Found problems: 913
1998 National High School Mathematics League, 1
If $a>1,b>1,\lg(a+b)=\lg a+\lg b$, then the value of $\lg(a-1)+\lg(b-1)$ is
$\text{(A)}\lg2\qquad\text{(B)}1\qquad\text{(C)}0\qquad\text{(D)}$ not sure
2004 IMC, 5
Prove that
\[ \int^1_0 \int^1_0 \frac { dx \ dy }{ \frac 1x + |\log y| -1 } \leq 1 . \]
2001 District Olympiad, 4
a)Prove that $\ln(1+x)\le x,\ (\forall)x\ge 0$.
b)Let $a>0$. Prove that:
\[\lim_{n\to \infty} n\int_0^1\frac{x^n}{a+x^n}dx=\ln \frac{a+1}{a}\]
[i]***[/i]
1994 China Team Selection Test, 1
Given $5n$ real numbers $r_i, s_i, t_i, u_i, v_i \geq 1 (1 \leq i \leq n)$, let $R = \frac {1}{n} \sum_{i=1}^{n} r_i$, $S = \frac {1}{n} \sum_{i=1}^{n} s_i$, $T = \frac {1}{n} \sum_{i=1}^{n} t_i$, $U = \frac {1}{n} \sum_{i=1}^{n} u_i$, $V = \frac {1}{n} \sum_{i=1}^{n} v_i$. Prove that $\prod_{i=1}^{n}\frac {r_i s_i
t_i u_i v_i + 1}{r_i s_i t_i u_i v_i - 1} \geq \left(\frac {RSTUV +1}{RSTUV - 1}\right)^n$.
2009 Today's Calculation Of Integral, 509
Evaluate $ \int_0^{\frac{\pi}{4}} \frac{\tan x}{1\plus{}\sin x}\ dx$.
2004 China Team Selection Test, 2
Let u be a fixed positive integer. Prove that the equation $n! = u^{\alpha} - u^{\beta}$ has a finite number of solutions $(n, \alpha, \beta).$
2005 Today's Calculation Of Integral, 11
Calculate the following indefinite integrals.
[1] $\int \frac{6x+1}{\sqrt{3x^2+x+4}}dx$
[2] $\int \frac{e^x}{e^x+e^{a-x}}dx$
[3] $\int \frac{(\sqrt{x}+1)^3}{\sqrt{x}}dx$
[4] $\int x\ln (x^2-1)dx$
[5] $\int \frac{2(x+2)}{x^2+4x+1}dx$
2011 Today's Calculation Of Integral, 710
Evaluate $\int_0^{\frac{\pi}{4}} \frac{\sin \theta (\sin \theta \cos \theta +2)}{\cos ^ 4 \theta}\ d\theta$.
2006 Hanoi Open Mathematics Competitions, 3
Suppose that $a^{\log_{b}c}+b^{\log_{c}a}=m$.
Find the value of $c^{\log_{b}a}+a^{\log_{c}b}$ .
2012 Kyoto University Entry Examination, 1
Answer the following questions:
(1) Let $a$ be positive real number. Find $\lim_{n\to\infty} (1+a^{n})^{\frac{1}{n}}.$
(2) Evaluate $\int_1^{\sqrt{3}} \frac{1}{x^2}\ln \sqrt{1+x^2}dx.$
35 points
2021 JHMT HS, 8
Find the unique integer $a > 1$ that satisfies
\[ \int_{a}^{a^2} \left(\frac{1}{\ln x} - \frac{2}{(\ln x)^3}\right) dx = \frac{a}{\ln a}. \]
2010 Malaysia National Olympiad, 8
Show that \[\log_{a}bc+\log_bca+\log_cab \ge 4(\log_{ab}c+\log_{bc}a+\log_{ca}b)\] for all $a,b,c$ greater than 1.
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 $\lim_{n\to\infty} \ln \left(\frac{a_{2n}}{a_n}\right)^{\frac{1}{n}}$.
2011 China Second Round Olympiad, 9
Let $f(x)=|\log(x+1)|$ and let $a,b$ be two real numbers ($a<b$) satisfying the equations $f(a)=f\left(-\frac{b+1}{a+1}\right)$ and $f\left(10a+6b+21\right)=4\log 2$. Find $a,b$.
