Found problems: 20
PEN P Problems, 12
The positive function $p(n)$ is defined as the number of ways that the positive integer $n$ can be written as a sum of positive integers. Show that, for all positive integers $n \ge 2$, \[2^{\lfloor \sqrt{n}\rfloor}< p(n) < n^{3 \lfloor\sqrt{n}\rfloor }.\]
1953 AMC 12/AHSME, 16
Adams plans a profit of $ 10\%$ on the selling price of an article and his expenses are $ 15\%$ of sales. The rate of markup on an article that sells for $ \$5.00$ is:
$ \textbf{(A)}\ 20\% \qquad\textbf{(B)}\ 25\% \qquad\textbf{(C)}\ 30\% \qquad\textbf{(D)}\ 33\frac {1}{3}\% \qquad\textbf{(E)}\ 35\%$
1951 AMC 12/AHSME, 8
The price of an article is cut $ 10\%$. To restore it to its former value, the new price must be increased by:
$ \textbf{(A)}\ 10\% \qquad\textbf{(B)}\ 9\% \qquad\textbf{(C)}\ 11\frac {1}{9}\% \qquad\textbf{(D)}\ 11\% \qquad\textbf{(E)}\ \text{none of these answers}$
2005 AMC 12/AHSME, 20
Let $ a,b,c,d,e,f,g$ and $ h$ be distinct elements in the set
\[ \{ \minus{} 7, \minus{} 5, \minus{} 3, \minus{} 2,2,4,6,13\}.
\]What is the minimum possible value of
\[ (a \plus{} b \plus{} c \plus{} d)^2 \plus{} (e \plus{} f \plus{} g \plus{} h)^2
\]$ \textbf{(A)}\ 30\qquad
\textbf{(B)}\ 32\qquad
\textbf{(C)}\ 34\qquad
\textbf{(D)}\ 40\qquad
\textbf{(E)}\ 50$
1961 AMC 12/AHSME, 15
If $x$ men working $x$ hours a day for $x$ days produce $x$ articles, then the number of articles (not necessarily an integer) produced by $y$ men working $y$ hours a day for $y$ days is:
${{ \textbf{(A)}\ \frac{x^3}{y^2} \qquad\textbf{(B)}\ \frac{y^3}{x^2} \qquad\textbf{(C)}\ \frac{x^2}{y^3} \qquad\textbf{(D)}\ \frac{y^2}{x^3} }\qquad\textbf{(E)}\ y} $
2008 Harvard-MIT Mathematics Tournament, 5
A Vandal and a Moderator are editing a Wikipedia article. The article originally is error-free. Each day, the Vandal introduces one new error into the Wikipedia article. At the end of the day, the moderator checks the article and has a $ 2/3$ chance of catching each individual error still in the article. After $ 3$ days, what is the probability that the article is error-free?
2003 AIME Problems, 15
In $\triangle ABC$, $AB = 360$, $BC = 507$, and $CA = 780$. Let $M$ be the midpoint of $\overline{CA}$, and let $D$ be the point on $\overline{CA}$ such that $\overline{BD}$ bisects angle $ABC$. Let $F$ be the point on $\overline{BC}$ such that $\overline{DF} \perp \overline{BD}$. Suppose that $\overline{DF}$ meets $\overline{BM}$ at $E$. The ratio $DE: EF$ can be written in the form $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2013 AMC 10, 18
Let points $ A = (0,0) , \ B = (1,2), \ C = (3,3), $ and $ D = (4,0) $. Quadrilateral $ ABCD $ is cut into equal area pieces by a line passing through $ A $. This line intersects $ \overline{CD} $ at point $ \left (\frac{p}{q}, \frac{r}{s} \right ) $, where these fractions are in lowest terms. What is $ p + q + r + s $?
$ \textbf{(A)} \ 54 \qquad \textbf{(B)} \ 58 \qquad \textbf{(C)} \ 62 \qquad \textbf{(D)} \ 70 \qquad \textbf{(E)} \ 75 $
1953 AMC 12/AHSME, 43
If the price of an article is increased by percent $ p$, then the decrease in percent of sales must not exceed $ d$ in order to yield the same income. The value of $ d$ is:
$ \textbf{(A)}\ \frac{1}{1\plus{}p} \qquad\textbf{(B)}\ \frac{1}{1\minus{}p} \qquad\textbf{(C)}\ \frac{p}{1\plus{}p} \qquad\textbf{(D)}\ \frac{p}{p\minus{}1} \qquad\textbf{(E)}\ \frac{1\minus{}p}{1\plus{}p}$
2013 Online Math Open Problems, 48
$\omega$ is a complex number such that $\omega^{2013} = 1$ and $\omega^m \neq 1$ for $m=1,2,\ldots,2012$. Find the number of ordered pairs of integers $(a,b)$ with $1 \le a, b \le 2013$ such that \[ \frac{(1 + \omega + \cdots + \omega^a)(1 + \omega + \cdots + \omega^b)}{3} \] is the root of some polynomial with integer coefficients and leading coefficient $1$. (Such complex numbers are called [i]algebraic integers[/i].)
