Found problems: 260
2025 AMC 8, 3
Buffalo Shuffle-o is a card game in which all the cards are distributed evenly among all players at the start of the game. When Annika and $3$ of her friends play Buffalo Shuffle-o, each player is dealt $15$ cards. Suppose $2$ more friends join the next game. How many cards will be dealt to each player?
$\textbf{(A) } 8\qquad\textbf{(B) } 9\qquad\textbf{(C) } 10\qquad\textbf{(D) } 11\qquad\textbf{(E) } 12$
ngl easily silliable
2005 AMC 8, 14
The Little Twelve Basketball Conference has two divisions, with six teams in each division. Each team plays each of the other teams in its own division twice and every team in the other division once. How many conference games are scheduled?
$ \textbf{(A)}\ 80\qquad\textbf{(B)}\ 96\qquad\textbf{(C)}\ 100\qquad\textbf{(D)}\ 108\qquad\textbf{(E)}\ 192 $
2018 AMC 8, 15
In the diagram below, a diameter of each of the two smaller circles is a radius of the larger circle. If the two smaller circles have a combined area of $1$ square unit, then what is the area of the shaded region, in square units?
[asy]
size(4cm);
filldraw(scale(2)*unitcircle,gray,black);
filldraw(shift(-1,0)*unitcircle,white,black);
filldraw(shift(1,0)*unitcircle,white,black);
[/asy]
$\textbf{(A) } \frac{1}{4} \qquad \textbf{(B) } \frac{1}{3} \qquad \textbf{(C) } \frac{1}{2} \qquad \textbf{(D) } 1 \qquad \textbf{(E) } \frac{\pi}{2}$
2024 AMC 8 -, 5
Aaliyah rolls two standard 6-sided dice. She notices that the product of the two numbers rolled is a multiple of 6. Which of the following integers [i]cannot[/i] be the sum of the two numbers?
$\textbf{(A) } 5\qquad\textbf{(B) } 6\qquad\textbf{(C) } 7\qquad\textbf{(D) } 8\qquad\textbf{(E) } 9$
2020 AMC 8 -, 12
For a positive integer $n,$ the factorial notation $n!$ represents the product of the integers from $n$ to $1.$ (For example, $6! = 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1.$) What value of $N$ satisfies the following equation?
$$5! \cdot 9! = 12 \cdot N!$$
$\textbf{(A) }10 \qquad \textbf{(B) }11 \qquad \textbf{(C) }12 \qquad \textbf{(D) }13 \qquad \textbf{(E) }14$
2022 AMC 8 -, 12
The arrows on the two spinners shown below are spun. Let the number $N$ equal 10 times the number on Spinner $A$, added to the number on Spinner $B$. What is the probability that $N$ is a perfect square number?
$\textbf{(A)} ~\dfrac{1}{16}\qquad\textbf{(B)} ~\dfrac{1}{8}\qquad\textbf{(C)} ~\dfrac{1}{4}\qquad\textbf{(D)} ~\dfrac{3}{8}\qquad\textbf{(E)} ~\dfrac{1}{2}\qquad$
[center]
[asy]
//diagram by pog give me 1 billion dollars for this
size(6cm);
usepackage("mathptmx");
filldraw(arc((0,0), r=4, angle1=0, angle2=90)--(0,0)--cycle,mediumgray*0.5+gray*0.5);
filldraw(arc((0,0), r=4, angle1=90, angle2=180)--(0,0)--cycle,lightgray);
filldraw(arc((0,0), r=4, angle1=180, angle2=270)--(0,0)--cycle,mediumgray);
filldraw(arc((0,0), r=4, angle1=270, angle2=360)--(0,0)--cycle,lightgray*0.5+mediumgray*0.5);
label("$5$", (-1.5,1.7));
label("$6$", (1.5,1.7));
label("$7$", (1.5,-1.7));
label("$8$", (-1.5,-1.7));
label("Spinner A", (0, -5.5));
filldraw(arc((12,0), r=4, angle1=0, angle2=90)--(12,0)--cycle,mediumgray*0.5+gray*0.5);
filldraw(arc((12,0), r=4, angle1=90, angle2=180)--(12,0)--cycle,lightgray);
filldraw(arc((12,0), r=4, angle1=180, angle2=270)--(12,0)--cycle,mediumgray);
filldraw(arc((12,0), r=4, angle1=270, angle2=360)--(12,0)--cycle,lightgray*0.5+mediumgray*0.5);
label("$1$", (10.5,1.7));
label("$2$", (13.5,1.7));
label("$3$", (13.5,-1.7));
label("$4$", (10.5,-1.7));
label("Spinner B", (12, -5.5));
[/asy]
[/center]
2016 AMC 8, 22
Rectangle $DEFA$ below is a $3 \times 4$ rectangle with $DC=CB=BA$. The area of the "bat wings" is
[asy]
size(180);
defaultpen(fontsize(11pt));
draw((0,0)--(3,0)--(3,4)--(0,4)--(0,0)--(2,4)--(3,0));
draw((3,0)--(1,4)--(0,0));
fill((0,0)--(1,4)--(1.5,3)--cycle, black);
fill((3,0)--(2,4)--(1.5,3)--cycle, black);
label("$D$",(0,4),NW);
label("$C$",(1,4),N);
label("$B$",(2,4),N);
label("$A$",(3,4),NE);
label("$E$",(0,0),SW);
label("$F$",(3,0),SE);[/asy]
$\textbf{(A) }2\qquad\textbf{(B) }2 \frac{1}{2}\qquad\textbf{(C) }3\qquad\textbf{(D) }3 \frac{1}{2}\qquad \textbf{(E) }5$
2019 AMC 8, 7
Shauna takes $5$ tests, each worth a maximum of a $100$ points. Her scores on the first three tests were $76$, $94$, and $87$. In order to average an $81$ on all five tests, what is the lowest score she could earn on one of the two tests?