2008 Putnam, A6
Prove that there exists a constant $ c>0$ such that in every nontrivial finite group $ G$ there exists a sequence of length at most $ c\ln |G|$ with the property that each element of $ G$ equals the product of some subsequence. (The elements of $ G$ in the sequence are not required to be distinct. A [i]subsequence[/i] of a sequence is obtained by selecting some of the terms, not necessarily consecutive, without reordering them; for example, $ 4,4,2$ is a subesequence of $ 2,4,6,4,2,$ but $ 2,2,4$ is not.)
2010 Today's Calculation Of Integral, 537
Evaluate $ \int_0^{\frac{\pi}{6}} \frac{\sqrt{1\plus{}\sin x}}{\cos x}\ dx$.
2014 USAMTS Problems, 3:
Let $a_1,a_2,a_3,...$ be a sequence of positive real numbers such that:
(i) For all positive integers $m,n$, we have $a_{mn}=a_ma_n$
(ii) There exists a positive real number $B$ such that for all positive integers $m,n$ with $m<n$, we have $a_m < Ba_n$
Find all possible values of $\log_{2015}(a_{2015}) - \log_{2014}(a_{2014})$
1974 Miklós Schweitzer, 6
Let $ f(x)\equal{}\sum_{n\equal{}1}^{\infty} a_n/(x\plus{}n^2), \;(x \geq 0)\ ,$ where $ \sum_{n\equal{}1}^{\infty} |a_n|n^{\minus{} \alpha} < \infty$ for some $ \alpha > 2$. Let us assume that for some $ \beta > 1/{\alpha}$, we have $ f(x)\equal{}O(e^{\minus{}x^{\beta}})$ as $ x \rightarrow \infty$. Prove that $ a_n$ is identically $ 0$.
[i]G. Halasz[/i]
1997 APMO, 5
Suppose that $n$ people $A_1$, $A_2$, $\ldots$, $A_n$, ($n \geq 3$) are seated in a circle and that $A_i$ has $a_i$ objects such that
\[ a_1 + a_2 + \cdots + a_n = nN \]
where $N$ is a positive integer. In order that each person has the same number of objects, each person $A_i$ is to give or to receive a certain number of objects to or from its two neighbours $A_{i-1}$ and $A_{i+1}$. (Here $A_{n+1}$ means $A_1$ and $A_n$ means $A_0$.) How should this redistribution be performed so that the total number of objects transferred is minimum?
2012 Bogdan Stan, 3
$ \lim_{n\to\infty }\frac{1}{\sqrt[n]{n!}}\left\lfloor \log_5 \sum_{k=2}^{1+5^n} \sqrt[5^n]{k} \right\rfloor $
[i]Taclit Daniela Nadia[/i]
2005 Today's Calculation Of Integral, 14
Calculate the following indefinite integrals.
[1] $\int \frac{\sin x\cos x}{1+\sin ^ 2 x}dx$
[2] $\int x\log_{10} x dx$
[3] $\int \frac{x}{\sqrt{2x-1}}dx$
[4] $\int (x^2+1)\ln x dx$
[5] $\int e^x\cos x dx$
2000 AMC 12/AHSME, 7
How many positive integers $ b$ have the property that $ \log_b729$ is a positive integer?
$ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$
2010 Today's Calculation Of Integral, 615
For $0\leq a\leq 2$, find the minimum value of $\int_0^2 \left|\frac{1}{1+e^x}-\frac{1}{1+e^a}\right|\ dx.$
[i]2010 Kyoto Institute of Technology entrance exam/Textile e.t.c.[/i]
2011 International Zhautykov Olympiad, 3
Let $\mathbb{N}$ denote the set of all positive integers. An ordered pair $(a;b)$ of numbers $a,b\in\mathbb{N}$ is called [i]interesting[/i], if for any $n\in\mathbb{N}$ there exists $k\in\mathbb{N}$ such that the number $a^k+b$ is divisible by $2^n$. Find all [i]interesting[/i] ordered pairs of numbers.
2012 Stanford Mathematics Tournament, 3
Given that $\log_{10}2 \approx 0.30103$, find the smallest positive integer $n$ such that the decimal representation of $2^{10n}$ does not begin with the digit $1$.