Found problems: 260
2004 AMC 8, 16
Two $600$ ml pitchers contain orange juice. One pitcher is $\frac{1}{3}$ full and the other pitcher is $\frac{2}{5}$ full. Water is added to fill each pitcher completely, then both pitchers are poured into one large container. What fraction of the mixture in the large container is orange juice?
$\textbf{(A)}\ \frac{1}{8}\qquad
\textbf{(B)}\ \frac{3}{16}\qquad
\textbf{(C)}\ \frac{11}{30}\qquad
\textbf{(D)}\ \frac{11}{19}\qquad
\textbf{(E)}\ \frac{11}{15}$
2010 Contests, 1
At Euclid High School, the mathematics teachers are Mrs. Germain, Mr. Newton, and Mrs. Young. There are $11$ students in Mrs. Germain's class, 8 in Mr. Newton, and $9$ in Mrs. Young's class are taking the AMC $8$ this year. How many mathematics students at Euclid High School are taking the contest?
$ \textbf{(A)}\ 26 \qquad\textbf{(B)}\ 27\qquad\textbf{(C)}\ 28\qquad\textbf{(D)}\ 29\qquad\textbf{(E)}\ 30 $
2018 AMC 8, 21
How many positive three-digit integers have a remainder of 2 when divided by 6, a remainder of 5 when divided by 9, and a remainder of 7 when divided by 11?
$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad \textbf{(E) }5$
2022 AMC 8 -, 21
Steph scored $15$ baskets out of $20$ attempts in the first half of a game, and $10$ baskets out of $10$ attempts in the second half. Candace took $12$ attempts in the first half and $18$ attempts in the second. In each half, Steph scored a higher percentage of baskets than Candace. Surprisingly they ended with the same overall percentage of baskets scored. How many more baskets did Candace score in the second half than in the first?
[asy]
size(7cm);
draw((-8,27)--(72,27));
draw((16,0)--(16,35));
draw((40,0)--(40,35));
label("12", (28,3));
draw((25,6.5)--(25,12)--(31,12)--(31,6.5)--cycle);
draw((25,5.5)--(31,5.5));
label("18", (56,3));
draw((53,6.5)--(53,12)--(59,12)--(59,6.5)--cycle);
draw((53,5.5)--(59,5.5));
draw((53,5.5)--(59,5.5));
label("20", (28,18));
label("15", (28,24));
draw((25,21)--(31,21));
label("10", (56,18));
label("10", (56,24));
draw((53,21)--(59,21));
label("First Half", (28,31));
label("Second Half", (56,31));
label("Candace", (2.35,6));
label("Steph", (0,21));
[/asy]
$\textbf{(A)} ~7\qquad\textbf{(B)} ~8\qquad\textbf{(C)} ~9\qquad\textbf{(D)} ~10\qquad\textbf{(E)} ~11$
2023 AMC 8, 8
Lola, Lolo, Tiya, and Tiyo participated in a ping pong tournament. Each player competed against each of the other three players exactly twice. Shown below are the win-loss records for the players. The numbers $1$ and $0$ represent a win or loss, respectively. For example, Lola won five matches and lost the fourth match. What was Tiyo’s win-loss record?
\[\begin{tabular}{c | c} Player & Result \\ \hline Lola & \texttt{111011}\\ Lolo & \texttt{101010}\\ Tiya & \texttt{010100}\\ Tiyo & \texttt{??????} \end{tabular}\]
$\textbf{(A)}\ \texttt{000101} \qquad \textbf{(B)}\ \texttt{001001} \qquad \textbf{(C)}\ \texttt{010000} \qquad \textbf{(D)}\ \texttt{010101} \qquad \textbf{(E)}\ \texttt{011000}$
2015 AMC 8, 21
In the given figure hexagon $ABCDEF$ is equiangular, $ABJI$ and $FEHG$ are squares with areas $18$ and $32$ respectively, $\triangle JBK$ is equilateral and $FE=BC$. What is the area of $\triangle KBC$?
