Found problems: 260
2022 AMC 8 -, 4
The letter [b]M[/b] in the figure below is first reflected over the line $q$ and then reflected over the line $p$. What is the resulting image?
[asy]
// pog diagram
usepackage("newtxtext");
size(3cm);
draw((-1,0)--(1,0)); draw((0,-1)--(0,1)); label("$\textbf{\textsf{M}}$",(0.25,0.6));
draw((-0.8,-0.8)--(0.8,0.8),linewidth(1.1)); label("$p$", (-1,0),NE); label("$q$", (-0.75,-0.75), N*1.5);
[/asy]
[asy]
// pog diagram
usepackage("newtxtext");
size(12.5cm);
draw((-1,0)--(1,0)); draw((0,-1)--(0,1)); label(rotate(90)*"$\textbf{\textsf{M}}$",(0.6,-0.25));
draw((-0.8,-0.8)--(0.8,0.8),linewidth(1.1));
label("$\textbf{(A)}$",(-1,1),W);
draw((2,0)--(4,0)); draw((3,-1)--(3,1)); label(rotate(270)*"$\textbf{\textsf{M}}$",(2.8,0.7));
draw((2.2,-0.8)--(3.8,0.8),linewidth(1.1));
label("$\textbf{(B)}$",(2,1),W);
draw((5,0)--(7,0)); draw((6,-1)--(6,1)); label(rotate(90)*"$\textbf{\textsf{M}}$",(5.4,0.2));
draw((5.2,-0.8)--(6.8,0.8),linewidth(1.1));
label("$\textbf{(C)}$",(5,1),W);
draw((-1,-2.5)--(1,-2.5)); draw((0,-3.5)--(0,-1.5)); label(rotate(180)*"$\textbf{\textsf{M}}$",(-0.25,-3.1));
draw((-0.8,-3.3)--(0.8,-1.7),linewidth(1.1));
label("$\textbf{(D)}$",(-1,-1.5),W);
draw((2,-2.5)--(4,-2.5)); draw((3,-3.5)--(3,-1.5)); label(rotate(270)*"$\textbf{\textsf{M}}$",(3.6,-2.75));
draw((2.2,-3.3)--(3.8,-1.7),linewidth(1.1));
label("$\textbf{(E)}$",(2,-1.5),W);
[/asy]
2022 AMC 8 -, 14
In how many ways can the letters in BEEKEEPER be rearranged so that two or more Es do not appear together?
$\textbf{(A)} ~1\qquad\textbf{(B)} ~4\qquad\textbf{(C)} ~12\qquad\textbf{(D)} ~24\qquad\textbf{(E)} ~120\qquad$
2020 AMC 8 -, 14
There are $20$ cities in the County of Newton. Their populations are shown in the bar chart below. The average population of all the cities is indicated by the horizontal dashed line. Which of the following is closest to the total population of all $20$ cities?
[asy]
// made by SirCalcsALot
size(300);
pen shortdashed=linetype(new real[] {6,6});
// axis
draw((0,0)--(0,9300), linewidth(1.25));
draw((0,0)--(11550,0), linewidth(1.25));
for (int i = 2000; i < 9000; i = i + 2000) {
draw((0,i)--(11550,i), linewidth(0.5)+1.5*grey);
label(string(i), (0,i), W);
}
for (int i = 500; i < 9300; i=i+500) {
draw((0,i)--(150,i),linewidth(1.25));
if (i % 2000 == 0) {
draw((0,i)--(250,i),linewidth(1.25));
}
}
int[] data = {8750, 3800, 5000, 2900, 6400, 7500, 4100, 1400, 2600, 1470, 2600, 7100, 4070, 7500, 7000, 8100, 1900, 1600, 5850, 5750};
int data_length = 20;
int r = 550;
for (int i = 0; i < data_length; ++i) {
fill(((i+1)*r,0)--((i+1)*r, data[i])--((i+1)*r,0)--((i+1)*r, data[i])--((i+1)*r,0)--((i+1)*r, data[i])--((i+2)*r-100, data[i])--((i+2)*r-100,0)--cycle, 1.5*grey);
draw(((i+1)*r,0)--((i+1)*r, data[i])--((i+1)*r,0)--((i+1)*r, data[i])--((i+1)*r,0)--((i+1)*r, data[i])--((i+2)*r-100, data[i])--((i+2)*r-100,0));
}
draw((0,4750)--(11450,4750),shortdashed);
label("Cities", (11450*0.5,0), S);
label(rotate(90)*"Population", (0,9000*0.5), 10*W);
[/asy]
$\textbf{(A) }65{,}000 \qquad \textbf{(B) }75{,}000 \qquad \textbf{(C) }85{,}000 \qquad \textbf{(D) }95{,}000 \qquad \textbf{(E) }105{,}000$
2023 AMC 8, 21
Alina writes the numbers $1, 2, \dots , 9$ on separate cards, one number per card. She wishes to divide the cards into $3$ groups of $3$ cards so that the sum of the number in each group will be the same. In how many ways can this be done?
