Found problems: 730
2024 AMC 10, 9
In how many ways can $6$ juniors and $6$ seniors form $3$ disjoint teams of $4$ people so that each team has $2$ juniors and $2$ seniors?
$
\textbf{(A) }720 \qquad
\textbf{(B) }1350 \qquad
\textbf{(C) }2700 \qquad
\textbf{(D) }3280 \qquad
\textbf{(E) }8100 \qquad
$
2017 AMC 12/AHSME, 11
Call a positive integer [i]monotonous[/i] if it is a one-digit number or its digits, when read from left to right, form either a strictly increasing or a strictly decreasing sequence. For example, 3, 23578, and 987620 are monotonous, but 88, 7434, and 23557 are not. How many monotonous positive integers are there?
$\textbf{(A)} \text{ 1024} \qquad \textbf{(B)} \text{ 1524} \qquad \textbf{(C)} \text{ 1533} \qquad \textbf{(D)} \text{ 1536} \qquad \textbf{(E)} \text{ 2048}$
2024 AMC 10, 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
$
2017 AMC 10, 3
Tamara has three rows of two 6-feet by 2-feet flower beds in her garden. The beds are separated and also surrounded by 1-foot-wide walkways, as shown on the diagram. What is the total area of the walkways, in square feet?
[asy]
unitsize(0.7cm);
path p1 = (0,0)--(15,0)--(15,10)--(0,10)--cycle;
fill(p1,lightgray);
draw(p1);
for (int i = 1; i <= 8; i += 7) {
for (int j = 1; j <= 7; j += 3 ) {
path p2 = (i,j)--(i+6,j)--(i+6,j+2)--(i,j+2)--cycle;
draw(p2);
fill(p2,white);
}
}
draw((0,8)--(1,8),Arrows);
label("1",(0.5,8),S);
draw((7,8)--(8,8),Arrows);
label("1",(7.5,8),S);
draw((14,8)--(15,8),Arrows);
label("1",(14.5,8),S);
draw((11,0)--(11,1),Arrows);
label("1",(11,0.5),W);
draw((11,3)--(11,4),Arrows);
label("1",(11,3.5),W);
draw((11,6)--(11,7),Arrows);
label("1",(11,6.5),W);
draw((11,9)--(11,10),Arrows);
label("1",(11,9.5),W);
label("6",(4,1),N);
label("2",(1,2),E);
[/asy]
$\textbf{(A) }72 \qquad
\textbf{(B) }78 \qquad
\textbf{(C) }90 \qquad
\textbf{(D) }120 \qquad
\textbf{(E) }150 $
2021 AMC 12/AHSME Fall, 5
Elmer the emu takes $44$ equal strides to walk between consecutive telephone poles on a rural road. Oscar the ostrich can cover the same distance in $12$ equal leaps. The telephone poles are evenly spaced, and the $41$st pole along this road is exactly one mile ($5280$ feet) from the first pole. How much longer, in feet, is Oscar's leap than Elmer's stride?
$\textbf{(A) }6\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }11\qquad\textbf{(E) }15$
2021 AMC 10 Spring, 11
For which of the following integers $b$ is the base-$b$ number $2021_b - 221_b$ not divisible by $3$?
$\textbf{(A) } 3 \qquad \textbf{(B) } 4 \qquad \textbf{(C) } 6 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 8$
2013 AMC 10, 24
Central High School is competing against Northern High School in a backgammon match. Each school has three players, and the contest rules require that each player play two games against each of the other's school's players. The match takes place in six rounds, with three games played simultaneously in each round. In how many different ways can the match be scheduled?
$\textbf{(A)} \ 540 \qquad \textbf{(B)} \ 600 \qquad \textbf{(C)} \ 720 \qquad \textbf{(D)} \ 810 \qquad \textbf{(E)} \ 900$
2022 AMC 10, 7
The least common multiple of a positive integer $n$ and 18 is 180, and the greatest common divisor of $n$ and 45 is 15. What is the sum of the digits of $n$?
$\textbf{(A) }3\qquad\textbf{(B) }6\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }12$
2019 AMC 10, 19
Let $S$ be the set of all positive integer divisors of $100,000.$ How many numbers are the product of two distinct elements of $S?$
$\textbf{(A) }98\qquad\textbf{(B) }100\qquad\textbf{(C) }117\qquad\textbf{(D) }119\qquad\textbf{(E) }121$
2016 AMC 10, 23
In regular hexagon $ABCDEF$, points $W$, $X$, $Y$, and $Z$ are chosen on sides $\overline{BC}$, $\overline{CD}$, $\overline{EF}$, and $\overline{FA}$ respectively, so lines $AB$, $ZW$, $YX$, and $ED$ are parallel and equally spaced. What is the ratio of the area of hexagon $WCXYFZ$ to the area of hexagon $ABCDEF$?
