Found problems: 3632
2000 AIME Problems, 11
Let $S$ be the sum of all numbers of the form $a/b,$ where $a$ and $b$ are relatively prime positive divisors of $1000.$ What is the greatest integer that does not exceed $S/10?$
2013 AMC 10, 14
Define $a\clubsuit b=a^2b-ab^2$. Which of the following describes the set of points $(x, y)$ for which $x\clubsuit y=y\clubsuit x$?
${ \textbf{(A)}\ \text{a finite set of points} \\ \qquad\textbf{(B)}\ \text{one line} \\ \qquad\textbf{(C)}\ \text{two parallel lines}\\ \qquad\textbf{(D}}\ \text{two intersecting lines}\\ \qquad\textbf{(E)}\ \text{three lines} $
2020 AMC 12/AHSME, 23
How many integers $n \geq 2$ are there such that whenever $z_1, z_2, ..., z_n$ are complex numbers such that
$$|z_1| = |z_2| = ... = |z_n| = 1 \text{ and } z_1 + z_2 + ... + z_n = 0,$$
then the numbers $z_1, z_2, ..., z_n$ are equally spaced on the unit circle in the complex plane?
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5$
2014 Contests, 1
What is $10 \cdot \left(\tfrac{1}{2} + \tfrac{1}{5} + \tfrac{1}{10}\right)^{-1}?$
${ \textbf{(A)}\ 3\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ \frac{25}{2}\qquad\textbf{(D)}}\ \frac{170}{3}\qquad\textbf{(E)}\ 170$
2021 AMC 10 Spring, 14
Three equally spaced parallel lines intersect a circle, creating three chords of lengths $38, 38,$ and $34.$ What is the distance between two adjacent parallel lines?
$\textbf{(A)}\ 5\frac{1}{2} \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 6\frac{1}{2} \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ 7\frac{1}{2}$
2013 AIME Problems, 15
Let $N$ be the number of ordered triples $(A,B,C)$ of integers satisfying the conditions
(a) $0\leq A<B<C\leq99$,
(b) there exist integers $a$, $b$, and $c$, and prime $p$ where $0\leq b < a < c < p$,
(c) $p$ divides $A-a$, $B-b$, and $C-c$, and
(d) each ordered triple $(A,B,C)$ and each ordered triple $(b,a,c)$ form arithmetic sequences.
Find $N$.
2022 AMC 12/AHSME, 24
The figure below depicts a regular 7-gon inscribed in a unit circle.
[asy]
import geometry;
unitsize(3cm);
draw(circle((0,0),1),linewidth(1.5));
for (int i = 0; i < 7; ++i) {
for (int j = 0; j < i; ++j) {
draw(dir(i * 360/7) -- dir(j * 360/7),linewidth(1.5));
}
}
for(int i = 0; i < 7; ++i) {
dot(dir(i * 360/7),5+black);
}
[/asy]
What is the sum of the 4th powers of the lengths of all 21 of its edges and diagonals?
$\textbf{(A)}49~\textbf{(B)}98~\textbf{(C)}147~\textbf{(D)}168~\textbf{(E)}196$
1972 AMC 12/AHSME, 15
A contractor estimated that one of his two bricklayers would take $9$ hours to build a certain wall and the other $10$ hours. However, he knew from experience that when they worked together, their combined output fell by $10$ bricks per hour. Being in a hurry, he put both men on the job and found that it took exactly 5 hours to build the wall. The number of bricks in the wall was
$\textbf{(A) }500\qquad\textbf{(B) }550\qquad\textbf{(C) }900\qquad\textbf{(D) }950\qquad \textbf{(E) }960$
2008 AMC 10, 8
Heather compares the price of a new computer at two different stores. Store A offers $ 15\%$ off the sticker price followed by a $ \$90$ rebate, and store B offers $ 25\%$ off the same sticker price with no rebate. Heather saves $ \$15$ by buying the computer at store A instead of store B. What is the sticker price of the computer, in dollars?
$ \textbf{(A)}\ 750 \qquad \textbf{(B)}\ 900 \qquad \textbf{(C)}\ 1000 \qquad \textbf{(D)}\ 1050 \qquad \textbf{(E)}\ 1500$
2022 AMC 10, 18
Let $T_k$ be the transformation of the coordinate plane that first rotates the plane $k$ degrees counterclockwise around the origin and then reflects the plane across the $y$-axis. What is the least positive integer $n$ such that performing the sequence of transformations transformations $T_1, T_2, T_3, \dots, T_n$ returns the point $(1,0)$ back to itself?