$\textbf{(A) } 48 \qquad\textbf{(B) } 52 \qquad\textbf{(C) } 66 \qquad\textbf{(D) } 70 \qquad\textbf{(E) } 74$
2024 AMC 8 -, 25
A small airplane has $4$ rows of seats with $3$ seats in each row. Eight passengers have boarded the plane and are distributed randomly among the seats. A married couple is next to board. What is the probability there will be 2 adjacent seats in the same row for the couple?
2003 AMC 8, 17
The six children listed below are from two families of three siblings each. Each child has blue or brown eyes and black or blond hair. Children from the same family have at least one of these characteristics in common. Which two children are Jim's siblings?
\[ \begin{array}{c|c|c}\text{Child}&\text{Eye Color}&\text{Hair Color}\\ \hline \text{Benjamin}& \text{Blue} & \text{Black} \\ \hline \text{Jim} & \text{Brown} & \text{Blonde} \\ \hline \text{Nadeen} & \text{Brown} & \text{Black}\\ \hline \text{Austin}& \text{Blue} & \text{Blonde}\\ \hline \text{Tevyn} & \text{Blue} & \text{Black} \\ \hline \text{Sue} & \text{Blue} & \text{Blonde} \\ \hline \end{array} \]
$\textbf{(A)}\ \text{Nadeen and Austin} \qquad
\textbf{(B)}\ \text{Benjamin and Sue}\qquad
\textbf{(C)}\ \text{Benjamin and Austin}\qquad$
$\textbf{(D)}\ \text{Nadeen and Tevyn} \qquad
\textbf{(E)}\ \text{Austin and Sue} $
2023 AMC 8, 16
The letters $P$, $Q$, and $R$ are entered in a $20\times 20$ grid according to the pattern shown below. How many $P$s, $Q$s, and $R$s will appear in the completed table?
[asy]
usepackage("mathdots");
size(5cm);
draw((0,0)--(6,0),linewidth(1.5)+mediumgray);
draw((0,1)--(6,1),linewidth(1.5)+mediumgray);
draw((0,2)--(6,2),linewidth(1.5)+mediumgray);
draw((0,3)--(6,3),linewidth(1.5)+mediumgray);
draw((0,4)--(6,4),linewidth(1.5)+mediumgray);
draw((0,5)--(6,5),linewidth(1.5)+mediumgray);
draw((0,0)--(0,6),linewidth(1.5)+mediumgray);
draw((1,0)--(1,6),linewidth(1.5)+mediumgray);
draw((2,0)--(2,6),linewidth(1.5)+mediumgray);
draw((3,0)--(3,6),linewidth(1.5)+mediumgray);
draw((4,0)--(4,6),linewidth(1.5)+mediumgray);
draw((5,0)--(5,6),linewidth(1.5)+mediumgray);
label(scale(.9)*"\textsf{P}", (.5,.5));
label(scale(.9)*"\textsf{Q}", (.5,1.5));
label(scale(.9)*"\textsf{R}", (.5,2.5));
label(scale(.9)*"\textsf{P}", (.5,3.5));
label(scale(.9)*"\textsf{Q}", (.5,4.5));
label("$\vdots$", (.5,5.6));
label(scale(.9)*"\textsf{Q}", (1.5,.5));
label(scale(.9)*"\textsf{R}", (1.5,1.5));
label(scale(.9)*"\textsf{P}", (1.5,2.5));
label(scale(.9)*"\textsf{Q}", (1.5,3.5));
label(scale(.9)*"\textsf{R}", (1.5,4.5));
label("$\vdots$", (1.5,5.6));
label(scale(.9)*"\textsf{R}", (2.5,.5));
label(scale(.9)*"\textsf{P}", (2.5,1.5));
label(scale(.9)*"\textsf{Q}", (2.5,2.5));
label(scale(.9)*"\textsf{R}", (2.5,3.5));
label(scale(.9)*"\textsf{P}", (2.5,4.5));
label("$\vdots$", (2.5,5.6));
label(scale(.9)*"\textsf{P}", (3.5,.5));
label(scale(.9)*"\textsf{Q}", (3.5,1.5));
label(scale(.9)*"\textsf{R}", (3.5,2.5));
label(scale(.9)*"\textsf{P}", (3.5,3.5));
label(scale(.9)*"\textsf{Q}", (3.5,4.5));
label("$\vdots$", (3.5,5.6));
label(scale(.9)*"\textsf{Q}", (4.5,.5));
label(scale(.9)*"\textsf{R}", (4.5,1.5));