$\textbf{(A) }6\sqrt{2}\qquad\textbf{(B) }9\qquad\textbf{(C) }12\qquad\textbf{(D) }9\sqrt{2}\qquad\textbf{(E) }32$
[asy]
draw((-4,6*sqrt(2))--(4,6*sqrt(2)));
draw((-4,-6*sqrt(2))--(4,-6*sqrt(2)));
draw((-8,0)--(-4,6*sqrt(2)));
draw((-8,0)--(-4,-6*sqrt(2)));
draw((4,6*sqrt(2))--(8,0));
draw((8,0)--(4,-6*sqrt(2)));
draw((-4,6*sqrt(2))--(4,6*sqrt(2))--(4,8+6*sqrt(2))--(-4,8+6*sqrt(2))--cycle);
draw((-8,0)--(-4,-6*sqrt(2))--(-4-6*sqrt(2),-4-6*sqrt(2))--(-8-6*sqrt(2),-4)--cycle);
label("$I$",(-4,8+6*sqrt(2)),dir(100)); label("$J$",(4,8+6*sqrt(2)),dir(80));
label("$A$",(-4,6*sqrt(2)),dir(280)); label("$B$",(4,6*sqrt(2)),dir(250));
label("$C$",(8,0),W); label("$D$",(4,-6*sqrt(2)),NW); label("$E$",(-4,-6*sqrt(2)),NE); label("$F$",(-8,0),E);
draw((4,8+6*sqrt(2))--(4,6*sqrt(2))--(4+4*sqrt(3),4+6*sqrt(2))--cycle);
label("$K$",(4+4*sqrt(3),4+6*sqrt(2)),E);
draw((4+4*sqrt(3),4+6*sqrt(2))--(8,0),dashed);
label("$H$",(-4-6*sqrt(2),-4-6*sqrt(2)),S);
label("$G$",(-8-6*sqrt(2),-4),W);
label("$32$",(-10,-8),N);
label("$18$",(0,6*sqrt(2)+2),N);
[/asy]
2023 AMC 8, 17
A [i]regular octahedron[/i] has eight equilateral triangle faces with four faces meeting at each vertex. Jun will make the regular octahedron shown on the right by folding the piece of paper shown on the left. Which numbered face will end up to the right of $Q$?
[asy]
// Note: This diagram was not made by me.
import graph;
// The Solid
// To save processing time, do not use three (dimensions)
// Project (roughly) to two
size(15cm);
pair Fr, Lf, Rt, Tp, Bt, Bk;
Lf=(0,0);
Rt=(12,1);
Fr=(7,-1);
Bk=(5,2);
Tp=(6,6.7);
Bt=(6,-5.2);
draw(Lf--Fr--Rt);
draw(Lf--Tp--Rt);
draw(Lf--Bt--Rt);
draw(Tp--Fr--Bt);
draw(Lf--Bk--Rt,dashed);
draw(Tp--Bk--Bt,dashed);
label(rotate(-8.13010235)*slant(0.1)*"$Q$", (4.2,1.6));
label(rotate(21.8014095)*slant(-0.2)*"$?$", (8.5,2.05));
pair g = (-8,0); // Define Gap transform
real a = 8;
draw(g+(-a/2,1)--g+(a/2,1), Arrow()); // Make arrow
// Time for the NET
pair DA,DB,DC,CD,O;
DA = (6.92820323028,0);
DB = (3.46410161514,6);
DC = (DA+DB)/3;
CD = conj(DC);
O=(0,0);
transform trf=shift(3g+(0,3));
path NET = O--(-2*DA)--(-2DB)--(-DB)--(2DA-DB)--DB--O--DA--(DA-DB)--O--(-DB)--(-DA)--(-DA-DB)--(-DB);
draw(trf*NET);
label("$7$",trf*DC);
label("$Q$",trf*DC+DA-DB);
label("$5$",trf*DC-DB);
label("$3$",trf*DC-DA-DB);
label("$6$",trf*CD);
label("$4$",trf*CD-DA);
label("$2$",trf*CD-DA-DB);
label("$1$",trf*CD-2DA);
[/asy]
$\textbf{(A)}~1\qquad\textbf{(B)}~2\qquad\textbf{(C)}~3\qquad\textbf{(D)}~4\qquad\textbf{(E)}~5\qquad$
2017 AMC 8, 6
If the degree measures of the angles of a triangle are in the ratio $3:3:4$, what is the degree measure of the largest angle of the triangle?
