Found problems: 913
2005 Baltic Way, 5
Let $a$, $b$, $c$ be positive real numbers such that $abc=1$. Prove that
\[\frac a{a^{2}+2}+\frac b{b^{2}+2}+\frac c{c^{2}+2}\leq 1 \]
2013 Romania Team Selection Test, 1
Fix a point $O$ in the plane and an integer $n\geq 3$. Consider a finite family $\mathcal{D}$ of closed unit discs in the plane such that:
(a) No disc in $\mathcal{D}$ contains the point $O$; and
(b) For each positive integer $k < n$, the closed disc of radius $k + 1$ centred at $O$ contains the centres of at least $k$ discs in $\mathcal{D}$.
Show that some line through $O$ stabs at least $\frac{2}{\pi} \log \frac{n+1}{2}$ discs in $\mathcal{D}$.
2007 Harvard-MIT Mathematics Tournament, 8
Let $A \text{ :}= \mathbb{Q}\setminus \{0,1\}$ denote the set of all rationals other than $0$ and $1$. A function $f:A\to \mathbb{R}$ has the property that for all $x\in A$, \[f(x)+f\left(1-\dfrac{1}{x}\right)=\log |x|.\] Compute the value of $f(2007)$.
2005 China Team Selection Test, 2
Determine whether $\sqrt{1001^2+1}+\sqrt{1002^2+1}+ \cdots + \sqrt{2000^2+1}$ be a rational number or not?
2007 Romania National Olympiad, 1
Let $\mathcal{F}$ be the set of functions $f: [0,1]\to\mathbb{R}$ that are differentiable, with continuous derivative, and $f(0)=0$, $f(1)=1$. Find the minimum of $\int_{0}^{1}\sqrt{1+x^{2}}\cdot \big(f'(x)\big)^{2}\ dx$ (where $f\in\mathcal{F}$) and find all functions $f\in\mathcal{F}$ for which this minimum is attained.
[hide="Comment"]
In the contest, this was the b) point of the problem. The a) point was simply ``Prove the Cauchy inequality in integral form''.
[/hide]
2007 Today's Calculation Of Integral, 175
Evaluate $\sum_{n=0}^{\infty}\frac{1}{(2n+1)2^{2n+1}}.$
2006 ISI B.Stat Entrance Exam, 8
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.
2009 District Olympiad, 3
Let $ A $ be the set of real solutions of the equation $ 3^x=x+2, $ and let be the set $ B $ of real solutions of the equation $ \log_3 (x+2) +\log_2 \left( 3^x-x \right) =3^x-1 . $ Prove the validity of the following subpoints:
[b]a)[/b] $ A\subset B. $
[b]b)[/b] $ B\not\subset\mathbb{Q} \wedge B\not\subset \mathbb{R}\setminus\mathbb{Q} . $
2011 Postal Coaching, 5
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.
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,$
\[ n^{n^2/2\minus{}Cn}e^{\minus{}n^2/4}\le f(n)\le n^{n^2/2\plus{}Cn}e^{\minus{}n^2/4}.\]
1970 AMC 12/AHSME, 8
If $a=\log_8225$ and $b=\log_215$, then
$\textbf{(A) }a=\frac{1}{2}b\qquad\textbf{(B) }a=\frac{2b}{3}\qquad\textbf{(C) }a=b\qquad\textbf{(D) }b=\frac{1}{2}a\qquad \textbf{(E) }a=\frac{3b}{2}$
1969 Miklós Schweitzer, 4
Show that the following inequality hold for all $ k \geq 1$, real numbers $ a_1,a_2,...,a_k$, and positive numbers $ x_1,x_2,...,x_k.$
\[ \ln \frac {\sum\limits_{i \equal{} 1}^kx_i}{\sum\limits_{i \equal{} 1}^kx_i^{1 \minus{} a_i}} \leq \frac {\sum\limits_{i \equal{} 1}^ka_ix_i \ln x_i}{\sum\limits_{i \equal{} 1}^kx_i} .
