Found problems: 3632
2014 AMC 10, 20
The product $(8)(888\dots8)$, where the second factor has $k$ digits, is an integer whose digits have a sum of $1000$. What is $k$?
${ \textbf{(A)}\ 901\qquad\textbf{(B)}\ 911\qquad\textbf{(C)}\ 919\qquad\textbf{(D)}}\ 991\qquad\textbf{(E)}\ 999 $
2011 AMC 12/AHSME, 17
Circles with radii $1, 2$, and $3$ are mutually externally tangent. What is the area of the triangle determined by the points of tangency?
$ \textbf{(A)}\ \frac{3}{5} \qquad
\textbf{(B)}\ \frac{4}{5} \qquad
\textbf{(C)}\ 1 \qquad
\textbf{(D)}\ \frac{6}{5} \qquad
\textbf{(E)}\ \frac{4}{3}
$
2014 AMC 8, 19
A cube with $3$-inch edges is to be constructed from $27$ smaller cubes with $1$-inch edges. Twenty-one of the cubes are colored red and $6$ are colored white. If the $3$-inch cube is constructed to have the smallest possible white surface area showing, what fraction of the surface area is white?
$\textbf{(A) }\frac{5}{54}\qquad\textbf{(B) }\frac{1}{9}\qquad\textbf{(C) }\frac{5}{27}\qquad\textbf{(D) }\frac{2}{9}\qquad \textbf{(E) }\frac{1}{3}$
2008 AMC 12/AHSME, 19
In the expansion of
\[ \left(1 \plus{} x \plus{} x^2 \plus{} \cdots \plus{} x^{27}\right)\left(1 \plus{} x \plus{} x^2 \plus{} \cdots \plus{} x^{14}\right)^2,
\]what is the coefficient of $ x^{28}$?
$ \textbf{(A)}\ 195 \qquad \textbf{(B)}\ 196 \qquad \textbf{(C)}\ 224 \qquad \textbf{(D)}\ 378 \qquad \textbf{(E)}\ 405$
1996 AMC 8, 21
How many subsets containing three different numbers can be selected from the set
\[\{ 89,95,99,132, 166,173 \}\]
so that the sum of the three numbers is even?
$\text{(A)}\ 6 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 10 \qquad \text{(D)}\ 12 \qquad \text{(E)}\ 24$
2008 AMC 12/AHSME, 5
A class collects $ \$50$ to buy flowers for a classmate who is in the hospital. Roses cost $ \$3$ each, and carnations cost $ \$2$ each. No other flowers are to be used. How many different bouquets could be purchased for exactly $ \$50$?
$ \textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 7 \qquad
\textbf{(C)}\ 9 \qquad
\textbf{(D)}\ 16 \qquad
\textbf{(E)}\ 17$
2018 AIME Problems, 10
Find the number of functions $f(x)$ from $\{1,2,3,4,5\}$ to $\{1,2,3,4,5\}$ that satisfy $f(f(x)) = f(f(f(x)))$ for all $x$ in $\{1,2,3,4,5\}$.
2010 AMC 10, 15
On a 50-question multiple choice math contest, students receive 4 points for a correct answer, 0 points for an answer left blank, and -1 point for an incorrect answer. Jesse's total score on the contest was 99. What is the maximum number of questions that Jesse could have answered correctly?
$ \textbf{(A)}\ 25\qquad\textbf{(B)}\ 27\qquad\textbf{(C)}\ 29\qquad\textbf{(D)}\ 31\qquad\textbf{(E)}\ 33$
2025 AIME, 13
Alex divides a disk into four quadrants with two perpendicular diameters intersecting at the center of the disk. He draws $25$ more lines segments through the disk, drawing each segment by selecting two points at random on the perimeter of the disk in different quadrants and connecting these two points. Find the expected number of regions into which these $27$ line segments divide the disk.
2008 AIME Problems, 5
A right circular cone has base radius $ r$ and height $ h$. The cone lies on its side on a flat table. As the cone rolls on the surface of the table without slipping, the point where the cone's base meets the table traces a circular arc centered at the point where the vertex touches the table. The cone first returns to its original position on the table after making $ 17$ complete rotations. The value of $ h/r$ can be written in the form $ 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$.
