Found problems: 3632
2007 AMC 10, 17
Suppose that $ m$ and $ n$ are positive integers such that $ 75m \equal{} n^{3}$. What is the minimum possible value of $ m \plus{} n$?
$ \textbf{(A)}\ 15 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 50 \qquad \textbf{(D)}\ 60 \qquad \textbf{(E)}\ 5700$
2007 AMC 10, 7
Last year Mr. John Q. Public received an inheritance. He paid $20\%$ in federal taxes on the inheritance, and paid $10\%$ of what he has left in state taxes. He paid a total of $ \$10,500$ for both taxes. How many dollars was the inheritance?
$ \textbf{(A)}\ 30,000 \qquad \textbf{(B)}\ 32,500 \qquad \textbf{(C)}\ 35,000 \qquad \textbf{(D)}\ 37,500 \qquad \textbf{(E)}\ 40,000$
1979 AMC 12/AHSME, 26
The function $f$ satisfies the functional equation \[f(x) +f(y) = f(x + y ) - xy - 1\] for every pair $x,~ y$ of real numbers. If $f( 1) = 1$, then the number of integers $n \neq 1$ for which $f ( n ) = n$ is
$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }\text{infinite}$
2014 AMC 12/AHSME, 19
There are exactly $N$ distinct rational numbers $k$ such that $|k|<200$ and \[5x^2+kx+12=0\] has at least one integer solution for $x$. What is $N$?
$\textbf{(A) }6\qquad
\textbf{(B) }12\qquad
\textbf{(C) }24\qquad
\textbf{(D) }48\qquad
\textbf{(E) }78\qquad$
2001 AMC 12/AHSME, 3
The state income tax where Kristin lives is levied at the rate of $ p \%$ of the first $ \$28000$ of annual income plus $ (p \plus{} 2) \%$ of any amount above $ \$28000$. Kristin noticed that the state income tax she paid amounted to $ (p \plus{} 0.25) \%$ of her annual income. What was her annual income?
$ \textbf{(A)} \ \$28000 \qquad \textbf{(B)} \ \$32000 \qquad \textbf{(C)} \ \$35000 \qquad \textbf{(D)} \ \$42000 \qquad \textbf{(E)} \ \$56000$
2006 AMC 12/AHSME, 5
Doug and Dave shared a pizza with $ 8$ equally-sized slices. Doug wanted a plain pizza, but Dave wanted anchovies on half the pizza. The cost of a plain pizza was $ \$8$, and there was an additional cost of $ \$2$ for putting anchovies on one half. Dave ate all the slices of anchovy pizza and one plain slice. Doug ate the remainder. Each paid for what he had eaten. How many more dollars did Dave pay than Doug?
$ \textbf{(A) } 1\qquad \textbf{(B) } 2\qquad \textbf{(C) } 3\qquad \textbf{(D) } 4\qquad \textbf{(E) } 5$
2012 AMC 10, 9
A pair of six-sided fair dice are labeled so that one die has only even numbers (two each of $2$, $4$, and $6$), and the other die has only odd numbers (two each of $1$, $3$, and $5$). The pair of dice is rolled. What is the probability that the sum of the numbers on top of the two dice is $7$?
$ \textbf{(A)}\ \dfrac{1}{6}
\qquad\textbf{(B)}\ \dfrac{1}{5}
\qquad\textbf{(C)}\ \dfrac{1}{4}
\qquad\textbf{(D)}\ \dfrac{1}{3}
\qquad\textbf{(E)}\ \dfrac{1}{2}
$
2014 AMC 12/AHSME, 24
Let $ABCDE$ be a pentagon inscribed in a circle such that $AB=CD=3$, $BC=DE=10$, and $AE=14$. The sum of the lengths of all diagonals of $ABCDE$ is equal to $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
$\textbf{(A) }129\qquad
\textbf{(B) }247\qquad
\textbf{(C) }353\qquad
\textbf{(D) }391\qquad
\textbf{(E) }421\qquad$
2024 AMC 12/AHSME, 9
A dartboard is the region B in the coordinate plane consisting of points $(x, y)$ such that $|x| + |y| \le 8$. A target T is the region where $(x^2 + y^2 - 25)^2 \le 49$. A dart is thrown at a random point in B. The probability that the dart lands in T can be expressed as $\frac{m}{n} \pi$, where $m$ and $n$ are relatively prime positive integers. What is $m + n$?