$\textbf{(A) }18\qquad\textbf{(B) }36\qquad\textbf{(C) }60\qquad\textbf{(D) }72\qquad\textbf{(E) }90$
2003 AMC 8, 9
$\textbf{Bake Sale}$
Four friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies di ffer, as shown.
$\circ$ Art's cookies are trapezoids:
[asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8));
draw(origin--(5,0)--(5,3)--(2,3)--cycle);
draw(rightanglemark((5,3), (5,0), origin));
label("5 in", (2.5,0), S);
label("3 in", (5,1.5), E);
label("3 in", (3.5,3), N);[/asy]
$\circ$ Roger's cookies are rectangles:
[asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8));
draw(origin--(4,0)--(4,2)--(0,2)--cycle);
draw(rightanglemark((4,2), (4,0), origin));
draw(rightanglemark((0,2), origin, (4,0)));
label("4 in", (2,0), S);
label("2 in", (4,1), E);[/asy]
$\circ$ Paul's cookies are parallelograms:
[asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8));
draw(origin--(3,0)--(2.5,2)--(-0.5,2)--cycle);
draw((2.5,2)--(2.5,0), dashed);
draw(rightanglemark((2.5,2),(2.5,0), origin));
label("3 in", (1.5,0), S);
label("2 in", (2.5,1), W);[/asy]
$\circ$ Trisha's cookies are triangles:
[asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8));
draw(origin--(3,0)--(3,4)--cycle);
draw(rightanglemark((3,4),(3,0), origin));
label("3 in", (1.5,0), S);
label("4 in", (3,2), E);[/asy]
Each friend uses the same amount of dough, and Art makes exactly 12 cookies. Art's cookies sell for 60 cents each. To earn the same amount from a single batch, how much should one of Roger's cookies cost in cents?
$ \textbf{(A)}\ 18\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 40\qquad\textbf{(D)}\ 75\qquad\textbf{(E)}\ 90$
2015 AMC 8, 23
Tom has twelve slips of paper which he wants to put into five cups labeled $A$, $B$, $C$, $D$, $E$. He wants the sum of the numbers on the slips in each cup to be an integer. Furthermore, he wants the five integers to be consecutive and increasing from $A$ to $E$. The numbers on the papers are 2, 2, 2, 2.5, 2.5, 3, 3, 3, 3, 3.5, 4, and 4.5. If a slip with 2 goes into cup $E$ and a slip with 3 goes into cup $B$, then the slip with 3.5 must go into what cup?
$
\textbf{(A) } A \qquad
\textbf{(B) } B \qquad
\textbf{(C) } C \qquad
\textbf{(D) } D \qquad
\textbf{(E) } E
$
1972 AMC 12/AHSME, 5
From among $2^{1/2},$ $3^{1/3},$ $8^{1/8},$ $9^{1/9}$ those which have the greatest and the next to the greatest values, in that order, are
\[ \begin{array}{rlrlrlrl} \hbox {(A)}& 3^{1/3},\ 2^{1/2} \quad & \hbox {(B)}& 3^{1/3},\ 8^{1/8} \quad & \hbox {(C)}& 3^{1/3},\ 9^{1/9} \quad & \hbox {(D)}& 8^{1/8},\ 9^{1/9} \\ \hbox {(E)}& \multicolumn{3}{l}{\hbox{None of these}} \end{array} \]
2020 AMC 8 -, 9
Akash's birthday cake is in the form of a $4 \times 4 \times 4$ inch cube. The cake has icing on the top and the four side faces, and no icing on the bottom. Suppose the cake is cut into $64$ smaller cubes, each measuring $1 \times 1 \times 1$ inch, as shown below. How many of the small pieces will have icing on exactly two sides?