2014 Contests, 1
The $8$ eyelets for the lace of a sneaker all lie on a rectangle, four equally spaced on each of the longer sides. The rectangle has a width of $50$ mm and a length of $80$ mm. There is one eyelet at each vertex of the rectangle. The lace itself must pass between the vertex eyelets along a width side of the rectangle and then crisscross between successive eyelets until it reaches the two eyelets at the other width side of the rectrangle as shown. After passing through these final eyelets, each of the ends of the lace must extend at least $200$ mm farther to allow a knot to be tied. Find the minimum length of the lace in millimeters.
[asy]
size(200);
defaultpen(linewidth(0.7));
path laceL=(-20,-30)..tension 0.75 ..(-90,-135)..(-102,-147)..(-152,-150)..tension 2 ..(-155,-140)..(-135,-40)..(-50,-4)..tension 0.8 ..origin;
path laceR=reflect((75,0),(75,-240))*laceL;
draw(origin--(0,-240)--(150,-240)--(150,0)--cycle,gray);
for(int i=0;i<=3;i=i+1)
{
path circ1=circle((0,-80*i),5),circ2=circle((150,-80*i),5);
unfill(circ1); draw(circ1);
unfill(circ2); draw(circ2);
}
draw(laceL--(150,-80)--(0,-160)--(150,-240)--(0,-240)--(150,-160)--(0,-80)--(150,0)^^laceR,linewidth(1));[/asy]
2014 AMC 10, 13
Six regular hexagons surround a regular hexagon of side length $1$ as shown. What is the area of $\triangle ABC$?
[asy]
for(int i = 0; i < 6; ++i){
for(int j = 0; j < 6; ++j){
draw(sqrt(3)*dir(60*i+30)+dir(60*j)--sqrt(3)*dir(60*i+30)+dir(60*j+60));
}
}
draw(2*dir(60)--2*dir(180)--2*dir(300)--cycle);
label("A",2*dir(180),dir(180));
label("B",2*dir(60),dir(60));
label("C",2*dir(300),dir(300));
[/asy]
$ \textbf {(A) } 2\sqrt{3} \qquad \textbf {(B) } 3\sqrt{3} \qquad \textbf {(C) } 1+3\sqrt{2} \qquad \textbf {(D) } 2+2\sqrt{3} \qquad \textbf {(E) } 3+2\sqrt{3} $
1963 AMC 12/AHSME, 10
Point $P$ is taken interior to a square with side-length $a$ and such that is it equally distant from two consecutive vertices and from the side opposite these vertices. If $d$ represents the common distance, then $d$ equals:
$\textbf{(A)}\ \dfrac{3a}{5} \qquad
\textbf{(B)}\ \dfrac{5a}{8} \qquad
\textbf{(C)}\ \dfrac{3a}{8} \qquad
\textbf{(D)}\ \dfrac{a\sqrt{2}}{2} \qquad
\textbf{(E)}\ \dfrac{a}{2}$
1968 AMC 12/AHSME, 11
If an arc of $60^\circ$ on circle I has the same length as an arc of $45^\circ$ on circle II, the ratio of the area of circle I to that of circle II is:
$\textbf{(A)}\ 16:9 \qquad
\textbf{(B)}\ 9:16 \qquad
\textbf{(C)}\ 4:3 \qquad
\textbf{(D)}\ 3:4 \qquad
\textbf{(E)}\ \text{None of these} $
1970 AMC 12/AHSME, 28
In triangle $ABC$, the median from vertex $A$ is perpendicular to the median from vertex $B$. If the lengths of sides $AC$ and $BC$ are $6$ and $7$ respectively, then the length of side $AB$ is
$\textbf{(A) }\sqrt{17}\qquad\textbf{(B) }4\qquad\textbf{(C) }4\dfrac{1}{2}\qquad\textbf{(D) }2\sqrt{5}\qquad \textbf{(E) }4\dfrac{1}{4}$
2018 AMC 12/AHSME, 12
Side $\overline{AB}$ of $\triangle ABC$ has length $10$. The bisector of angle $A$ meets $\overline{BC}$ at $D$, and $CD = 3$. The set of all possible values of $AC$ is an open interval $(m,n)$. What is $m+n$?