$
\textbf{(A) }39 \qquad
\textbf{(B) }71 \qquad
\textbf{(C) }73 \qquad
\textbf{(D) }75 \qquad
\textbf{(E) }135 \qquad
$
2022 AIME Problems, 10
Find the remainder when $$\binom{\binom{3}{2}}{2} + \binom{\binom{4}{2}}{2} + \dots + \binom{\binom{40}{2}}{2}$$ is divided by $1000$.
2018 AMC 12/AHSME, 8
Line segment $\overline{AB}$ is a diameter of a circle with $AB=24$. Point $C$, not equal to $A$ or $B$, lies on the circle. As point $C$ moves around the circle, the centroid (center of mass) of $\triangle{ABC}$ traces out a closed curve missing two points. To the nearest positive integer, what is the area of the region bounded by this curve?
$\textbf{(A)} \text{ 25} \qquad \textbf{(B)} \text{ 38} \qquad \textbf{(C)} \text{ 50} \qquad \textbf{(D)} \text{ 63} \qquad \textbf{(E)} \text{ 75}$
1986 AMC 12/AHSME, 21
In the configuration below, $\theta$ is measured in radians, $C$ is the center of the circle, $BCD$ and $ACE$ are line segments and $AB$ is tangent to the circle at $A$.
[asy]
defaultpen(fontsize(10pt)+linewidth(.8pt));
pair A=(0,-1), E=(0,1), C=(0,0), D=dir(10), F=dir(190), B=(-1/sin(10*pi/180))*dir(10);
fill(Arc((0,0),1,10,90)--C--D--cycle,mediumgray);
fill(Arc((0,0),1,190,270)--B--F--cycle,mediumgray);
draw(unitcircle);
draw(A--B--D^^A--E);
label("$A$",A,S);
label("$B$",B,W);
label("$C$",C,SE);
label("$\theta$",C,SW);
label("$D$",D,NE);
label("$E$",E,N);
[/asy]
A necessary and sufficient condition for the equality of the two shaded areas, given $0 < \theta < \frac{\pi}{2}$, is
$ \textbf{(A)}\ \tan \theta = \theta\qquad\textbf{(B)}\ \tan \theta = 2\theta\qquad\textbf{(C)}\ \tan \theta = 4\theta\qquad\textbf{(D)}\ \tan 2\theta = \theta\qquad \\ \textbf{(E)}\ \tan \frac{\theta}{2} = \theta$
1996 AIME Problems, 6
In triangle $ ABC$ the medians $ \overline{AD}$ and $ \overline{CE}$ have lengths 18 and 27, respectively, and $ AB \equal{} 24$. Extend $ \overline{CE}$ to intersect the circumcircle of $ ABC$ at $ F$. The area of triangle $ AFB$ is $ m\sqrt {n}$, where $ m$ and $ n$ are positive integers and $ n$ is not divisible by the square of any prime. Find $ m \plus{} n$.
2011 AMC 12/AHSME, 7
A majority of the $30$ students in Ms. Demeanor's class bought pencils at the school bookstore. Each of these students bought the same number of pencils, and this number was greater than $1$. The cost of a pencil in cents was greater than the number of pencils each student bought, and the total cost of all the pencils was $\$17.71$. What was the cost of a pencil in cents?
$ \textbf{(A)}\ 7 \qquad
\textbf{(B)}\ 11 \qquad
\textbf{(C)}\ 17 \qquad
\textbf{(D)}\ 23 \qquad
\textbf{(E)}\ 77
$
2008 Tuymaada Olympiad, 2
Is it possible to arrange on a circle all composite positive integers not exceeding $ 10^6$, so that no two neighbouring numbers are coprime?
[i]Author: L. Emelyanov[/i]
[hide="Tuymaada 2008, Junior League, First Day, Problem 2."]Prove that all composite positive integers not exceeding $ 10^6$
may be arranged on a circle so that no two neighbouring numbers are coprime. [/hide]
1986 AIME Problems, 1
What is the sum of the solutions to the equation $\sqrt[4]x =\displaystyle \frac{12}{7-\sqrt[4]x}$?