$\textbf{(A) }0 \qquad \textbf{(B) } 1 \qquad \textbf{(C) } 2 \qquad \textbf{(D) } 3 \qquad \textbf{(E) } 4$
2020 AMC 8 -, 5
Three fourths of a pitcher is filled with pineapple juice. The pitcher is emptied by pouring an equal amount of juice into each of $5$ cups. What percent of the total capacity of the pitcher did each cup receive?
$\textbf{(A) }5 \qquad \textbf{(B) }10 \qquad \textbf{(C) }15 \qquad \textbf{(D) }20 \qquad \textbf{(E) }25$
2016 AMC 8, 16
Annie and Bonnie are running laps around a 400-meter oval track. They started together, but Annie has pulled ahead because she is $25 \%$ faster than Bonnie. How many laps will Annie have run when she first passes Bonnie?
$\textbf{(A) }1 \frac{1}{4}\qquad\textbf{(B) }3 \frac{1}{3}\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad \textbf{(E) }25$
2020 AMC 8 -, 24
A large square region is paved with $n^2$ gray square tiles, each measuring $s$ inches on a side. A border $d$ inches wide surrounds each tile. The figure below shows the case for $n = 3$. When $n = 24$, the $576$ gray tiles cover $64\%$ of the area of the large square region. What is the ratio $\frac{d}{s}$ for this larger value of $n$?
[asy]
draw((0,0)--(13,0)--(13,13)--(0,13)--cycle);
filldraw((1,1)--(4,1)--(4,4)--(1,4)--cycle, mediumgray);
filldraw((1,5)--(4,5)--(4,8)--(1,8)--cycle, mediumgray);
filldraw((1,9)--(4,9)--(4,12)--(1,12)--cycle, mediumgray);
filldraw((5,1)--(8,1)--(8,4)--(5,4)--cycle, mediumgray);
filldraw((5,5)--(8,5)--(8,8)--(5,8)--cycle, mediumgray);
filldraw((5,9)--(8,9)--(8,12)--(5,12)--cycle, mediumgray);
filldraw((9,1)--(12,1)--(12,4)--(9,4)--cycle, mediumgray);
filldraw((9,5)--(12,5)--(12,8)--(9,8)--cycle, mediumgray);
filldraw((9,9)--(12,9)--(12,12)--(9,12)--cycle, mediumgray);
[/asy]
$\textbf{(A) }\frac6{25} \qquad \textbf{(B) }\frac14 \qquad \textbf{(C) }\frac9{25} \qquad \textbf{(D) }\frac7{16} \qquad \textbf{(E) }\frac9{16}$
2016 AMC 8, 8
Find the value of the expression
$$100-98+96-94+92-90+\cdots+8-6+4-2.$$
$\textbf{(A) }20\qquad\textbf{(B) }40\qquad\textbf{(C) }50\qquad\textbf{(D) }80\qquad \textbf{(E) }100$
2023 AMC 8, 14
Nicolas is planning to send a package to his friend Anton, who is a stamp collector. To pay for the postage, Nicolas would like to cover the package with a large number of stamps. Suppose he has a collection of 5-cent, 10-cent, and 25-cent stamps, with exactly 20 of each type. What is the greatest number of stamps Nicolas can use to make exactly $\$7.10$ in postage?
