, 17
Suppose that $p, q \in \mathbb{N}$ satisfy the inequality \[\exp(1)\cdot( \sqrt{p+q}-\sqrt{q})^{2}<1.\] Show that $\ln \left(1+\frac{p}{q}\right)$ is irrational.
1966 AMC 12/AHSME, 24
If $\log_MN=\log_NM$, $M\ne N$, $MN>0$, $M\ne 1$, $N\ne 1$, then $MN$ equals:
$\text{(A)} \ \frac12 \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ 10 \qquad \text{(E)} \ \text{a number greater than 2 and less than 10}$
1997 Finnish National High School Mathematics Competition, 1
Determine the real numbers $a$ such that the equation $a 3^x + 3^{-x} = 3$ has exactly one solution $x.$
1984 IMO Longlists, 60
Determine all pairs $(a, b)$ of positive real numbers with $a \neq 1$ such that
\[\log_a b < \log_{a+1} (b + 1).\]
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{} ?$
$\textbf{(A)}\ 1998 \qquad\textbf{(B)}\ 1999 \qquad\textbf{(C)}\ 2000 \qquad\textbf{(D)}\ 3996 \qquad\textbf{(E)}\ 3998$
1953 AMC 12/AHSME, 39
The product, $ \log_a b \cdot \log_b a$ is equal to:
$ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ a \qquad\textbf{(C)}\ b \qquad\textbf{(D)}\ ab \qquad\textbf{(E)}\ \text{none of these}$
2013 Today's Calculation Of Integral, 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.
2018 AMC 12/AHSME, 14
The solution to the equation $\log_{3x} 4 = \log_{2x} 8$, where $x$ is a positive real number other than $\tfrac{1}{3}$ or $\tfrac{1}{2}$, can be written as $\tfrac {p}{q}$ where $p$ and $q$ are relatively prime positive integers. What is $p + q$?
$\textbf{(A) } 5 \qquad
\textbf{(B) } 13 \qquad
\textbf{(C) } 17 \qquad
\textbf{(D) } 31 \qquad
\textbf{(E) } 35 $
2018 Bosnia And Herzegovina - Regional Olympiad, 3
If numbers $x_1$, $x_2$,...,$x_n$ are from interval $\left( \frac{1}{4},1 \right)$ prove the inequality:
$\log _{x_1} {\left(x_2-\frac{1}{4} \right)} + \log _{x_2} {\left(x_3-\frac{1}{4} \right)}+ ... + \log _{x_{n-1}} {\left(x_n-\frac{1}{4} \right)} + \log _{x_n} {\left(x_1-\frac{1}{4} \right)} \geq 2n$
2007 Putnam, 6
For each positive integer $ n,$ let $ f(n)$ be the number of ways to make $ n!$ cents using an unordered collection of coins, each worth $ k!$ cents for some $ k,\ 1\le k\le n.$ Prove that for some constant $ C,$ independent of $ n,$
\[ n^{n^2/2\minus{}Cn}e^{\minus{}n^2/4}\le f(n)\le n^{n^2/2\plus{}Cn}e^{\minus{}n^2/4}.\]
2010 Victor Vâlcovici, 2
Let $ f:[2,\infty )\rightarrow\mathbb{R} $ be a differentiable function satisfying $ f(2)=0 $ and
$$ \frac{df}{dx}=\frac{2}{x^2+f^4{x}} , $$
for any $ x\in [2,\infty ) . $ Show that there exists $ \lim_{x\to\infty } f(x) $ and is at most $ \ln 3. $
[i]Gabriel Daniilescu[/i]
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\}$
1998 Vietnam National Olympiad, 1
Let $a\geq 1$ be a real number. Put $x_{1}=a,x_{n+1}=1+\ln{(\frac{x_{n}^{2}}{1+\ln{x_{n}}})}(n=1,2,...)$. Prove that the sequence $\{x_{n}\}$ converges and find its limit.
2007 Baltic Way, 1
For a positive integer $n$ consider any partition of the set $\{ 1,2,\ldots ,2n \}$ into $n$ two-element subsets $P_1,P_2\ldots,P_n$. In each subset $P_i$, let $p_i$ be the product of the two numbers in $P_i$. Prove that
\[\frac{1}{p_1}+\frac{1}{p_2}+\ldots + \frac{1}{p_n}<1 \]
1971 IMO Longlists, 44
Let $m$ and $n$ denote integers greater than $1$, and let $\nu (n)$ be the number of primes less than or equal to $n$. Show that if the equation $\frac{n}{\nu(n)}=m$ has a solution, then so does the equation $\frac{n}{\nu(n)}=m-1$.
2013 Stanford Mathematics Tournament, 8
According to Moor's Law, the number of shoes in Moor's room doubles every year. In 2013, Moor's room starts out having exactly one pair of shoes. If shoes always come in unique, matching pairs, what is the earliest year when Moor has the ability to wear at least 500 mismatches pairs of shoes? Note that left and right shoes are distinct, and Moor must always wear one of each.
2021 Purple Comet Problems, 17
For real numbers $x$ let $$f(x)=\frac{4^x}{25^{x+1}}+\frac{5^x}{2^{x+1}}.$$ Then $f\left(\frac{1}{1-\log_{10}4}\right)=\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.