1972 AMC 12/AHSME, 2
If a dealer could get his goods for $8\%$ less while keeping his selling price fixed, his profit, based on cost, would be increased to $(x+10)\%$ from his present profit of $x\%$, which is
$\textbf{(A) }12\%\qquad\textbf{(B) }15\%\qquad\textbf{(C) }30\%\qquad\textbf{(D) }50\%\qquad \textbf{(E) }75\%$
1993 AMC 8, 1
Which pair of numbers does NOT have a product equal to $36$?
$\text{(A)}\ \{ -4,-9\} \qquad \text{(B)}\ \{ -3,-12\} \qquad \text{(C)}\ \left\{ \dfrac{1}{2},-72\right\} \qquad \text{(D)}\ \{ 1,36\} \qquad \text{(E)}\ \left\{\dfrac{3}{2},24\right\}$
2014 AMC 12/AHSME, 14
A rectangular box has a total surface area of $94$ square inches. The sum of the lengths of all its edges is $48$ inches. What is the sum of the lengths in inches of all of its interior diagonals?
${ \textbf{(A)}\ 8\sqrt{3}\qquad\textbf{(B)}\ 10\sqrt{2}\qquad\textbf{(C)}\ 16\sqrt{3}\qquad\textbf{(D)}}\ 20\sqrt{2}\qquad\textbf{(E)}\ 40\sqrt{2} $
2015 AMC 12/AHSME, 11
On a sheet of paper, Isabella draws a circle of radius $2$, a circle of radius $3$, and all possible lines simultaneously tangent to both circles. Isabella notices that she has drawn exactly $k\geq 0$ lines. How many different values of $k$ are possible?
$\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6$
2022 AMC 12/AHSME, 20
Isosceles trapezoid $ABCD$ has parallel sides $\overline{AD}$ and $\overline{BC},$ with $BC < AD$ and $AB = CD.$ There is a point $P$ in the plane such that $PA=1, PB=2, PC=3,$ and $PD=4.$ What is $\tfrac{BC}{AD}?$
$\textbf{(A) }\frac{1}{4}\qquad\textbf{(B) }\frac{1}{3}\qquad\textbf{(C) }\frac{1}{2}\qquad\textbf{(D) }\frac{2}{3}\qquad\textbf{(E) }\frac{3}{4}$
1987 AMC 12/AHSME, 18
It takes $A$ algebra books (all the same thickness) and $H$ geometry books (all the same thickness, which is greater than that of an algebra book) to completely fill a certain shelf. Also, $S$ of the algebra books and $M$ of the geometry books would fill the same shelf. Finally, $E$ of the algebra books alone would fill this shelf. Given that $A, H, S, M, E$ are distinct positive integers, it follows that $E$ is
$ \textbf{(A)}\ \frac{AM+SH}{M+H} \qquad\textbf{(B)}\ \frac{AM^2+SH^2}{M^2+H^2} \qquad\textbf{(C)}\ \frac{AH-SM}{M-H} \qquad\textbf{(D)}\ \frac{AM-SH}{M-H} \qquad\textbf{(E)}\ \frac{AM^2-SH^2}{M^2-H^2} $
2010 AMC 10, 25
Jim starts with a positive integer $ n$ and creates a sequence of numbers. Each successive number is obtained by subtracting the largest possible integer square less than or equal to the current number until zero is reached. For example, if Jim starts with $ n=55$, then his sequence contains $ 5$ numbers:
\begin{align*}
&55\\
55-7^2=&\ 6\\
6-2^2=&\ 2\\
2-1^2=&\ 1\\
1-1^2=&\ 0
\end{align*}Let $ N$ be the smallest number for which Jim's sequence has 8 numbers. What is the units digit of $ N$?
$ \textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 3 \qquad
\textbf{(C)}\ 5 \qquad
\textbf{(D)}\ 7 \qquad
\textbf{(E)}\ 9$
1993 AMC 12/AHSME, 19
How many ordered pairs $(m,n)$ of positive integers are solutions to $\frac{4}{m}+\frac{2}{n}=1$?
$ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{more than}\ 4 $
1963 AMC 12/AHSME, 2
Let $n=x-y^{x-y}$. Find $n$ when $x=2$ and $y=-2$.
$\textbf{(A)}\ -14 \qquad
\textbf{(B)}\ 0 \qquad
\textbf{(C)}\ 1 \qquad
\textbf{(D)}\ 18 \qquad
\textbf{(E)}\ 256$