Found problems: 3632
1974 AMC 12/AHSME, 1
If $x\neq 0$ or $4$ and $y \neq 0$ or $6$, then $\frac{2}{x}+\frac{3}{y}=\frac{1}{2}$ is equivalent to
$ \textbf{(A)}\ 4x+3y=xy \qquad\textbf{(B)}\ y=\frac{4x}{6-y} \qquad\textbf{(C)}\ \frac{x}{2}+\frac{y}{3}=2 \\ \qquad\textbf{(D)}\ \frac{4y}{y-6}=x \qquad\textbf{(E)}\ \text{none of these} $
2011 AIME Problems, 1
Gary purchased a large beverage, but drank only $m/n$ of this beverage, where $m$ and $n$ are relatively prime positive integers. If Gary had purchased only half as much and drunk twice as much, he would have wasted only $\frac{2}{9}$ as much beverage. Find $m+n$.
2012 AMC 8, 14
In the BIG N, a middle school football conference, each team plays every other team exactly once. If a total of 21 conference games were played during the 2012 season, how many teams were members of the BIG N conference?
$\textbf{(A)}\hspace{.05in}6 \qquad \textbf{(B)}\hspace{.05in}7 \qquad \textbf{(C)}\hspace{.05in}8 \qquad \textbf{(D)}\hspace{.05in}9 \qquad \textbf{(E)}\hspace{.05in}10 $
1986 USAMO, 2
During a certain lecture, each of five mathematicians fell asleep exactly twice. For each pair of mathematicians, there was some moment when both were asleep simultaneously. Prove that, at some moment, three of them were sleeping simultaneously.
2017 AMC 12/AHSME, 24
Quadrilateral $ABCD$ has right angles at $B$ and $C$, $\triangle ABC \sim \triangle BCD$, and $AB > BC$. There is a point $E$ in the interior of $ABCD$ such that $\triangle ABC \sim \triangle CEB$ and the area of $\triangle AED$ is $17$ times the area of $\triangle CEB$. What is $\tfrac{AB}{BC}$?
$\textbf{(A) \ } 1+\sqrt{2} \qquad \textbf{(B) \ } 2+\sqrt{2}\qquad \textbf{(C) \ } \sqrt{17}\qquad \textbf{(D) \ } 2+\sqrt{5} \qquad \textbf{(E) \ } 1+2\sqrt{3}$
2014 AMC 8, 2
Paul owes Paula $35$ cents and has a pocket full of $5$-cent coins, $10$-cent coins, and $25$-cent coins that he can use to pay her. What is the difference between the largest and the smallest number of coins he can use to pay her?
$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad \textbf{(E) }5$
1960 AMC 12/AHSME, 10
Given the following six statements:
$\text{(1) All women are good drivers}$
$\text{(2) Some women are good drivers}$
$\text{(3) No men are good drivers}$
$\text{(4) All men are bad drivers}$
$\text{(5) At least one man is a bad driver}$
$\text{(6) All men are good drivers.}$
The statement that negates statement $\text{(6)}$ is:
$ \textbf{(A) }(1)\qquad\textbf{(B) }(2)\qquad\textbf{(C) }(3)\qquad\textbf{(D) }(4)\qquad\textbf{(E) }(5) $
2019 AMC 10, 21
Debra flips a fair coin repeatedly, keeping track of how many heads and how many tails she has seen in total, until she gets either two heads in a row or two tails in a row, at which point she stops flipping. What is the probability that she gets two heads in a row but she sees a second tail before she sees a second head?
$\textbf{(A) } \frac{1}{36} \qquad \textbf{(B) } \frac{1}{24} \qquad \textbf{(C) } \frac{1}{18} \qquad \textbf{(D) } \frac{1}{12} \qquad \textbf{(E) } \frac{1}{6}$
1976 AMC 12/AHSME, 25
For a sequence $u_1,u_2\dots,$ define $\Delta^1(u_n)=u_{n+1}-u_n$ and, for all integer $k>1$, $\Delta^k(u_n)=\Delta^1(\Delta^{k-1}(u_n))$. If $u_n=n^3+n$, then $\Delta^k(u_n)=0$ for all $n$
$\textbf{(A) }\text{if }k=1\qquad$
$\textbf{(B) }\text{if }k=2,\text{ but not if }k=1\qquad$
$\textbf{(C) }\text{if }k=3,\text{ but not if }k=2\qquad$
$\textbf{(D) }\text{if }k=4,\text{ but not if }k=3\qquad$
$\textbf{(E) }\text{for no value of }k$
2012 AMC 10, 20
Bernado and Silvia play the following game. An integer between 0 and 999, inclusive, is selected and given to Bernado. Whenever Bernado receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she adds 50 to it and passes the result to Bernado. The winner is the last person who produces a number less than 1000. Let $N$ be the smallest initial number that results in a win for Bernado. What is the sum of the digits of $N$?
$\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ 11$
2010 AIME Problems, 3
Let $ K$ be the product of all factors $ (b\minus{}a)$ (not necessarily distinct) where $ a$ and $ b$ are integers satisfying $ 1\le a < b \le 20$. Find the greatest positive integer $ n$ such that $ 2^n$ divides $ K$.
2011 AMC 12/AHSME, 11
Circles $A$, $B$, and $C$ each have radius $1$. Circles $A$ and $B$ share one point of tangency. Circle $C$ has a point of tangency with the midpoint of $\overline{AB}$. What is the area inside circle $C$ but outside circle $A$ and circle $B$?