[asy]
/*
Created by SirCalcsALot and sonone
Code modfied from https://artofproblemsolving.com/community/c3114h2152994_the_old__aops_logo_with_asymptote
*/
import three;
currentprojection=orthographic(1.75,7,2);
//++++ edit colors, names are self-explainatory ++++
//pen top=rgb(27/255, 135/255, 212/255);
//pen right=rgb(254/255,245/255,182/255);
//pen left=rgb(153/255,200/255,99/255);
pen top = rgb(170/255, 170/255, 170/255);
pen left = rgb(81/255, 81/255, 81/255);
pen right = rgb(165/255, 165/255, 165/255);
pen edges=black;
int max_side = 4;
//+++++++++++++++++++++++++++++++++++++++
path3 leftface=(1,0,0)--(1,1,0)--(1,1,1)--(1,0,1)--cycle;
path3 rightface=(0,1,0)--(1,1,0)--(1,1,1)--(0,1,1)--cycle;
path3 topface=(0,0,1)--(1,0,1)--(1,1,1)--(0,1,1)--cycle;
for(int i=0; i<max_side; ++i){
for(int j=0; j<max_side; ++j){
draw(shift(i,j,-1)*surface(topface),top);
draw(shift(i,j,-1)*topface,edges);
draw(shift(i,-1,j)*surface(rightface),right);
draw(shift(i,-1,j)*rightface,edges);
draw(shift(-1,j,i)*surface(leftface),left);
draw(shift(-1,j,i)*leftface,edges);
}
}
picture CUBE;
draw(CUBE,surface(leftface),left,nolight);
draw(CUBE,surface(rightface),right,nolight);
draw(CUBE,surface(topface),top,nolight);
draw(CUBE,topface,edges);
draw(CUBE,leftface,edges);
draw(CUBE,rightface,edges);
// begin made by SirCalcsALot
int[][] heights = {{4,4,4,4},{4,4,4,4},{4,4,4,4},{4,4,4,4}};
for (int i = 0; i < max_side; ++i) {
for (int j = 0; j < max_side; ++j) {
for (int k = 0; k < min(heights[i][j], max_side); ++k) {
add(shift(i,j,k)*CUBE);
}
}
}
[/asy]
$\textbf{(A)}\ 12\qquad~~\textbf{(B)}\ 16\qquad~~\textbf{(C)}\ 18\qquad~~\textbf{(D)}\ 20\qquad~~\textbf{(E)}\ 24$
2003 AMC 8, 22
The following figures are composed of squares and circles. Which figure has a shaded region with largest area?
[asy]/* AMC8 2003 #22 Problem */
size(3inch, 2inch);
unitsize(1cm);
pen outline = black+linewidth(1);
filldraw((0,0)--(2,0)--(2,2)--(0,2)--cycle, mediumgrey, outline);
filldraw(shift(3,0)*((0,0)--(2,0)--(2,2)--(0,2)--cycle), mediumgrey, outline);
filldraw(Circle((7,1), 1), mediumgrey, black+linewidth(1));
filldraw(Circle((1,1), 1), white, outline);
filldraw(Circle((3.5,.5), .5), white, outline);
filldraw(Circle((4.5,.5), .5), white, outline);
filldraw(Circle((3.5,1.5), .5), white, outline);
filldraw(Circle((4.5,1.5), .5), white, outline);
filldraw((6.3,.3)--(7.7,.3)--(7.7,1.7)--(6.3,1.7)--cycle, white, outline);
label("A", (1, 2), N);
label("B", (4, 2), N);
label("C", (7, 2), N);
draw((0,-.5)--(.5,-.5), BeginArrow);
draw((1.5, -.5)--(2, -.5), EndArrow);
label("2 cm", (1, -.5));
draw((3,-.5)--(3.5,-.5), BeginArrow);
draw((4.5, -.5)--(5, -.5), EndArrow);
label("2 cm", (4, -.5));
draw((6,-.5)--(6.5,-.5), BeginArrow);
draw((7.5, -.5)--(8, -.5), EndArrow);
label("2 cm", (7, -.5));
draw((6,1)--(6,-.5), linetype("4 4"));
draw((8,1)--(8,-.5), linetype("4 4"));[/asy]
$ \textbf{(A)}\ \text{A only}\qquad\textbf{(B)}\ \text{B only}\qquad\textbf{(C)}\ \text{C only}\qquad\textbf{(D)}\ \text{both A and B}\qquad\textbf{(E)}\ \text{all are equal}$
2018 AMC 8, 18
How many positive factors does $23,232$ have?