(Note: The amount $\$7.10$ corresponds to 7 dollars and 10 cents. One dollar is worth 100 cents.)
$\textbf{(A)}\ 45 \qquad \textbf{(B)}\ 46 \qquad \textbf{(C)}\ 51 \qquad \textbf{(D)}\ 54 \qquad \textbf{(E)}\ 55$
1972 AMC 12/AHSME, 1
The lengths in inches of the three sides of each of four triangles I, II, III, and IV are as follows:
\[ \begin{array}{rlrl} \hbox {I}& 3,\ 4,\ \hbox{and}\ 5 \qquad & \hbox{III}& 7,\ 24,\ \hbox{and}\ 25 \\ \hbox{II}& 4,\ 7\frac{1}{2},\ \hbox{and}\ 8\frac{1}{2} \qquad & \hbox{IV}& 3\frac{1}{2},\ 4\frac{1}{2},\ \hbox{and}\ 5\frac{1}{2}. \end{array} \]
Of these four given triangles, the only right triangles are
\[ \begin{tabular}{rlrlrl} (A) & I and II \qquad & (B) & I and III \qquad & (C) & I and IV \\ (D) & I, II, and III \qquad & (E) & I, II, and IV & \end{tabular} \]
2022 AMC 8 -, 3
When three positive integers $a, b$, and $c$ are multiplied together, their product is $100$. Suppose $a < b < c$. In how many ways can the numbers be chosen?
$\textbf{(A)} ~0\qquad\textbf{(B)} ~1\qquad\textbf{(C)} ~2\qquad\textbf{(D)} ~3\qquad\textbf{(E)} ~4\qquad$
2016 AMC 8, 6
The following bar graph represents the length (in letters) of the names of 19 people. What is the median length of these names? $\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad \textbf{(E) }7$
[asy] unitsize(0.9cm); draw((-0.5,0)--(10,0), linewidth(1.5)); draw((-0.5,1)--(10,1)); draw((-0.5,2)--(10,2)); draw((-0.5,3)--(10,3)); draw((-0.5,4)--(10,4)); draw((-0.5,5)--(10,5)); draw((-0.5,6)--(10,6)); draw((-0.5,7)--(10,7)); label("frequency",(-0.5,8)); label("0", (-1, 0)); label("1", (-1, 1)); label("2", (-1, 2)); label("3", (-1, 3)); label("4", (-1, 4)); label("5", (-1, 5)); label("6", (-1, 6)); label("7", (-1, 7)); filldraw((0,0)--(0,7)--(1,7)--(1,0)--cycle, black); filldraw((2,0)--(2,3)--(3,3)--(3,0)--cycle, black); filldraw((4,0)--(4,1)--(5,1)--(5,0)--cycle, black); filldraw((6,0)--(6,4)--(7,4)--(7,0)--cycle, black); filldraw((8,0)--(8,4)--(9,4)--(9,0)--cycle, black); label("3", (0.5, -0.5)); label("4", (2.5, -0.5)); label("5", (4.5, -0.5)); label("6", (6.5, -0.5)); label("7", (8.5, -0.5)); label("name length", (4.5,-1.5)); [/asy]
2024 AMC 8 -, 3
Four squares of side length $4, 7, 9,$ and $10$ are arranged in increasing size order so that their left edges and bottom edges align. The squares alternate in color white-gray-white-gray, respectively, as shown in the figure. What is the area of the visible gray region in square units?