$
\textbf{(A) }16 \qquad
\textbf{(B) }17 \qquad
\textbf{(C) }18 \qquad
\textbf{(D) }19 \qquad
\textbf{(E) }20 \qquad
$
1996 AMC 8, 25
A point is chosen at random from within a circular region. What is the probability that the point is closer to the center of the region than it is to the boundary of the region?
$\text{(A)}\ 1/4 \qquad \text{(B)}\ 1/3 \qquad \text{(C)}\ 1/2 \qquad \text{(D)}\ 2/3 \qquad \text{(E)}\ 3/4$
1997 AMC 8, 6
In the number $74982.1035$ the value of the ''place'' occupied by the digit 9 is how many times as great as the value of the ''place'' occupied by the digit 3?
$\textbf{(A)}\ 1,000 \qquad \textbf{(B)}\ 10,000 \qquad \textbf{(C)}\ 100,000 \qquad \textbf{(D)}\ 1,000,000 \qquad \textbf{(E)}\ 10,000,000$
2024 AMC 12/AHSME, 1
In a long line of people, the 1013th person from the left is also the 1010th person from the right. How many people are in the line?
$
\textbf{(A) }2021 \qquad
\textbf{(B) }2022 \qquad
\textbf{(C) }2023 \qquad
\textbf{(D) }2024 \qquad
\textbf{(E) }2025 \qquad
$
2025 AIME, 3
The 9 members of a baseball team went to an ice-cream parlor after their game. Each player had a singlescoop cone of chocolate, vanilla, or strawberry ice cream. At least one player chose each flavor, and the number of players who chose chocolate was greater than the number of players who chose vanilla, which was greater than the number of players who chose strawberry. Let $N$ be the number of different assignments of flavors to players that meet these conditions. Find the remainder when $N$ is divided by $1000.$
2009 AMC 12/AHSME, 5
One dimension of a cube is increased by $ 1$, another is decreased by $ 1$, and the third is left unchanged. The volume of the new rectangular solid is $ 5$ less than that of the cube. What was the volume of the cube?
$ \textbf{(A)}\ 8 \qquad
\textbf{(B)}\ 27 \qquad
\textbf{(C)}\ 64 \qquad
\textbf{(D)}\ 125 \qquad
\textbf{(E)}\ 216$
1959 AMC 12/AHSME, 7
The sides of a right triangle are $a, a+d,$ and $a+2d$, with $a$ and $d$ both positive. The ratio of $a$ to $d$ is:
$ \textbf{(A)}\ 1:3 \qquad\textbf{(B)}\ 1:4 \qquad\textbf{(C)}\ 2:1\qquad\textbf{(D)}\ 3:1\qquad\textbf{(E)}\ 3:4 $
2010 AMC 12/AHSME, 19
A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neither team scored more than $ 100$ points. What was the total number of points scored by the two teams in the first half?
$ \textbf{(A)}\ 30 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\ 33 \qquad \textbf{(E)}\ 34$
1967 AMC 12/AHSME, 39
Given the sets of consecutive integers $\{1\}$,$ \{2, 3\}$,$ \{4,5,6\}$,$ \{7,8,9,10\}$,$\; \cdots \; $, where each set contains one more element than the preceding one, and where the first element of each set is one more than the last element of the preceding set. Let $S_n$ be the sum of the elements in the $N$th set. Then $S_{21}$ equals:
$\textbf{(A)}\ 1113\qquad
\textbf{(B)}\ 4641 \qquad
\textbf{(C)}\ 5082\qquad
\textbf{(D)}\ 53361\qquad
\textbf{(E)}\ \text{none of these}$
1995 AMC 12/AHSME, 18
Two rays with common endpoint $O$ forms a $30^\circ$ angle. Point $A$ lies on one ray, point $B$ on the other ray, and $AB = 1$. The maximum possible length of $OB$ is
$\textbf{(A)}\ 1 \qquad
\textbf{(B)}\ \dfrac{1+\sqrt{3}}{\sqrt{2}} \qquad
\textbf{(C)}\ \sqrt{3} \qquad
\textbf{(D)}\ 2 \qquad
\textbf{(E)}\ \dfrac{4}{\sqrt{3}}$