$\textbf{(A) }9\qquad\textbf{(B) }12\qquad\textbf{(C) }28\qquad\textbf{(D) }36\qquad\textbf{(E) }42$
2025 AMC 8, 14
A number N is inserted into the list 2, 6, 7, 7, 28. The mean is now twice as great as the median. What is N?
$\textbf{(A) } 7\qquad\textbf{(B) } 14\qquad\textbf{(C) } 20\qquad\textbf{(D) } 28\qquad\textbf{(E) } 34$
2017 AMC 8, 20
An integer between $1000$ and $9999$, inclusive, is chosen at random. What is the probability that it is an odd integer whose digits are all distinct?
$\textbf{(A) }\frac{14}{75}\qquad\textbf{(B) }\frac{56}{225}\qquad\textbf{(C) }\frac{107}{400}\qquad\textbf{(D) }\frac{7}{25}\qquad\textbf{(E) }\frac{9}{25}$
2024 AMC 8 -, 17
A chess king is said to ''attack'' all squares one step away from it (basically any square right next to it in any direction), horizontally, vertically, or diagonally. For instance, a king on the center square of a 3 x 3 grid attacks all 8 other squares, as shown below. Suppose a white king and a black king are placed on different squares of 3 x 3 grid so that they do not attack each other. In how many ways can this be done?
[asy]
/* AMC8 P17 2024, revised by Teacher David */
unitsize(29pt);
import math;
add(grid(3,3));
pair [] a = {(0.5,0.5), (0.5, 1.5), (0.5, 2.5), (1.5, 2.5), (2.5,2.5), (2.5,1.5), (2.5,0.5), (1.5,0.5)};
for (int i=0; i<a.length; ++i) {
pair x = (1.5,1.5) + 0.4*dir(225-45*i);
draw(x -- a[i], arrow=EndArrow());
}
label("$K$", (1.5,1.5));
[/asy]
$\textbf{(A)}\ 20 \qquad \textbf{(B)}\ 24 \qquad \textbf{(C)}\ 27 \qquad \textbf{(D)}\ 28 \qquad \textbf{(E)}\ 32$
2015 AMC 8, 20
Ralph went to the store and bought 12 pairs of socks for a total of \$24. Some of the socks he bought cost \$1 a pair, some of the socks he bought cost \$3 a pair, and some of the socks he bought cost \$4 a pair. If he bought at least one pair of each type, how many pairs of \$1 socks did Ralph buy?
$
\textbf{(A) } 4 \qquad
\textbf{(B) } 5 \qquad
\textbf{(C) } 6 \qquad
\textbf{(D) } 7 \qquad
\textbf{(E) } 8
$
2016 AMC 8, 19
The sum of $25$ consecutive even integers is $10,000$. What is the largest of these $25$ consecutive integers?
$\textbf{(A)}\mbox{ }360\qquad\textbf{(B)}\mbox{ }388\qquad\textbf{(C)}\mbox{ }412\qquad\textbf{(D)}\mbox{ }416\qquad\textbf{(E)}\mbox{ }424$
2017 AMC 8, 17
Starting with some gold coins and some empty treasure chests, I tried to put 9 gold coins in each treasure chest, but that left 2 treasure chests empty. So instead I put 6 gold coins in each treasure chest, but then I had 3 gold coins left over. How many gold coins did I have?
