Found problems: 141
2018 AMC 10, 8
Joe has a collection of 23 coins, consisting of 5-cent coins, 10-cent coins, and 25-cent coins. He has 3 more 10-cent coins than 5-cent coins, and the total value of his collection is 320 cents. How many more 25-cent coins does Joe have than 5-cent coins?
$\textbf{(A) } 0 \qquad \textbf{(B) } 1 \qquad \textbf{(C) } 2 \qquad \textbf{(D) } 3 \qquad \textbf{(E) } 4 $
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 $3$?
$\textbf{(A) } 3 \qquad \textbf{(B) } 4 \qquad \textbf{(C) } 6 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 8$
2018 AMC 10, 4
How many ways can a student schedule 3 mathematics courses -- algebra, geometry, and number theory -- in a 6-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other 3 periods is of no concern here.)
$\textbf{(A) }3\qquad\textbf{(B) }6\qquad\textbf{(C) }12\qquad\textbf{(D) }18\qquad\textbf{(E) }24$
2017 AMC 10, 19
Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of $5$ chairs under these conditions?
$\textbf{(A)}\ 12\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 28\qquad\textbf{(D)}\ 32\qquad\textbf{(E)}\ 40$
2018 AMC 12/AHSME, 4
Alice, Bob, and Charlie were on a hike and were wondering how far away the nearest town was. When Alice said, "We are at least 6 miles away," Bob replied, "We are at most 5 miles away." Charlie then remarked, "Actually the nearest town is at most 4 miles away." It turned out that none of the three statements were true. Let $d$ be the distance in miles to the nearest town. Which of the following intervals is the set of all possible values of $d$?
$\textbf{(A) } (0,4) \qquad \textbf{(B) } (4,5) \qquad \textbf{(C) } (4,6) \qquad \textbf{(D) } (5,6) \qquad \textbf{(E) } (5,\infty) $
2018 AMC 10, 11
When 7 fair standard 6-sided dice are thrown, the probability that the sum of the numbers on the top faces is 10 can be written as $$\frac{n}{6^7},$$where $n$ is a positive integer. What is $n$?
$\textbf{(A) } 42 \qquad \textbf{(B) } 49 \qquad \textbf{(C) } 56 \qquad \textbf{(D) } 63 \qquad \textbf{(E) } 84 $
2021 AMC 10 Spring, 6
Chantal and Jean start hiking from a trailhead toward a fire tower. Jean is wearing a heavy backpack and walks slower. Chantal starts walking at $4$ miles per hour. Halfway to the tower, the trail becomes really steep, and Chantal slows down to $2$ miles per hour. After reaching the tower, she immediately turns around and descends the steep part of the trail at $3$ miles per hour. She meets Jean at the halfway point. What was Jean's average speed, in miles per hour, until they meet?
$\textbf{(A)}~\frac{12}{13}\qquad \textbf{(B)}~1\qquad \textbf{(C)}~\frac{13}{12}\qquad \textbf{(D)}~\frac{24}{13}\qquad \textbf{(E)}~2$
2023 AMC 10, 6
An integer is assigned to each vertex of a cube. The value of an edge is defined to be the sum of the values of the two vertices it touches, and the value of a face is defined to be the sum of the values of the four edges surrounding it. The value of the cube is defined as the sum of the values of its six faces. Suppose the sum of the integers assigned to the vertices is $21$. What is the value of the cube?
$\textbf{(A)}~42\qquad\textbf{(B)}~63\qquad\textbf{(C)}~84\qquad\textbf{(D)}~126\qquad\textbf{(E)}~252$
2013 AMC 12/AHSME, 1
Square $ ABCD $ has side length $ 10 $. Point $ E $ is on $ \overline{BC} $, and the area of $ \bigtriangleup ABE $ is $ 40 $. What is $ BE $?
$\textbf{(A)} \ 4 \qquad \textbf{(B)} \ 5 \qquad \textbf{(C)} \ 6 \qquad \textbf{(D)} \ 7 \qquad \textbf{(E)} \ 8 \qquad $
[asy]
pair A,B,C,D,E;
A=(0,0);
B=(0,50);
C=(50,50);
D=(50,0);
E = (30,50);
draw(A--B);
draw(B--E);
draw(E--C);
draw(C--D);
draw(D--A);
draw(A--E);
dot(A);
dot(B);
dot(C);
dot(D);
dot(E);
label("A",A,SW);
label("B",B,NW);
label("C",C,NE);
label("D",D,SE);
label("E",E,N);
[/asy]
2021 AMC 10 Fall, 24
Each of the $12$ edges of a cube is labeled $0$ or $1$. Two labelings are considered different even if one can be obtained from the other by a sequence of one or more rotations and/or reflections. For how many such labelings is the sum of the labels on the edges of each of the $6$ faces of the cube equal to $2?$
$\textbf{(A) }8\qquad\textbf{(B) }10\qquad\textbf{(C) }12\qquad\textbf{(D) }16\qquad\textbf{(E) }20$
2017 AMC 12/AHSME, 1
Pablo buys popsicles for his friends. The store sells single popsicles for $\$1$ each, 3-popsicle boxes for $\$2$, and 5-popsicle boxes for $\$3$. What is the greatest number of popsicles that Pablo can buy with $\$8$?
$\textbf{(A)}\ 8\qquad\textbf{(B)}\ 11\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 13\qquad\textbf{(E)}\ 15$
2023 AMC 10, 13
Abdul and Chiang are standing $48$ feet apart in a field. Bharat is standing in the same field as far from Abdul as possible so that the angle formed by his lines of sight to Abdul and Chiang measures $60^{\circ}.$ What is the square of the distance (in feet) between Abdul and Bharat?