label(scale(.9)*"\textsf{P}", (4.5,2.5));
label(scale(.9)*"\textsf{Q}", (4.5,3.5));
label(scale(.9)*"\textsf{R}", (4.5,4.5));
label("$\vdots$", (4.5,5.6));
label(scale(.9)*"$\dots$", (5.5,.5));
label(scale(.9)*"$\dots$", (5.5,1.5));
label(scale(.9)*"$\dots$", (5.5,2.5));
label(scale(.9)*"$\dots$", (5.5,3.5));
label(scale(.9)*"$\dots$", (5.5,4.5));
label(scale(.9)*"$\iddots$", (5.5,5.6));
[/asy]
$\textbf{(A)}~132~\text{Ps}, 134~\text{Qs}, 134~\text{Rs}\qquad\textbf{(B)}~133~\text{Ps}, 133~\text{Qs}, 134~\text{Rs}\qquad\textbf{(C)}~133~\text{Ps}, 134~\text{Qs}, 133~\text{Rs}$\\
$\textbf{(D)}~134~\text{Ps}, 132~\text{Qs}, 134~\text{Rs}\qquad\textbf{(E)}~134~\text{Ps}, 133~\text{Qs}, 133~\text{Rs}\qquad$
2016 AMC 8, 4
When Cheenu was a boy he could run $15$ miles in $3$ hours and $30$ minutes. As an old man he can now walk $10$ miles in $4$ hours. How many minutes longer does it take for him to walk a mile now compared to when he was a boy?
$\textbf{(A) }6\qquad\textbf{(B) }10\qquad\textbf{(C) }15\qquad\textbf{(D) }18\qquad \textbf{(E) }30$
2016 AMC 8, 9
What is the sum of the distinct prime integer divisors of $2016$?
$\textbf{(A) }9\qquad\textbf{(B) }12\qquad\textbf{(C) }16\qquad\textbf{(D) }49\qquad \textbf{(E) }63$
2005 AMC 8, 18
How many three-digit numbers are divisible by 13?
$ \textbf{(A)}\ 7\qquad\textbf{(B)}\ 67\qquad\textbf{(C)}\ 69\qquad\textbf{(D)}\ 76\qquad\textbf{(E)}\ 77$
2015 AMC 8, 22
On June 1, a group of students is standing in rows, with 15 students in each row. On June 2, the same group is standing with all of the students in one long row. On June 3, the same group is standing with just one student in each row. On June 4, the same group is standing with 6 students in each row. This process continues through June 12 with a different number of students per row each day. However, on June 13, they cannot find a new way of organizing the students. What is the smallest possible number of students in the group?
$
\textbf{(A) } 21 \qquad
\textbf{(B) } 30 \qquad
\textbf{(C) } 60 \qquad
\textbf{(D) } 90 \qquad
\textbf{(E) } 1080
$
2010 AMC 8, 4
What is the sum of the mean, median, and mode of the numbers, $2,3,0,3,1,4,0,3$?
$ \textbf{(A)}\ 6.5 \qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 7.5\qquad\textbf{(D)}\ 8.5\qquad\textbf{(E)}\ 9 $
1999 AMC 8, 17
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.
Walter can buy eggs by the half-dozen. How many half-dozens should he buy to make enough cookies? (Some eggs and some cookies may be left over.)
$ \text{(A)}\ 1\qquad\text{(B)}\ 2\qquad\text{(C)}\ 5\qquad\text{(D)}\ 7\qquad\text{(E)}\ 15 $
2018 AMC 8, 2
What is the value of the product$$\left(1+\frac{1}{1}\right)\cdot\left(1+\frac{1}{2}\right)\cdot\left(1+\frac{1}{3}\right)\cdot\left(1+\frac{1}{4}\right)\cdot\left(1+\frac{1}{5}\right)\cdot\left(1+\frac{1}{6}\right)?$$
$\textbf{(A) }\frac{7}{6}\qquad\textbf{(B) }\frac{4}{3}\qquad\textbf{(C) }\frac{7}{2}\qquad\textbf{(D) }7\qquad\textbf{(E) }8$
2025 AMC 8, 13
Each of the even numbers $2, 4, 6, \ldots, 50$ is divided by $7$. The remainders are recorded. Which histogram displays the number of times each remainder appears?