[i]Victor Wang[/i]
2008 Stanford Mathematics Tournament, 7
At the Rice Mathematics Tournament, 80% of contestants wear blue jeans, 70% wear tennis shoes, and 80% of those who wear blue jeans also wear tennis shoes. What fraction of people wearing tennis shoes are wearing blue jeans?
2009 AIME Problems, 13
The terms of the sequence $ (a_i)$ defined by $ a_{n \plus{} 2} \equal{} \frac {a_n \plus{} 2009} {1 \plus{} a_{n \plus{} 1}}$ for $ n \ge 1$ are positive integers. Find the minimum possible value of $ a_1 \plus{} a_2$.
1991 Arnold's Trivium, 77
Find the eigenvalues and their multiplicities of the Laplace operator $\Delta = \text{div grad}$ on a sphere of radius $R$ in Euclidean space of dimension $n$.
2005 AMC 12/AHSME, 22
A sequence of complex numbers $ z_0,z_1,z_2,....$ is defined by the rule
\[ z_{n \plus{} 1} \equal{} \frac {i z_n}{\overline{z_n}}
\]where $ \overline{z_n}$ is the complex conjugate of $ z_n$ and $ i^2 \equal{} \minus{} 1$. Suppose that $ |z_0| \equal{} 1$ and $ z_{2005} \equal{} 1$. How many possible values are there for $ z_0$?
$ \textbf{(A)}\ 1\qquad
\textbf{(B)}\ 2\qquad
\textbf{(C)}\ 4\qquad
\textbf{(D)}\ 2005\qquad
\textbf{(E)}\ 2^{2005}$
PEN E Problems, 2
Let $a, b, c, d$ be integers with $a>b>c>d>0$. Suppose that $ac+bd=(b+d+a-c)(b+d-a+c)$. Prove that $ab+cd$ is not prime.
1954 AMC 12/AHSME, 23
If the margin made on an article costing $ C$ dollars and selling for $ S$ dollars is $ M\equal{}\frac{1}{n}C$, then the margin is given by:
$ \textbf{(A)}\ M\equal{}\frac{1}{n\minus{}1}S \qquad
\textbf{(B)}\ M\equal{}\frac{1}{n}S \qquad
\textbf{(C)}\ M\equal{}\frac{n}{n\plus{}1}S \\
\textbf{(D)}\ M\equal{}\frac{1}{n\plus{}1}S \qquad
\textbf{(E)}\ M\equal{}\frac{n}{n\minus{}1}S$
1952 AMC 12/AHSME, 34
The price of an article was increased $ p\%$. Later the new price was decreased $ p\%$. If the last price was one dollar, the original price was:
$ \textbf{(A)}\ \frac {1 \minus{} p^2}{200} \qquad\textbf{(B)}\ \frac {\sqrt {1 \minus{} p^2}}{100} \qquad\textbf{(C)}\ \text{one dollar} \qquad\textbf{(D)}\ 1 \minus{} \frac {p^2}{10000 \minus{} p^2}$
$ \textbf{(E)}\ \frac {10000}{10000 \minus{} p^2}$
1965 AMC 12/AHSME, 32
An article costing $ C$ dollars is sold for $ \$100$ at a lostt of $ x$ percent of the selling price. It is then resold at a profit of $ x$ percent of the new selling price $ S'$. If the difference between $ S'$ and $ C$ is $ 1\frac {1}{9}$ dollars, then $ x$ is:
$ \textbf{(A)}\ \text{undetermined} \qquad \textbf{(B)}\ \frac {80}{9} \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ \frac {95}{9} \qquad \textbf{(E)}\ \frac {100}{9}$
1954 AMC 12/AHSME, 10
The sum of the numerical coefficients in the expansion of the binomial $ (a\plus{}b)^8$ is:
$ \textbf{(A)}\ 32 \qquad
\textbf{(B)}\ 16 \qquad
\textbf{(C)}\ 64 \qquad
\textbf{(D)}\ 48 \qquad
\textbf{(E)}\ 7$
2013 AMC 12/AHSME, 13
Let points $ A = (0,0) , \ B = (1,2), \ C = (3,3), $ and $ D = (4,0) $. Quadrilateral $ ABCD $ is cut into equal area pieces by a line passing through $ A $. This line intersects $ \overline{CD} $ at point $ \left (\frac{p}{q}, \frac{r}{s} \right ) $, where these fractions are in lowest terms. What is $ p + q + r + s $?
$ \textbf{(A)} \ 54 \qquad \textbf{(B)} \ 58 \qquad \textbf{(C)} \ 62 \qquad \textbf{(D)} \ 70 \qquad \textbf{(E)} \ 75 $