$\textbf{(A) } 1728 \qquad\textbf{(B) } 2601 \qquad\textbf{(C) } 3072 \qquad\textbf{(D) } 4608 \qquad\textbf{(E) } 6912$
2016 AMC 12/AHSME, 13
Let $N$ be a positive multiple of $5$. One red ball and $N$ green balls are arranged in a line in random order. Let $P(N)$ be the probability that at least $\tfrac{3}{5}$ of the green balls are on the same side of the red ball. Observe that $P(5)=1$ and that $P(N)$ approaches $\tfrac{4}{5}$ as $N$ grows large. What is the sum of the digits of the least value of $N$ such that $P(N) < \tfrac{321}{400}$?
$\textbf{(A) } 12 \qquad \textbf{(B) } 14 \qquad \textbf{(C) }16 \qquad \textbf{(D) } 18 \qquad \textbf{(E) } 20$
2020 AMC 10, 17
Define $$P(x) =(x-1^2)(x-2^2)\cdots(x-100^2).$$
How many integers $n$ are there such that $P(n)\leq 0$?
$\textbf{(A) } 4900 \qquad \textbf{(B) } 4950\qquad \textbf{(C) } 5000\qquad \textbf{(D) } 5050 \qquad \textbf{(E) } 5100$
2018 AMC 10, 14
What is the greatest integer less than or equal to $$\frac{3^{100}+2^{100}}{3^{96}+2^{96}}?$$
$
\textbf{(A) }80\qquad
\textbf{(B) }81 \qquad
\textbf{(C) }96 \qquad
\textbf{(D) }97 \qquad
\textbf{(E) }625\qquad
$
2010 AMC 12/AHSME, 23
The number obtained from the last two nonzero digits of $ 90!$ is equal to $ n$. What is $ n$?
$ \textbf{(A)}\ 12 \qquad
\textbf{(B)}\ 32 \qquad
\textbf{(C)}\ 48 \qquad
\textbf{(D)}\ 52 \qquad
\textbf{(E)}\ 68$
2019 AMC 10, 6
For how many of the following types of quadrilaterals does there exist a point in the plane of the quadrilateral that is equidistant from all four vertices of the quadrilateral?
[list]
[*] a square
[*]a rectangle that is not a square
[*] a rhombus that is not a square
[*] a parallelogram that is not a rectangle or a rhombus
[*] an isosceles trapezoid that is not a parallelogram
[/list]
$\textbf{(A) } 1 \qquad\textbf{(B) } 2 \qquad\textbf{(C) } 3 \qquad\textbf{(D) } 4 \qquad\textbf{(E) } 5$
2016 AMC 10, 19
In rectangle $ABCD$, $AB=6$ and $BC=3$. Point $E$ between $B$ and $C$, and point $F$ between $E$ and $C$ are such that $BE=EF=FC$. Segments $\overline{AE}$ and $\overline{AF}$ intersect $\overline{BD}$ at $P$ and $Q$, respectively. The ratio $BP:PQ:QD$ can be written as $r:s:t$, where the greatest common factor of $r,s$ and $t$ is $1$. What is $r+s+t$?
$\textbf{(A) } 7
\qquad \textbf{(B) } 9
\qquad \textbf{(C) } 12
\qquad \textbf{(D) } 15
\qquad \textbf{(E) } 20$
2016 AMC 12/AHSME, 6
A triangular array of $2016$ coins has $1$ coin in the first row, $2$ coins in the second row, $3$ coins in the third row, and so on up to $N$ coins in the $N$th row. What is the sum of the digits of $N$?
$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10$
2021 AMC 10 Fall, 13
Each of $6$ balls is randomly and independently painted either black or white with equal probability. What is the probability that every ball is different in color from more than half of the other $5$ balls?
$\textbf{(A) }\dfrac1{64}\qquad\textbf{(B) }\dfrac16\qquad\textbf{(C) }\dfrac14\qquad\textbf{(D) }\dfrac5{16}\qquad\textbf{(E) }\dfrac12$
2018 AMC 10, 25
For a positive integer $n$ and nonzero digits $a$, $b$, and $c$, let $A_n$ be the $n$-digit integer each of whose digits is equal to $a$; let $B_n$ be the $n$-digit integer each of whose digits is equal to $b$, and let $C_n$ be the $2n$-digit (not $n$-digit) integer each of whose digits is equal to $c$. What is the greatest possible value of $a + b + c$ for which there are at least two values of $n$ such that $C_n - B_n = A_n^2$?
$\textbf{(A)} \text{ 12} \qquad \textbf{(B)} \text{ 14} \qquad \textbf{(C)} \text{ 16} \qquad \textbf{(D)} \text{ 18} \qquad \textbf{(E)} \text{ 20}$
2021 AMC 10 Spring, 4
A cart rolls down a hill, traveling 5 inches the first second and accelerating so that each successive 1-second time
interval, it travels 7 inches more than during the previous 1-second interval. The cart takes 30 seconds to reach the
bottom of the hill. How far, in inches, does it travel?
$\textbf{(A) }215 \qquad \textbf{(B) }360 \qquad \textbf{(C) }2992 \qquad \textbf{(D) }3195 \qquad \textbf{(E) }3242$
2017 AMC 10, 4
Mia is “helping” her mom pick up $30$ toys that are strewn on the floor. Mia’s mom manages to put $3$ toys into the toy box every $30$ seconds, but each time immediately after those $30$ seconds have elapsed, Mia takes $2$ toys out of the box. How much time, in minutes, will it take Mia and her mom to put all $30$ toys into the box for the first time?
$\textbf{(A)}\ 13.5\qquad\textbf{(B)}\ 14\qquad\textbf{(C)}\ 14.5\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 15.5$