Found problems: 894
2013 USAMTS Problems, 5
Let $S$ be a planar region. A $\emph{domino-tiling}$ of $S$ is a partition of $S$ into $1\times2$ rectangles. (For example, a $2\times3$ rectangle has exactly $3$ domino-tilings, as shown below.)
[asy]
import graph; size(7cm);
pen dps = linewidth(0.7); defaultpen(dps);
draw((0,0)--(3,0)--(3,2)--(0,2)--cycle, linewidth(2));
draw((4,0)--(4,2)--(7,2)--(7,0)--cycle, linewidth(2));
draw((8,0)--(8,2)--(11,2)--(11,0)--cycle, linewidth(2));
draw((1,0)--(1,2));
draw((2,1)--(3,1));
draw((0,1)--(2,1), linewidth(2));
draw((2,0)--(2,2), linewidth(2));
draw((4,1)--(7,1));
draw((5,0)--(5,2), linewidth(2));
draw((6,0)--(6,2), linewidth(2));
draw((8,1)--(9,1));
draw((10,0)--(10,2));
draw((9,0)--(9,2), linewidth(2));
draw((9,1)--(11,1), linewidth(2));
[/asy]
The rectangles in the partition of $S$ are called $\emph{dominoes}$.
(a) For any given positive integer $n$, find a region $S_n$ with area at most $2n$ that has exactly $n$ domino-tilings.
(b) Find a region $T$ with area less than $50000$ that has exactly $100002013$ domino-tilings.
1954 AMC 12/AHSME, 15
$ \log 125$ equals:
$ \textbf{(A)}\ 100 \log 1.25 \qquad \textbf{(B)}\ 5 \log 3 \qquad \textbf{(C)}\ 3 \log 25$
$ \textbf{(D)}\ 3 \minus{} 3\log 2 \qquad \textbf{(E)}\ (\log 25)(\log 5)$
2002 AIME Problems, 6
The solutions to the system of equations
\begin{eqnarray*} \log_{225}{x}+\log_{64}{y} &=& 4\\ \log_x{225}-\log_y{64} &=& 1 \end{eqnarray*}
are $(x_1,y_1)$ and $(x_2, y_2).$ Find $\log_{30}{(x_1y_1x_2y_2)}.$
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.
2003 Purple Comet Problems, 6
Evaluate:
\[\frac{1}{\log_2 (\frac{1}{6})} - \frac{1}{\log_3 (\frac{1}{6})} - \frac{1}{\log_4 (\frac{1}{6})}\]
2010 AMC 12/AHSME, 11
The solution of the equation $ 7^{x\plus{}7}\equal{}8^x$ can be expressed in the form $ x\equal{}\log_b 7^7$. What is $ b$?
$ \textbf{(A)}\ \frac{7}{15} \qquad
\textbf{(B)}\ \frac{7}{8} \qquad
\textbf{(C)}\ \frac{8}{7} \qquad
\textbf{(D)}\ \frac{15}{8} \qquad
\textbf{(E)}\ \frac{15}{7}$
2010 Putnam, A6
Let $f:[0,\infty)\to\mathbb{R}$ be a strictly decreasing continuous function such that $\lim_{x\to\infty}f(x)=0.$ Prove that $\displaystyle\int_0^{\infty}\frac{f(x)-f(x+1)}{f(x)}\,dx$ diverges.
1991 Vietnam Team Selection Test, 3
Let $\{x\}$ be a sequence of positive reals $x_1, x_2, \ldots, x_n$, defined by: $x_1 = 1, x_2 = 9, x_3=9, x_4=1$. And for $n \geq 1$ we have:
\[x_{n+4} = \sqrt[4]{x_{n} \cdot x_{n+1} \cdot x_{n+2} \cdot x_{n+3}}.\]
Show that this sequence has a finite limit. Determine this limit.
2010 Today's Calculation Of Integral, 602
Prove the following inequality.
\[\frac{e-1}{n+1}\leqq\int^e_1(\log x)^n dx\leqq\frac{(n+1)e+1}{(n+1)(n+2)}\ (n=1,2,\cdot\cdot\cdot) \]
1994 Kyoto University entrance exam/Science
2005 AMC 12/AHSME, 13
Suppose that $ 4^{x_1} \equal{} 5, 5^{x_2} \equal{} 6, 6^{x_3} \equal{} 7,...,127^{x_{124}} \equal{} 128$. What is $ x_1x_2 \cdots x_{124}$?
$ \textbf{(A)}\ 2\qquad
\textbf{(B)}\ \frac {5}{2}\qquad
\textbf{(C)}\ 3\qquad
\textbf{(D)}\ \frac {7}{2}\qquad
\textbf{(E)}\ 4$
Today's calculation of integrals, 893
Find the minimum value of $f(x)=\int_0^{\frac{\pi}{4}} |\tan t-x|dt.$
2010 Contests, 1
Prove that $ 7^{2^{20}} + 7^{2^{19}} + 1 $ has at least $ 21 $ distinct prime divisors.
2013 ELMO Shortlist, 9
Let $a, b, c$ be positive reals, and let $\sqrt[2013]{\frac{3}{a^{2013}+b^{2013}+c^{2013}}}=P$. Prove that
\[\prod_{\text{cyc}}\left(\frac{(2P+\frac{1}{2a+b})(2P+\frac{1}{a+2b})}{(2P+\frac{1}{a+b+c})^2}\right)\ge \prod_{\text{cyc}}\left(\frac{(P+\frac{1}{4a+b+c})(P+\frac{1}{3b+3c})}{(P+\frac{1}{3a+2b+c})(P+\frac{1}{3a+b+2c})}\right).\][i]Proposed by David Stoner[/i]
2013 Today's Calculation Of Integral, 887
For the function $f(x)=\int_0^x \frac{dt}{1+t^2}$, answer the questions as follows.