\]
[i]L. Losonczi[/i]
2007 IberoAmerican Olympiad For University Students, 3
Let $f:\mathbb{R}\to\mathbb{R}^+$ be a continuous and periodic function. Prove that for all $\alpha\in\mathbb{R}$ the following inequality holds:
$\int_0^T\frac{f(x)}{f(x+\alpha)}dx\ge T$,
where $T$ is the period of $f(x)$.
1978 IMO Longlists, 37
Simplify
\[\frac{1}{\log_a(abc)}+\frac{1}{\log_b(abc)}+\frac{1}{\log_c(abc)},\]
where $a, b, c$ are positive real numbers.
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 $a^i$.
2009 Indonesia TST, 2
Find the value of real parameter $ a$ such that $ 2$ is the smallest integer solution of \[ \frac{x\plus{}\log_2 (2^x\minus{}3a)}{1\plus{}\log_2 a} >2.\]
2007 Today's Calculation Of Integral, 219
Let $ f(x)\equal{}\left(1\plus{}\frac{1}{x}\right)^{x}\ (x>0)$.
Find $ \lim_{n\to\infty}\left\{f\left(\frac{1}{n}\right)f\left(\frac{2}{n}\right)f\left(\frac{3}{n}\right)\cdots\cdots f\left(\frac{n}{n}\right)\right\}^{\frac{1}{n}}$.
2012 Iran MO (3rd Round), 3
Prove that for each $n \in \mathbb N$ there exist natural numbers $a_1<a_2<...<a_n$ such that $\phi(a_1)>\phi(a_2)>...>\phi(a_n)$.
[i]Proposed by Amirhossein Gorzi[/i]
2011 Today's Calculation Of Integral, 763
Evaluate $\int_1^4 \frac{x-2}{(x^2+4)\sqrt{x}}dx.$
1991 USAMO, 2
For any nonempty set $\,S\,$ of numbers, let $\,\sigma(S)\,$ and $\,\pi(S)\,$ denote the sum and product, respectively, of the elements of $\,S\,$. Prove that
\[ \sum \frac{\sigma(S)}{\pi(S)} = (n^2 + 2n) - \left(1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} \right) (n+1), \]
where ``$\Sigma$'' denotes a sum involving all nonempty subsets $S$ of $\{1,2,3, \ldots,n\}$.
2012 Today's Calculation Of Integral, 774
Find the real number $a$ such that $\int_0^a \frac{e^x+e^{-x}}{2}dx=\frac{12}{5}.$
2010 Today's Calculation Of Integral, 547
Find the minimum value of $ \int_0^1 |e^{ \minus{} x} \minus{} a|dx\ ( \minus{} \infty < a < \infty)$.
2009 Today's Calculation Of Integral, 456
Find $ \lim_{n\to\infty} \frac{\pi}{n}\left\{\frac{1}{\sin \frac{\pi (n\plus{}1)}{4n}}\plus{}\frac{1}{\sin \frac{\pi (n\plus{}2)}{4n}}\plus{}\cdots \plus{}\frac{1}{\sin \frac{\pi (n\plus{}n)}{4n}}\right\}$
2012 Today's Calculation Of Integral, 796
Answer the following questions:
(1) Let $a$ be non-zero constant. Find $\int x^2 \cos (a\ln x)dx.$
(2) Find the volume of the solid generated by a rotation of the figures enclosed by the curve $y=x\cos (\ln x)$, the $x$-axis and
the lines $x=1,\ x=e^{\frac{\pi}{4}}$ about the $x$-axis.
2008 Federal Competition For Advanced Students, Part 2, 1
Prove the inequality
\[ \sqrt {a^{1 \minus{} a}b^{1 \minus{} b}c^{1 \minus{} c}} \le \frac {1}{3}
\]
holds for all positive real numbers $ a$, $ b$ and $ c$ with $ a \plus{} b \plus{} c \equal{} 1$.