[asy]
size(150);
filldraw((0,0)--(10,0)--(10,10)--(0,10)--cycle,gray(0.7),linewidth(1));
filldraw((0,0)--(9,0)--(9,9)--(0,9)--cycle,white,linewidth(1));
filldraw((0,0)--(7,0)--(7,7)--(0,7)--cycle,gray(0.7),linewidth(1));
filldraw((0,0)--(4,0)--(4,4)--(0,4)--cycle,white,linewidth(1));
draw((11,0)--(11,4),linewidth(1));
draw((11,6)--(11,10),linewidth(1));
label("$10$",(11,5),fontsize(14pt));
draw((10.75,0)--(11.25,0),linewidth(1));
draw((10.75,10)--(11.25,10),linewidth(1));
draw((0,11)--(3,11),linewidth(1));
draw((5,11)--(9,11),linewidth(1));
draw((0,11.25)--(0,10.75),linewidth(1));
draw((9,11.25)--(9,10.75),linewidth(1));
label("$9$",(4,11),fontsize(14pt));
draw((-1,0)--(-1,1),linewidth(1));
draw((-1,3)--(-1,7),linewidth(1));
draw((-1.25,0)--(-0.75,0),linewidth(1));
draw((-1.25,7)--(-0.75,7),linewidth(1));
label("$7$",(-1,2),fontsize(14pt));
draw((0,-1)--(1,-1),linewidth(1));
draw((3,-1)--(4,-1),linewidth(1));
draw((0,-1.25)--(0,-.75),linewidth(1));
draw((4,-1.25)--(4,-.75),linewidth(1));
label("$4$",(2,-1),fontsize(14pt));
[/asy]
$\textbf{(A)}\ 42 \qquad \textbf{(B)}\ 45\qquad \textbf{(C)}\ 49\qquad \textbf{(D)}\ 50\qquad \textbf{(E)}\ 52$
2023 AMC 8, 13
Along the route of a bicycle race, $7$ water stations are evenly spaced between the start and finish lines, as shown in the figure below. There are also $2$ repair stations evenly spaced between the start and finish lines. The $3$rd water station is located $2$ miles after the $1$st repair station. How long is the race in miles?
[asy]
size(10cm);
filldraw((11,4.5)--(171,4.5)--(171,17.5)--(11,17.5)--cycle,mediumgray);
draw((11,11)--(171,11),linetype("4 4")+white+linewidth(1.5));
draw((0,0)--(11,0)--(11,22)--(0,22)--cycle,linewidth(1.125));
draw((171,0)--(182,0)--(182,22)--(171,22)--cycle,linewidth(1.125));
draw((31,4.5)--(31,0));
draw((51,4.5)--(51,0));
draw((151,4.5)--(151,0));
label(scale(.9)*rotate(45)*"Water 1", (23,-13.5));
label(scale(.9)*rotate(45)*"Water 2", (43,-13.5));
label(scale(.9)*rotate(45)*"Water 7", (143,-13.5));
filldraw(circle((101,-13.5),.3));
filldraw(circle((97,-13.5),.3));
filldraw(circle((93,-13.5),.3));
filldraw(circle((89,-13.5),.3));
filldraw(circle((85,-13.5),.3));
label(scale(.9)*rotate(90)*"Start", (5.5,11));
label(scale(.9)*rotate(270)*"Finish", (176.5,11));
[/asy]
$\textbf{(A) } 8\qquad\textbf{(B) } 16\qquad\textbf{(C) } 24\qquad\textbf{(D) } 48\qquad\textbf{(E) } 96$
2002 AMC 8, 10
$\textbf{Juan's Old Stamping Grounds}$
Juan organizes the stamps in his collection by country and by the decade in which they were issued. The prices he paid for them at a stamp shop were: Brazil and France, 6 cents each, Peru 4 cents each, and Spain 5 cents each. (Brazil and Peru are South American countries and France and Spain are in Europe.)