[img]https://i.imgur.com/f1oQExa.png[/img]
2017 AMC 8, 9
All of Marcy's marbles are blue, red, green, or yellow. One third of her marbles are blue, one fourth of them are red, and six of them are green. What is the smallest number of yellow marbles that Marcy could have?
$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }5$
2017 AMC 8, 14
Chloe and Zoe are both students in Ms. Demeanor's math class. Last night they each solved half of the problems in their homework assignment alone and then solved the other half together. Chloe had correct answers to only $80\%$ of the problems she solved alone, but overall $88\%$ of her answers were correct. Zoe had correct answers to $90\%$ of the problems she solved alone. What was Zoe's overall percentage of correct answers?
$\textbf{(A) }89\qquad\textbf{(B) }92\qquad\textbf{(C) }93\qquad\textbf{(D) }96\qquad\textbf{(E) }98$
2017 AMC 8, 13
Peter, Emma, and Kyler played chess with each other. Peter won 4 games and lost 2 games. Emma won 3 games and lost 3 games. If Kyler lost 3 games, how many games did he win?
$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }4$
2015 AMC 8, 19
A triangle with vertices as $A=(1,3)$, $B=(5,1)$, and $C=(4,4)$ is plotted on a $6\times5$ grid. What fraction of the grid is covered by the triangle?
$\textbf{(A) }\frac{1}{6} \qquad \textbf{(B) }\frac{1}{5} \qquad \textbf{(C) }\frac{1}{4} \qquad \textbf{(D) }\frac{1}{3} \qquad \textbf{(E) }\frac{1}{2}$
[asy]
draw((1,0)--(1,5),linewidth(.5));
draw((2,0)--(2,5),linewidth(.5));
draw((3,0)--(3,5),linewidth(.5));
draw((4,0)--(4,5),linewidth(.5));
draw((5,0)--(5,5),linewidth(.5));
draw((6,0)--(6,5),linewidth(.5));
draw((0,1)--(6,1),linewidth(.5));
draw((0,2)--(6,2),linewidth(.5));
draw((0,3)--(6,3),linewidth(.5));
draw((0,4)--(6,4),linewidth(.5));
draw((0,5)--(6,5),linewidth(.5));
draw((0,0)--(0,6),EndArrow);
draw((0,0)--(7,0),EndArrow);
draw((1,3)--(4,4)--(5,1)--cycle);
label("$y$",(0,6),W); label("$x$",(7,0),S);
label("$A$",(1,3),dir(230)); label("$B$",(5,1),SE); label("$C$",(4,4),dir(50));
[/asy]
2022 AMC 8 -, 13
How many positive integers can fill the blank in the sentence below?
"One positive integer is $\underline{~~~~~}$ more than twice another, and the sum of the two numbers is 28."
$\textbf{(A)} ~6\qquad\textbf{(B)} ~7\qquad\textbf{(C)} ~8\qquad\textbf{(D)} ~9\qquad\textbf{(E)} ~10\qquad$
2020 AMC 8 -, 16
Each of the points $A$, $B$, $C$, $D$, $E$, and $F$ in the figure below represent a different digit from 1 to 6. Each of the five lines shown passes through some of these points. The digits along the line each are added to produce 5 sums, one for each line. The total of the sums is $47$. What is the digit represented by $B$?
[asy]
size(200);
dotfactor = 10;
pair p1 = (-28,0);
pair p2 = (-111,213);
draw(p1--p2,linewidth(1));
pair p3 = (-160,0);
pair p4 = (-244,213);
draw(p3--p4,linewidth(1));
pair p5 = (-316,0);
pair p6 = (-67,213);
draw(p5--p6,linewidth(1));
pair p7 = (0, 68);
pair p8 = (-350,10);
draw(p7--p8,linewidth(1));
pair p9 = (0, 150);
pair p10 = (-350, 62);
draw(p9--p10,linewidth(1));
pair A = intersectionpoint(p1--p2, p5--p6);
dot("$A$", A, 2*W);
pair B = intersectionpoint(p5--p6, p3--p4);
dot("$B$", B, 2*WNW);
pair C = intersectionpoint(p7--p8, p5--p6);
dot("$C$", C, 1.5*NW);
pair D = intersectionpoint(p3--p4, p7--p8);
dot("$D$", D, 2*NNE);
pair EE = intersectionpoint(p1--p2, p7--p8);
dot("$E$", EE, 2*NNE);
pair F = intersectionpoint(p1--p2, p9--p10);
dot("$F$", F, 2*NNE);
[/asy]
$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$