$\textbf{(A)}\ \frac{1}{3}\qquad\textbf{(B)}\ \frac{10}{27}\qquad\textbf{(C)}\ \frac{11}{27}\qquad\textbf{(D)}\ \frac{4}{9}\qquad\textbf{(E)}\ \frac{13}{27}$
2016 AMC 12/AHSME, 18
What is the area of the region enclosed by the graph of the equation $x^2+y^2=|x|+|y|?$
$\textbf{(A)}\ \pi+\sqrt{2} \qquad\textbf{(B)}\ \pi+2 \qquad\textbf{(C)}\ \pi+2\sqrt{2} \qquad\textbf{(D)}\ 2\pi+\sqrt{2} \qquad\textbf{(E)}\ 2\pi+2\sqrt{2}$
2017 AMC 10, 18
In the figure below, $3$ of the $6$ disks are to be painted blue, $2$ are to be painted red, and $1$ is to be painted green. Two paintings that can be obtained from one another by a rotation or a reflection of the entire figure are considered the same. How many different paintings are possible?
[asy]
size(100);
pair A, B, C, D, E, F;
A = (0,0);
B = (1,0);
C = (2,0);
D = rotate(60, A)*B;
E = B + D;
F = rotate(60, A)*C;
draw(Circle(A, 0.5));
draw(Circle(B, 0.5));
draw(Circle(C, 0.5));
draw(Circle(D, 0.5));
draw(Circle(E, 0.5));
draw(Circle(F, 0.5));
[/asy]
$\textbf{(A) } 6 \qquad \textbf{(B) } 8 \qquad \textbf{(C) } 9 \qquad \textbf{(D) } 12 \qquad \textbf{(E) } 15$
2018 AMC 12/AHSME, 19
Mary chose an even $4$-digit number $n$. She wrote down all the divisors of $n$ in increasing order from left to right: $1,2,...,\tfrac{n}{2},n$. At some moment Mary wrote $323$ as a divisor of $n$. What is the smallest possible value of the next divisor written to the right of $323$?
$\textbf{(A) } 324 \qquad \textbf{(B) } 330 \qquad \textbf{(C) } 340 \qquad \textbf{(D) } 361 \qquad \textbf{(E) } 646$
2012 AMC 12/AHSME, 14
The closed curve in the figure is made up of $9$ congruent circular arcs each of length $\frac{2\pi}{3}$, where each of the centers of the corresponding circles is among the vertices of a regular hexagon of side $2$. What is the area enclosed by the curve?
[asy]
size(170);
defaultpen(fontsize(6pt));
dotfactor=4;
label("$\circ$",(0,1));
label("$\circ$",(0.865,0.5));
label("$\circ$",(-0.865,0.5));
label("$\circ$",(0.865,-0.5));
label("$\circ$",(-0.865,-0.5));
label("$\circ$",(0,-1));
dot((0,1.5));
dot((-0.4325,0.75));
dot((0.4325,0.75));
dot((-0.4325,-0.75));
dot((0.4325,-0.75));
dot((-0.865,0));
dot((0.865,0));
dot((-1.2975,-0.75));
dot((1.2975,-0.75));
draw(Arc((0,1),0.5,210,-30));
draw(Arc((0.865,0.5),0.5,150,270));
draw(Arc((0.865,-0.5),0.5,90,-150));
draw(Arc((0.865,-0.5),0.5,90,-150));
draw(Arc((0,-1),0.5,30,150));
draw(Arc((-0.865,-0.5),0.5,330,90));
draw(Arc((-0.865,0.5),0.5,-90,30));
[/asy]
$ \textbf{(A)}\ 2\pi+6\qquad\textbf{(B)}\ 2\pi+4\sqrt3 \qquad\textbf{(C)}\ 3\pi+4 \qquad\textbf{(D)}\ 2\pi+3\sqrt3+2 \qquad\textbf{(E)}\ \pi+6\sqrt3 $
2009 AMC 10, 24
The keystone arch is an ancient architectural feature. It is composed of congruent isosceles trapezoids fitted together along the non-parallel sides, as shown. The bottom sides of the two end trapezoids are horizontal. In an arch made with $ 9$ trapezoids, let $ x$ be the angle measure in degrees of the larger interior angle of the trapezoid. What is $ x$?
[asy]unitsize(4mm);
defaultpen(linewidth(.8pt));
int i;
real r=5, R=6;
path t=r*dir(0)--r*dir(20)--R*dir(20)--R*dir(0);
for(i=0; i<9; ++i)
{
draw(rotate(20*i)*t);
}
draw((-r,0)--(R+1,0));
draw((-R,0)--(-R-1,0));[/asy]$ \textbf{(A)}\ 100 \qquad
\textbf{(B)}\ 102 \qquad
\textbf{(C)}\ 104 \qquad
\textbf{(D)}\ 106 \qquad
\textbf{(E)}\ 108$
2016 AMC 10, 20
A dilation of the plane—that is, a size transformation with a positive scale factor—sends the circle of radius $2$ centered at $A(2,2)$ to the circle of radius $3$ centered at $A’(5,6)$. What distance does the origin $O(0,0)$, move under this transformation?