[asy]
/* AMC8 2002 #8, 9, 10 Problem */
size(3inch, 1.5inch);
for ( int y = 0; y <= 5; ++y )
{
draw((0,y)--(18,y));
}
draw((0,0)--(0,5));
draw((6,0)--(6,5));
draw((9,0)--(9,5));
draw((12,0)--(12,5));
draw((15,0)--(15,5));
draw((18,0)--(18,5));
draw(scale(0.8)*"50s", (7.5,4.5));
draw(scale(0.8)*"4", (7.5,3.5));
draw(scale(0.8)*"8", (7.5,2.5));
draw(scale(0.8)*"6", (7.5,1.5));
draw(scale(0.8)*"3", (7.5,0.5));
draw(scale(0.8)*"60s", (10.5,4.5));
draw(scale(0.8)*"7", (10.5,3.5));
draw(scale(0.8)*"4", (10.5,2.5));
draw(scale(0.8)*"4", (10.5,1.5));
draw(scale(0.8)*"9", (10.5,0.5));
draw(scale(0.8)*"70s", (13.5,4.5));
draw(scale(0.8)*"12", (13.5,3.5));
draw(scale(0.8)*"12", (13.5,2.5));
draw(scale(0.8)*"6", (13.5,1.5));
draw(scale(0.8)*"13", (13.5,0.5));
draw(scale(0.8)*"80s", (16.5,4.5));
draw(scale(0.8)*"8", (16.5,3.5));
draw(scale(0.8)*"15", (16.5,2.5));
draw(scale(0.8)*"10", (16.5,1.5));
draw(scale(0.8)*"9", (16.5,0.5));
label(scale(0.8)*"Country", (3,4.5));
label(scale(0.8)*"Brazil", (3,3.5));
label(scale(0.8)*"France", (3,2.5));
label(scale(0.8)*"Peru", (3,1.5));
label(scale(0.8)*"Spain", (3,0.5));
label(scale(0.9)*"Juan's Stamp Collection", (9,0), S);
label(scale(0.9)*"Number of Stamps by Decade", (9,5), N);
[/asy]
The average price of his '70s stamps is closest to
$\text{(A)}\ 3.5 \text{ cents} \qquad \text{(B)}\ 4 \text{ cents} \qquad \text{(C)}\ 4.5 \text{ cents} \qquad \text{(D)}\ 5 \text{ cents} \qquad \text{(E)}\ 5.5 \text{ cents}$
2023 AMC 8, 6
The digits $2$, $0$, $2$, and $3$ are placed in the expression below, one digit per box. What is the maximum possible value of the expression?
[asy]
// Diagram by TheMathGuyd. I can compress this later
size(5cm);
real w=2.2;
pair O,I,J;
O=(0,0);I=(1,0);J=(0,1);
path bsqb = O--I;
path bsqr = I--I+J;
path bsqt = I+J--J;
path bsql = J--O;
path lsqb = shift((1.2,0.75))*scale(0.5)*bsqb;
path lsqr = shift((1.2,0.75))*scale(0.5)*bsqr;
path lsqt = shift((1.2,0.75))*scale(0.5)*bsqt;
path lsql = shift((1.2,0.75))*scale(0.5)*bsql;
draw(bsqb,dashed);
draw(bsqr,dashed);
draw(bsqt,dashed);
draw(bsql,dashed);
draw(lsqb,dashed);
draw(lsqr,dashed);
draw(lsqt,dashed);
draw(lsql,dashed);
label(scale(3)*"$\times$",(w,1/3));
draw(shift(1.3w,0)*bsqb,dashed);
draw(shift(1.3w,0)*bsqr,dashed);
draw(shift(1.3w,0)*bsqt,dashed);
draw(shift(1.3w,0)*bsql,dashed);
draw(shift(1.3w,0)*lsqb,dashed);
draw(shift(1.3w,0)*lsqr,dashed);
draw(shift(1.3w,0)*lsqt,dashed);
draw(shift(1.3w,0)*lsql,dashed);
[/asy]$\textbf{(A) } 0\qquad\textbf{(B) } 8\qquad\textbf{(C) } 9\qquad\textbf{(D) } 16\qquad\textbf{(E) } 18$
2014 AMC 8, 25
A straight one-mile stretch of highway, $40$ feet wide, is closed. Robert rides his bike on a path composed of semicircles as shown. If he rides at $5$ miles per hour, how many hours will it take to cover the one-mile stretch?