$\textbf{(A) }9\qquad\textbf{(B) }27\qquad\textbf{(C) }45\qquad\textbf{(D) }63\qquad\textbf{(E) }81$
2018 AMC 8, 5
What is the value of $1+3+5+\cdots+2017+2019-2-4-6-\cdots-2016-2018$?
$\textbf{(A) }-1010\qquad\textbf{(B) }-1009\qquad\textbf{(C) }1008\qquad\textbf{(D) }1009\qquad \textbf{(E) }1010$
1986 AMC 8, 21
[asy]draw((0,0)--(1,0)--(1,1)--(2,1)--(2,2)--(3,2)--(3,3)--(2,3)--(2,4)--(1,4)--(1,5)--(0,5)--(0,4)--(-1,4)--(-1,1)--(0,1)--cycle);
draw((0,1)--(1,1));
draw((-1,2)--(2,2));
draw((-1,3)--(2,3));
draw((0,4)--(1,4));
draw((0,1)--(0,4));
draw((1,1)--(1,4));
draw((2,2)--(2,3));
draw((0,1)--(1,1)--(1,2)--(2,2)--(2,3)--(1,3)--(1,4)--(0,4)--cycle);
draw((0,1)--(1,1)--(1,2)--(2,2)--(2,3)--(1,3)--(1,4)--(0,4)--cycle);
draw((0,1)--(1,1)--(1,2)--(2,2)--(2,3)--(1,3)--(1,4)--(0,4)--cycle);
draw((0,1)--(1,1)--(1,2)--(2,2)--(2,3)--(1,3)--(1,4)--(0,4)--cycle);
draw((0,1)--(1,1)--(1,2)--(2,2)--(2,3)--(1,3)--(1,4)--(0,4)--cycle);
draw((0,1)--(1,1)--(1,2)--(2,2)--(2,3)--(1,3)--(1,4)--(0,4)--cycle);
label("H",(0.5,0.2),N);
label("G",(1.5,1.2),N);
label("F",(-0.5,1.2),N);
label("E",(2.5,2.2),N);
label("D",(-0.5,2.2),N);
label("C",(1.5,3.2),N);
label("B",(-0.5,3.2),N);
label("A",(0.5,4.2),N);[/asy]
Suppose one of the eight lettered identical squares is included with the four squares in the T-shaped figure outlined. How many of the resulting figures can be folded into a topless cubical box?
\[ \textbf{(A)}\ 2 \qquad
\textbf{(B)}\ 3 \qquad
\textbf{(C)}\ 4 \qquad
\textbf{(D)}\ 5 \qquad
\textbf{(E)}\ 6
\]
1985 AMC 8, 22
Assume every $ 7$-digit whole number is a possible telephone number except those which begin with $ 0$ or $ 1$. What fraction of telephone numbers begin with $ 9$ and end with $ 0$?
\[ \textbf{(A)}\ \frac{1}{63} \qquad
\textbf{(B)}\ \frac{1}{80} \qquad
\textbf{(C)}\ \frac{1}{81} \qquad
\textbf{(D)}\ \frac{1}{90} \qquad
\textbf{(E)}\ \frac{1}{100}
\]
2024 AMC 8 -, 9
All the marbles in Maria's collection are red, green, or blue. Maria has half as many red marbles as green marbles and twice as many blue marbles as green marbles. Which of the following could be the total number of marbles in Maria's collection?
$\textbf{(A) } 24\qquad\textbf{(B) } 25\qquad\textbf{(C) } 26\qquad\textbf{(D) } 27\qquad\textbf{(E) } 28$
2015 AMC 8, 5
Billy's basketball team scored the following points over the course of the first 11 games of the season:
\[42, 47, 53, 53, 58, 58, 58, 61, 64, 65, 73\]
If his team scores 40 in the 12th game, which of the following statistics will show an increase?
$
\textbf{(A) } \text{range} \qquad
\textbf{(B) } \text{median} \qquad
\textbf{(C) } \text{mean} \qquad
\textbf{(D) } \text{mode} \qquad
\textbf{(E) } \text{mid-range}
$