Note : Please solve the problems without using directly the formula $\int \frac{1}{1+x^2}\ dx=\tan^{-1}x +C$ for Japanese High School students those who don't study arc sin x, arc cos x, arc tanx.
(1) Find $f(\sqrt{3})$
(2) Find $\int_0^{\sqrt{3}} xf(x)\ dx$
(3) Prove that for $x>0$. $f(x)+f\left(\frac{1}{x}\right)$ is constant, then find the value.
2019 Korea USCM, 5
A sequence $\{a_n\}_{n\geq 1}$ is defined by a recurrence relation
$$a_1 = 1,\quad a_{n+1} = \log \frac{e^{a_n}-1}{a_n}$$
And a sequence $\{b_n\}_{n\geq 1}$ is defined as $b_n = \prod\limits_{i=1}^n a_i$. Evaluate an infinite series $\sum\limits_{n=1}^\infty b_n$.
1999 USAMTS Problems, 2
Let $a$ be a positive real number, $n$ a positive integer, and define the [i]power tower[/i] $a\uparrow n$ recursively with $a\uparrow 1=a$, and $a\uparrow(i+1)=a^{a\uparrow i}$ for $i=1,2,3,\ldots$. For example, we have $4\uparrow 3=4^{(4^4)}=4^{256}$, a number which has $155$ digits. For each positive integer $k$, let $x_k$ denote the unique positive real number solution of the equation $x\uparrow k=10\uparrow (k+1)$. Which is larger: $x_{42}$ or $x_{43}$?
1991 Romania Team Selection Test, 3
Prove the following identity for every $ n\in N$:
$ \sum_{j\plus{}h\equal{}n,j\geq h}\frac{(\minus{}1)^h2^{j\minus{}h}\binom{j}{h}}{j}\equal{}\frac{2}{n}$
2009 Indonesia TST, 1
2008 persons take part in a programming contest. In one round, the 2008 programmers are divided into two groups. Find the minimum number of groups such that every two programmers ever be in the same group.
2009 Putnam, B1
Show that every positive rational number can be written as a quotient of products of factorials of (not necessarily distinct) primes. For example, $ \frac{10}9\equal{}\frac{2!\cdot 5!}{3!\cdot 3!\cdot 3!}.$
2005 Today's Calculation Of Integral, 23
Evaluate
\[\lim_{a\rightarrow \frac{\pi}{2}-0}\ \int_0^a\ (\cos x)\ln (\cos x)\ dx\ \left(0\leqq a <\frac{\pi}{2}\right)\]
2012 AIME Problems, 9
Let $x$, $y$, and $z$ be positive real numbers that satisfy \[ 2\log_x(2y) = 2\log_{2x}(4z) = \log_{2x^4}(8yz) \neq 0. \] The value of $xy^5z$ can be expressed in the form $\frac{1}{2^{p/q}}$, where $p$ and $q$ are relatively prime integers. Find $p+q$.
1992 Hungary-Israel Binational, 3
We examine the following two sequences: The Fibonacci sequence: $F_{0}= 0, F_{1}= 1, F_{n}= F_{n-1}+F_{n-2 }$ for $n \geq 2$; The Lucas sequence: $L_{0}= 2, L_{1}= 1, L_{n}= L_{n-1}+L_{n-2}$ for $n \geq 2$. It is known that for all $n \geq 0$ \[F_{n}=\frac{\alpha^{n}-\beta^{n}}{\sqrt{5}},L_{n}=\alpha^{n}+\beta^{n},\] where $\alpha=\frac{1+\sqrt{5}}{2},\beta=\frac{1-\sqrt{5}}{2}$. These formulae can be used without proof.
We call a nonnegative integer $r$-Fibonacci number if it is a sum of $r$ (not necessarily distinct) Fibonacci numbers. Show that there infinitely many positive integers that are not $r$-Fibonacci numbers for any $r, 1 \leq r\leq 5.$
2006 District Olympiad, 1
Let $ a,b,c\in (0,1)$ and $ x,y,z\in (0, \plus{} \infty)$ be six real numbers such that
\[ a^x \equal{} bc , \quad b^y \equal{} ca , \quad c^z \equal{} ab .\]
Prove that
\[ \frac 1{2 \plus{} x} \plus{} \frac 1{2 \plus{} y} \plus{} \frac 1{2 \plus{} z} \leq \frac 34 .\]
[i]Cezar Lupu[/i]
1965 AMC 12/AHSME, 31
The number of real values of $ x$ satisfying the equality $ (\log_2x)(\log_bx) \equal{} \log_ab$, where $ a > 0$, $ b > 0$, $ a \neq 1$, $ b \neq 1$, is:
$ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ \text{a finite integer greater than 2} \qquad \textbf{(E)}\ \text{not finite}$
2011 AMC 12/AHSME, 17
Let $f\left(x\right)=10^{10x}, g\left(x\right)=\log_{10}\left(\frac{x}{10}\right), h_1\left(x\right)=g\left(f\left(x\right)\right),$ and $h_n\left(x\right)=h_1\left(h_{n-1}\left(x\right)\right)$ for integers $n \ge 2$. What is the sum of the digits of $h_{2011}\left(1\right)$?
$ \textbf{(A)}\ 16,081 \qquad
\textbf{(B)}\ 16,089 \qquad
\textbf{(C)}\ 18,089 \qquad
\textbf{(D)}\ 18,098 \qquad
\textbf{(E)}\ 18,099 $