Note: $1$ mile= $5280$ feet
[asy]size(10cm); pathpen=black; pointpen=black;
D(arc((-2,0),1,300,360));
D(arc((0,0),1,0,180));
D(arc((2,0),1,180,360));
D(arc((4,0),1,0,180));
D(arc((6,0),1,180,240));
D((-1.5,1)--(5.5,1));
D((-1.5,0)--(5.5,0),dashed);
D((-1.5,-1)--(5.5,-1));
[/asy]
$\textbf{(A) }\frac{\pi}{11}\qquad\textbf{(B) }\frac{\pi}{10}\qquad\textbf{(C) }\frac{\pi}{5}\qquad\textbf{(D) }\frac{2\pi}{5}\qquad \textbf{(E) }\frac{2\pi}{3}$
2020 AMC 8 -, 2
Four friends do yardwork for their neighbors over the weekend, earning $\$15$, $\$20$, $\$25$, and $\$40$, respectively. They decide to split their earnings equally among themselves. In total how much will the friend who earned $\$40$ give to the others?
$\textbf{(A)}\ \$5 \qquad \textbf{(B)}\ \$10 \qquad \textbf{(C)}\ \$15\qquad \textbf{(D)}\ \$20 \qquad \textbf{(E)}\ \$25$
2003 AMC 8, 16
Ali, Bonnie, Carlo, and Dianna are going to drive together to a nearby theme park. The car they are using has $4$ seats: $1$ Driver seat, $1$ front passenger seat, and $2$ back passenger seat. Bonnie and Carlo are the only ones who know how to drive the car. How many possible seating arrangements are there?
$\textbf{(A)}\ 2 \qquad
\textbf{(B)}\ 4 \qquad
\textbf{(C)}\ 6 \qquad
\textbf{(D)}\ 12 \qquad
\textbf{(E)}\ 24$
2022 AMC 8 -, 10
One sunny day, Ling decided to take a hike in the mountains. She left her house at $8 \, \textsc{am}$, drove at a constant speed of $45$ miles per hour, and arrived at the hiking trail at $10 \, \textsc{am}$. After hiking for $3$ hours, Ling drove home at a constant speed of $60$ miles per hour. Which of the following graphs best illustrates the distance between Ling’s car and her house over the course of her trip?
[asy]
unitsize(12);
usepackage("mathptmx");
defaultpen(fontsize(8)+linewidth(.7));
int mod12(int i) {if (i<13) {return i;} else {return i-12;}}
void drawgraph(pair sh,string lab) {
for (int i=0;i<11;++i) {
for (int j=0;j<6;++j) {
draw(shift(sh+(i,j))*unitsquare,mediumgray);
}
}
draw(shift(sh)*((-1,0)--(11,0)),EndArrow(angle=20,size=8));
draw(shift(sh)*((0,-1)--(0,6)),EndArrow(angle=20,size=8));
for (int i=1;i<10;++i) {
draw(shift(sh)*((i,-.2)--(i,.2)));
}
label("8\tiny{\textsc{am}}",sh+(1,-.2),S);
for (int i=2;i<9;++i) {
label(string(mod12(i+7)),sh+(i,-.2),S);
}
label("4\tiny{\textsc{pm}}",sh+(9,-.2),S);
for (int i=1;i<6;++i) {
label(string(30*i),sh+(0,i),2*W);
}
draw(rotate(90)*"Distance (miles)",sh+(-2.1,3),fontsize(10));
label("$\textbf{("+lab+")}$",sh+(-2.1,6.8),fontsize(10));
}
drawgraph((0,0),"A");
drawgraph((15,0),"B");
drawgraph((0,-10),"C");
drawgraph((15,-10),"D");
drawgraph((0,-20),"E");
dotfactor=6;
draw((1,0)--(3,3)--(6,3)--(8,0),linewidth(.9));
dot((1,0)^^(3,3)^^(6,3)^^(8,0));
pair sh = (15,0);
draw(shift(sh)*((1,0)--(3,1.5)--(6,1.5)--(8,0)),linewidth(.9));
dot(sh+(1,0)^^sh+(3,1.5)^^sh+(6,1.5)^^sh+(8,0));
pair sh = (0,-10);
draw(shift(sh)*((1,0)--(3,1.5)--(6,1.5)--(7.5,0)),linewidth(.9));
dot(sh+(1,0)^^sh+(3,1.5)^^sh+(6,1.5)^^sh+(7.5,0));
pair sh = (15,-10);
draw(shift(sh)*((1,0)--(3,4)--(6,4)--(9.3,0)),linewidth(.9));
dot(sh+(1,0)^^sh+(3,4)^^sh+(6,4)^^sh+(9.3,0));
pair sh = (0,-20);
draw(shift(sh)*((1,0)--(3,3)--(6,3)--(7.5,0)),linewidth(.9));
dot(sh+(1,0)^^sh+(3,3)^^sh+(6,3)^^sh+(7.5,0));
[/asy]
2022 AMC 8 -, 9
A cup of boiling water ($212^{\circ}\text{F}$) is placed to cool in a room whose temperature remains constant at $68^{\circ}\text{F}$. Suppose the difference between the water temperature and the room temperature is halved every $5$ minutes. What is the water temperature, in degrees Fahrenheit, after $15$ minutes?