$\textbf{(A)}\ 0\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ \sqrt{13}\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5$
2002 AMC 10, 18
For how many positive integers $n$ is $n^3-8n^2+20n-13$ a prime number?
$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }\text{more than 4}$
2023 AMC 10, 5
Maddy and Lara see a list of numbers written on a blackboard. Maddy adds $3$ to each number in the list and finds that the sum of her new numbers is $45$. Lara multiplies each number in the list by $3$ and finds that the sum of her new numbers is also $45$. How many numbers are written on the blackboard?
$\textbf{(A) }10\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }8\qquad\textbf{(E) }9$
2023 AMC 12/AHSME, 9
What is the area of the region in the coordinate plane defined by the inequality \[\left||x|-1\right|+\left||y|-1\right|\leq 1?\]
$\textbf{(A)}~4\qquad\textbf{(B)}~8\qquad\textbf{(C)}~10\qquad\textbf{(D)}~12\qquad\textbf{(E)}~15$
2023 AMC 12/AHSME, 21
If $A$ and $B$ are vertices of a polyhedron, define the [i]distance[/i] $d(A, B)$ to be the minimum number of edges of the polyhedron one must traverse in order to connect $A$ and $B$. For example, if $\overline{AB}$ is an edge of the polyhedron, then $d(A, B) = 1$, but if $\overline{AC}$ and $\overline{CB}$ are edges and $\overline{AB}$ is not an edge, then $d(A, B) = 2$. Let $Q$, $R$, and $S$ be randomly chosen distinct vertices of a regular icosahedron (regular polyhedron made up of 20 equilateral triangles). What is the probability that $d(Q, R) > d(R, S)$?
$\textbf{(A)}~\frac{7}{22}\qquad\textbf{(B)}~\frac13\qquad\textbf{(C)}~\frac38\qquad\textbf{(D)}~\frac5{12}\qquad\textbf{(E)}~\frac12$
2019 AMC 12/AHSME, 14
Let $S$ be the set of all positive integer divisors of $100,000.$ How many numbers are the product of two distinct elements of $S?$
$\textbf{(A) }98\qquad\textbf{(B) }100\qquad\textbf{(C) }117\qquad\textbf{(D) }119\qquad\textbf{(E) }121$
2019 AMC 10, 5
What is the greatest number of consecutive integers whose sum is $45 ?$
$\textbf{(A) } 9 \qquad\textbf{(B) } 25 \qquad\textbf{(C) } 45 \qquad\textbf{(D) } 90 \qquad\textbf{(E) } 120$
2018 AMC 12/AHSME, 14
Joey and Chloe and their daughter Zoe all have the same birthday. Joey is 1 year older than Chloe, and Zoe is exactly 1 year old today. Today is the first of the 9 birthdays on which Chloe's age will be an integral multiple of Zoe's age. What will be the sum of the two digits of Joey's age the next time his age is a multiple of Zoe's age?
$\textbf{(A) } 7 \qquad \textbf{(B) } 8 \qquad \textbf{(C) } 9 \qquad \textbf{(D) } 10 \qquad \textbf{(E) } 11 $
2017 AMC 10, 19
Let $ABC$ be an equilateral triangle. Extend side $\overline{AB}$ beyond $B$ to a point $B'$ so that $BB' = 3AB$. Similarly, extend side $\overline{BC}$ beyond $C$ to a point $C'$ so that $CC' = 3BC$, and extend side $\overline{CA}$ beyond $A$ to a point $A'$ so that $AA' = 3CA$. What is the ratio of the area of $\triangle A'B'C'$ to the area of $\triangle ABC$?
$\textbf{(A) }9:1\qquad\textbf{(B) }16:1\qquad\textbf{(C) }25:1\qquad\textbf{(D) }36:1\qquad\textbf{(E) }37:1$
2022 AMC 12/AHSME, 23
Let $x_{0}$, $x_{1}$, $x_{2}$, $\cdots$ be a sequence of numbers, where each $x_{k}$ is either $0$ or $1$. For each positive integer $n$, define
\[S_{n} = \displaystyle\sum^{n-1}_{k=0}{x_{k}2^{k}}\]
Suppose $7S_{n} \equiv 1\pmod {2^{n}}$ for all $n\geq 1$. What is the value of the sum
\[x_{2019}+2x_{2020}+4x_{2021}+8x_{2022}?\]
$ \textbf{(A)}\ 6 \qquad
\textbf{(B)}\ 7 \qquad
\textbf{(C)}\ 12 \qquad
\textbf{(D)}\ 14 \qquad
\textbf{(E)}\ 15$