$\textbf{(A) } 359 \qquad \textbf{(B) } 360\qquad \textbf{(C) } 719 \qquad \textbf{(D) } 720 \qquad \textbf{(E) } 721$
2003 AIME Problems, 6
In triangle $ABC,$ $AB=13,$ $BC=14,$ $AC=15,$ and point $G$ is the intersection of the medians. Points $A',$ $B',$ and $C',$ are the images of $A,$ $B,$ and $C,$ respectively, after a $180^\circ$ rotation about $G.$ What is the area if the union of the two regions enclosed by the triangles $ABC$ and $A'B'C'?$
1991 AMC 12/AHSME, 15
A circular table has exactly 60 chairs around it. There are $N$ people seated at this table in such a way that the next person to be seated must sit next to someone. The smallest possible value of $N$ is
$ \textbf{(A)}\ 15\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 30\qquad\textbf{(D)}\ 40\qquad\textbf{(E)}\ 58 $
1984 AIME Problems, 9
In tetrahedron $ABCD$, edge $AB$ has length 3 cm. The area of face $ABC$ is 15 $\text{cm}^2$ and the area of face $ABD$ is 12 $\text{cm}^2$. These two faces meet each other at a $30^\circ$ angle. Find the volume of the tetrahedron in $\text{cm}^3$.
2024 AMC 8 -, 9
All the marbles in Maria's collection are red, green, or blue. Maria has half as many red marbles as green marbles and twice as many blue marbles as green marbles. Which of the following could be the total number of marbles in Maria's collection?
$\textbf{(A) } 24\qquad\textbf{(B) } 25\qquad\textbf{(C) } 26\qquad\textbf{(D) } 27\qquad\textbf{(E) } 28$
2020 AMC 10, 22
What is the remainder when $2^{202} +202$ is divided by $2^{101}+2^{51}+1$?
$\textbf{(A) } 100 \qquad\textbf{(B) } 101 \qquad\textbf{(C) } 200 \qquad\textbf{(D) } 201 \qquad\textbf{(E) } 202$
2024 AIME, 15
Find the number of rectangles that can be formed inside a fixed regular dodecagon ($12$-gon) where each side of the rectangle lies on either a side or a diagonal of the dodecagon. The diagram below shows three of those rectangles.
[asy]
unitsize(40);
real r = pi/6;
pair A1 = (cos(r),sin(r));
pair A2 = (cos(2r),sin(2r));
pair A3 = (cos(3r),sin(3r));
pair A4 = (cos(4r),sin(4r));
pair A5 = (cos(5r),sin(5r));
pair A6 = (cos(6r),sin(6r));
pair A7 = (cos(7r),sin(7r));
pair A8 = (cos(8r),sin(8r));
pair A9 = (cos(9r),sin(9r));
pair A10 = (cos(10r),sin(10r));
pair A11 = (cos(11r),sin(11r));
pair A12 = (cos(12r),sin(12r));
draw(A1--A2--A3--A4--A5--A6--A7--A8--A9--A10--A11--A12--cycle);
filldraw(A2--A1--A8--A7--cycle, mediumgray, linewidth(1.2));
draw(A4--A11);
draw(0.365*A3--0.365*A12, linewidth(1.2));
dot(A1);
dot(A2);
dot(A3);
dot(A4);
dot(A5);
dot(A6);
dot(A7);
dot(A8);
dot(A9);
dot(A10);
dot(A11);
dot(A12);
[/asy]
2012 AMC 10, 2
A square with side length $8$ is cut in half, creating two congruent rectangles. What are the dimensions of one of these rectangles?
$ \textbf{(A)}\ 2\text{ by }4
\qquad\textbf{(B)}\ 2\text{ by }6
\qquad\textbf{(C)}\ 2\text{ by }8
\qquad\textbf{(D)}\ 4\text{ by }4
\qquad\textbf{(E)}\ 4\text{ by }8
$
2019 AMC 10, 9
The function $f$ is defined by $$f(x) = \Big\lfloor \lvert x \rvert \Big\rfloor - \Big\lvert \lfloor x \rfloor \Big\rvert$$for all real numbers $x$, where $\lfloor r \rfloor$ denotes the greatest integer less than or equal to the real number $r$. What is the range of $f$?