$\textbf{(A)} ~77\qquad\textbf{(B)} ~86\qquad\textbf{(C)} ~92\qquad\textbf{(D)} ~98\qquad\textbf{(E)} ~104\qquad$
1999 AMC 8, 19
Problems 17, 18, and 19 refer to the following:
At Central Middle School the 108 students who take the AMC8 meet in the evening to talk about problems and eat an average of two cookies apiece. Walter and Gretel are baking Bonnie's Best Bar Cookies this year. Their recipe, which makes a pan of 15 cookies, lists this items: 1.5 cups flour, 2 eggs, 3 tablespoons butter, 3/4 cups sugar, and 1 package of chocolate drops. They will make only full recipes, not partial recipes.
The drummer gets sick. The concert is cancelled. Walter and Gretel must make enough pans of cookies to supply 216 cookies. There are 8 tablespoons in a stick of butter. How many sticks of butter will be needed? (Some butter may be left over, of course.)
$ \text{(A)}\ 5\qquad\text{(B)}\ 6\qquad\text{(C)}\ 7\qquad\text{(D)}\ 8\qquad\text{(E)}\ 9 $
2024 AMC 8 -, 22
A roll of tape is $4$ inches in diameter and is wrapped around a ring that is $2$ inches in diameter. A cross section of the tape is shown in the figure below. The tape is $0.015$ inches thick. If the tape is completely unrolled, approximately how long would it be? Round your answer to the nearest $100$ inches.
[asy]
/* AMC8 P22 2024, revised by Teacher David */
size(120);
pair o = (0,0);
real r1 = 1;
real r2 = 2;
filldraw(circle(o, r2), mediumgray, linewidth(1pt));
filldraw(circle(o, r1), white, linewidth(1pt));
draw((-2,-2.6)--(-2,-2.4));
draw((2,-2.6)--(2,-2.4));
draw((-2,-2.5)--(2,-2.5), L=Label("4 in."));
draw((-1,0)--(1,0), L=Label("2 in.", align=(0,1)), arrow=Arrows());
draw((2,0)--(2,-1.3), linewidth(1pt));
[/asy]
$\textbf{(A) } 300\qquad\textbf{(B) } 600\qquad\textbf{(C) } 1200\qquad\textbf{(D) } 1500\qquad\textbf{(E) } 1800$
2006 AMC 8, 2
On the AMC 8 contest Billy answers 13 questions correctly, answers 7 questions incorrectly and doesn't answer the last 5. What is his score?
$ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 13 \qquad \textbf{(D)}\ 19 \qquad \textbf{(E)}\ 26$
2024 AMC 8 -, 15
Let the letters $F$, $L$, $Y$, $B$, $U$, $G$ represent different digits. Suppose $\underline{F}\underline{L}\underline{Y}\underline{F}\underline{L}\underline{Y}$ is the largest number that satisfies the equation $$8 \cdot \underline{F}\underline{L}\underline{Y}\underline{F}\underline{L}\underline{Y} = \underline{B}\underline{U}\underline{G}\underline{B}\underline{U}\underline{G}.$$ What is the value of $\underline{F}\underline{L}\underline{Y} + \underline{B}\underline{U}\underline{G}$?
$\textbf{(A) } 1089\qquad\textbf{(B) } 1098\qquad\textbf{(C) } 1107\qquad\textbf{(D) } 1116\qquad\textbf{(E) } 1125$