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$ \textbf{(A)}\ 3-\frac{\pi}{2} \qquad
\textbf{(B)}\ \frac{\pi}{2} \qquad
\textbf{(C)}\ 2 \qquad
\textbf{(D)}\ \frac{3\pi}{4} \qquad
\textbf{(E)}\ 1+\frac{\pi}{2}$
2022 AMC 10, 9
The sum
\[\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+\dots+\frac{2021}{2022!}\] can be expressed as $a-\frac{1}{b!}$, where $a$ and $b$ are positive integers. What is $a+b$?
$ \textbf{(A)}\ 2020 \qquad\textbf{(B)}\ 2021 \qquad\textbf{(C)}\ 2022 \qquad\textbf{(D)}\ 2023 \qquad\textbf{(E)}\ 2024$
2006 AMC 10, 3
A football game was played between two teams, the Cougars and the Panthers. The two teams scored a total of 34 points, and the Cougars won by a margin of 14 points. How many points did the Panthers score?
$ \textbf{(A) } 10 \qquad \textbf{(B) } 14 \qquad \textbf{(C) } 17 \qquad \textbf{(D) } 20 \qquad \textbf{(E) } 24$
2007 USAMO, 5
Prove that for every nonnegative integer $n$, the number $7^{7^{n}}+1$ is the product of at least $2n+3$ (not necessarily distinct) primes.
2003 USAMO, 1
Prove that for every positive integer $n$ there exists an $n$-digit number divisible by $5^n$ all of whose digits are odd.
1960 AMC 12/AHSME, 19
Consider equation I: $x+y+z=46$ where $x, y,$ and $z$ are positive integers, and equation II: $x+y+z+w=46$, where $x, y, z,$ and $w$ are positive integers. Then
$ \textbf{(A)}\ \text{I can be solved in consecutive integers} \qquad$
$\textbf{(B)}\ \text{I can be solved in consecutive even integers} \qquad$
$\textbf{(C)}\ \text{II can be solved in consecutive integers} \qquad$
$\textbf{(D)}\ \text{II can be solved in consecutive even integers} \qquad$
$\textbf{(E)}\ \text{II can be solved in consecutive odd integers} $
2009 AMC 12/AHSME, 6
By inserting parentheses, it is possible to give the expression
\[ 2\times3\plus{}4\times5
\]several values. How many different values can be obtained?
$ \textbf{(A)}\ 2\qquad
\textbf{(B)}\ 3\qquad
\textbf{(C)}\ 4\qquad
\textbf{(D)}\ 5\qquad
\textbf{(E)}\ 6$
2022 AMC 10, 20
Let $ABCD$ be a rhombus with $\angle{ADC} = 46^{\circ}$. Let $E$ be the midpoint of $\overline{CD}$, and let $F$ be the point on $\overline{BE}$ such that $\overline{AF}$ is perpendicular to $\overline{BE}$. What is the degree measure of $\angle{BFC}$?
$\textbf{(A)}\ 110 \qquad \textbf{(B)}\ 111 \qquad \textbf{(C)}\ 112 \qquad \textbf{(D)}\ 113 \qquad \textbf{(E)}\ 114$
1969 AMC 12/AHSME, 3
If $N$, written in base $2$, is $11000$, the integer immediately preceeding $N$, written in base $2$, is:
$\textbf{(A) }10001\qquad
\textbf{(B) }10010\qquad
\textbf{(C) }10011\qquad
\textbf{(D) }10110\qquad
\textbf{(E) }10111$
1988 USAMO, 2
The cubic equation $x^3 + ax^2 + bx + c = 0$ has three real roots. Show that $a^2-3b\geq 0$, and that $\sqrt{a^2-3b}$ is less than or equal to the difference between the largest and smallest roots.
2008 AIME Problems, 12
There are two distinguishable flagpoles, and there are $ 19$ flags, of which $ 10$ are identical blue flags, and $ 9$ are identical green flags. Let $ N$ be the number of distinguishable arrangements using all of the flags in which each flagpole has at least one flag and no two green flags on either pole are adjacent. Find the remainder when $ N$ is divided by $ 1000$.
2010 AMC 12/AHSME, 4
A month with 31 days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month?
$ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6$
1986 AMC 12/AHSME, 27
In the adjoining figure, $AB$ is a diameter of the circle, $CD$ is a chord parallel to $AB$, and $AC$ intersects $BD$ at $E$, with $\angle AED = \alpha$. The ratio of the area of $\triangle CDE$ to that of $\triangle ABE$ is
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$ \textbf{(A)}\ \cos\ \alpha\qquad\textbf{(B)}\ \sin\ \alpha\qquad\textbf{(C)}\ \cos^2\alpha\qquad\textbf{(D)}\ \sin^2\alpha\qquad\textbf{(E)}\ 1 - \sin\ \alpha $
2012 AMC 12/AHSME, 15
Jesse cuts a circular paper disk of radius $12$ along two radii to form two sectors, the smaller having a central angle of $120$ degrees. He makes two circular cones, using each sector to form the lateral surface of a cone. What is the ratio of the volume of the smaller cone to that of the larger?
$ \textbf{(A)}\ \frac{1}{8} \qquad\textbf{(B)}\ \frac{1}{4} \qquad\textbf{(C)}\ \frac{\sqrt{10}}{10} \qquad\textbf{(D)}\ \frac{\sqrt{5}}{6} \qquad\textbf{(E)}\ \frac{\sqrt{10}}{5} $