$\textbf{(A) } \{-1, 0\} \qquad\textbf{(B) } \text{The set of nonpositive integers} \qquad\textbf{(C) } \{-1, 0, 1\}$
$\textbf{(D) } \{0\} \qquad\textbf{(E) } \text{The set of nonnegative integers} $
1968 AMC 12/AHSME, 5
If $f(n)=\tfrac{1}{3}n(n1)(n+2)$, then $f(r)-f(r-1)$ equals:
$\textbf{(A)}\ r(r+1) \qquad
\textbf{(B)}\ (r+1)(r+2) \qquad
\textbf{(C)}\ \tfrac{1}{3}r(r+1) \qquad\\
\textbf{(D)}\ \tfrac{1}{3}(r+1)(r+2) \qquad
\textbf{(E)}\ \tfrac{1}{3}r(r+1)(r+2) $
2019 AMC 12/AHSME, 10
The figure below shows $13$ circles of radius $1$ within a larger circle. All the intersections occur at points of tangency. What is the area of the region, shaded in the figure, inside the larger circle but outside all the circles of radius $1 ?$
[asy]unitsize(20);filldraw(circle((0,0),2*sqrt(3)+1),rgb(0.5,0.5,0.5));filldraw(circle((-2,0),1),white);filldraw(circle((0,0),1),white);filldraw(circle((2,0),1),white);filldraw(circle((1,sqrt(3)),1),white);filldraw(circle((3,sqrt(3)),1),white);filldraw(circle((-1,sqrt(3)),1),white);filldraw(circle((-3,sqrt(3)),1),white);filldraw(circle((1,-1*sqrt(3)),1),white);filldraw(circle((3,-1*sqrt(3)),1),white);filldraw(circle((-1,-1*sqrt(3)),1),white);filldraw(circle((-3,-1*sqrt(3)),1),white);filldraw(circle((0,2*sqrt(3)),1),white);filldraw(circle((0,-2*sqrt(3)),1),white);[/asy]
$\textbf{(A) } 4 \pi \sqrt{3} \qquad\textbf{(B) } 7 \pi \qquad\textbf{(C) } \pi(3\sqrt{3} +2) \qquad\textbf{(D) } 10 \pi (\sqrt{3} - 1) \qquad\textbf{(E) } \pi(\sqrt{3} + 6)$
2013 AMC 10, 11
A student council must select a two-person welcoming committee and a three-person planning committee from among its members. There are exactly $10$ ways to select a two-person team for the welcoming committee. It is possible for students to serve on both committees. In how many different ways can a three-person planning committee be selected?
$ \textbf{(A)}\ 10\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 25$
2012 AIME Problems, 4
Ana, Bob, and Cao bike at constant rates of $8.6$ meters per second, $6.2$ meters per second, and $5$ meters per second, respectively. They all begin biking at the same time from the northeast corner of a rectangular field whose longer side runs due west. Ana starts biking along the edge of the field, initially heading west, Bob starts biking along the edge of the field, initially heading south, and Cao bikes in a straight line across the field to a point D on the south edge of the field. Cao arrives at point D at the same time that Ana and Bob arrive at D for the first time. The ratio of the field's length to the field's width to the distance from point D to the southeast corner of the field can be represented as $p : q : r$, where $p$, $q$, and $r$ are positive integers with p and q relatively prime. Find $p + q + r$.
1987 AMC 12/AHSME, 12
In an office, at various times during the day the boss gives the secretary a letter to type, each time putting the letter on top of the pile in the secretary's in-box. When there is time, the secretary takes the top letter off the pile and types it. If there are five letters in all, and the boss delivers them in the order $1\ 2\ 3\ 4\ 5$, which of the following could [b]not[/b] be the order in which the secretary types them?
$ \textbf{(A)}\ 1\ 2\ 3\ 4\ 5 \qquad\textbf{(B)}\ 2\ 4\ 3\ 5\ 1 \qquad\textbf{(C)}\ 3\ 2\ 4\ 1\ 5 \qquad\textbf{(D)}\ 4\ 5\ 2\ 3\ 1 \\ \qquad\textbf{(E)}\ 5\ 4\ 3\ 2\ 1 $
2012 AIME Problems, 15
There are $n$ mathematicians seated around a circular table with $n$ seats numbered $1,2,3,\cdots,n$ in clockwise order. After a break they again sit around the table. The mathematicians note that there is a positive integer $a$ such that
(1) for each $k$, the mathematician who was seated in seat $k$ before the break is seated in seat $ka$ after the break (where seat $i+n$ is seat $i$);
(2) for every pair of mathematicians, the number of mathematicians sitting between them after the break, counting in both the clockwise and the counterclockwise directions, is different from either of the number of mathematicians sitting between them before the break.
Find the number of possible values of $n$ with $1<n<1000$.
1997 AIME Problems, 15
The sides of rectangle $ABCD$ have lengths 10 and 11. An equilateral triangle is drawn so that no point of the triangle lies outside $ABCD.$ The maximum possible area of such a triangle can be written in the form $p\sqrt{q}-r,$ where $p, q,$ and $r$ are positive integers, and $q$ is not divisible by the square of any prime number. Find $p+q+r.$