Found problems: 260
2022 AMC 8 -, 5
Anna and Bella are celebrating their birthdays together. Five years ago, when Bella turned $6$ years old, she received a newborn kitten as a birthday present. Today the sum of the ages of the two children and the kitten is $30$ years. How many years older than Bella is Anna?
$\textbf{(A)} ~1\qquad\textbf{(B)} ~2\qquad\textbf{(C)} ~3\qquad\textbf{(D)} ~4\qquad\textbf{(E)} ~5\qquad$
2024 AMC 8 -, 24
Jean made a piece of stained glass art in the shape of two mountains, as shown in the figure below. One mountain peak is $8$ feet high and the other peak is $12$ feet high. Each peak forms a $90^\circ$ angle, and the straight sides of the mountains form $45^\circ$ with the ground. The artwork has an area of $183$ square feet. The sides of the mountains meet at an intersection point near the center of the artwork, $h$ feet above the ground. What is the value of $h$?
[asy]
unitsize(.3cm);
filldraw((0,0)--(8,8)--(11,5)--(18,12)--(30,0)--cycle,gray(0.7),linewidth(1));
draw((-1,0)--(-1,8),linewidth(.75));
draw((-1.4,0)--(-.6,0),linewidth(.75));
draw((-1.4,8)--(-.6,8),linewidth(.75));
label("$8$",(-1,4),W);
label("$12$",(31,6),E);
draw((-1,8)--(8,8),dashed);
draw((31,0)--(31,12),linewidth(.75));
draw((30.6,0)--(31.4,0),linewidth(.75));
draw((30.6,12)--(31.4,12),linewidth(.75));
draw((31,12)--(18,12),dashed);
label("$45^{\circ}$",(.75,0),NE,fontsize(10pt));
label("$45^{\circ}$",(29.25,0),NW,fontsize(10pt));
draw((8,8)--(7.5,7.5)--(8,7)--(8.5,7.5)--cycle);
draw((18,12)--(17.5,11.5)--(18,11)--(18.5,11.5)--cycle);
draw((11,5)--(11,0),dashed);
label("$h$",(11,2.5),E);
[/asy]
$\textbf{(A)}~4 \qquad \textbf{(B)}~5 \qquad \textbf{(C)}~4 \sqrt{2} \qquad \textbf{(D)}~6 \qquad \textbf{(E)}~5 \sqrt{2}$
2023 AMC 8, 24
Isosceles $\triangle$ $ABC$ has equal side lengths $AB$ and $BC$. In the figure below, segments are drawn parallel to $\overline{AC}$ so that the shaded portions of $\triangle$ $ABC$ have the same area. The heights of the two unshaded portions are 11 and 5 units, respectively. What is the height of $h$ of $\triangle$ $ABC$?
[asy]
size(12cm);
draw((5,10)--(5,6.7),dashed+gray+linewidth(.5));
draw((5,3)--(5,5.3),dashed+gray+linewidth(.5));
filldraw((1.5,3)--(8.5,3)--(10,0)--(0,0)--cycle,lightgray);
draw((0,0)--(10,0)--(5,10)--cycle,linewidth(1.3));
dot((0,0));
dot((5,10));
dot((10,0));
label(scale(.8)*"$11$", (5,6.5),S);
dot((17.5,0));
dot((27.5,0));
dot((22.5,10));
draw((22.5,1.3)--(22.5,0),dashed+gray+linewidth(.5));
draw((22.5,2.5)--(22.5,3.6),dashed+gray+linewidth(.5));
draw((17.5,0)--(27.5,0)--(22.5,10)--cycle,linewidth(1.3));
filldraw((19.3,3.6)--(25.7,3.6)--(22.5,10)--cycle,lightgray);
label(scale(.8)*"$5$", (22.5,1.9));
draw((5,10)--(22.5,10),dashed+gray+linewidth(.5));
draw((10,0)--(17.5,0),dashed+gray+linewidth(.5));
draw((13.75,4.3)--(13.75,0),dashed+gray+linewidth(.5));
draw((13.75,5.7)--(13.75,10),dashed+gray+linewidth(.5));
label(scale(.8)*"$h$", (13.75,5));
label(scale(.7)*"$A$", (0,0), S);
label(scale(.7)*"$C$", (10,0), S);
label(scale(.7)*"$B$", (5,10), N);
label(scale(.7)*"$A$", (17.5,0), S);
label(scale(.7)*"$C$", (27.5,0), S);
label(scale(.7)*"$B$", (22.5,10), N);
[/asy]
$\textbf{(A) } 14.6 \qquad \textbf{(B) } 14.8 \qquad \textbf{(C) } 15 \qquad \textbf{(D) } 15.2 \qquad \textbf{(E) } 15.4$
2015 AMC 8, 9
On her first day of work, Janabel sold one widget. On day two, she sold three widgets. On day three, she sold five widgets, and on each succeeding day, she sold two more widgets than she had sold on the previous day. How many widgets in total had Janabel sold after working $20$ days?
$\textbf{(A) }39\qquad\textbf{(B) }40\qquad\textbf{(C) }210\qquad\textbf{(D) }400\qquad \textbf{(E) }401$
2002 AMC 8, 16
Right isosceles triangles are constructed on the sides of a 3-4-5 right triangle, as shown. A capital letter represents the area of each triangle. Which one of the following is true?
[asy]/* AMC8 2002 #16 Problem */
draw((0,0)--(4,0)--(4,3)--cycle);
draw((4,3)--(-4,4)--(0,0));
draw((-0.15,0.1)--(0,0.25)--(.15,0.1));
draw((0,0)--(4,-4)--(4,0));
draw((4,0.2)--(3.8,0.2)--(3.8,-0.2)--(4,-0.2));
draw((4,0)--(7,3)--(4,3));
draw((4,2.8)--(4.2,2.8)--(4.2,3));
label(scale(0.8)*"$Z$", (0, 3), S);
label(scale(0.8)*"$Y$", (3,-2));
label(scale(0.8)*"$X$", (5.5, 2.5));
label(scale(0.8)*"$W$", (2.6,1));
label(scale(0.65)*"5", (2,2));
label(scale(0.65)*"4", (2.3,-0.4));
label(scale(0.65)*"3", (4.3,1.5));[/asy]
$ \textbf{(A)}\ X\plus{}Z\equal{}W\plus{}Y \qquad \textbf{(B)}\ W\plus{}X\equal{}Z \qquad\textbf{(C)}\ 3X\plus{}4Y\equal{}5Z \qquad $
$\textbf{(D)}\ X\plus{}W\equal{}\frac{1}{2}(Y\plus{}Z) \qquad\textbf{(E)}\ X\plus{}Y\equal{}Z$
2016 AMC 8, 14
Karl's car uses a gallon of gas every $35$ miles, and his gas tank holds $14$ gallons when it is full. One day, Karl started with a full tank of gas, drove $350$ miles, bought $8$ gallons of gas, and continued driving to his destination. When he arrived, his gas tank was half full. How many miles did Karl drive that day?
$\textbf{(A)}\mbox{ }525\qquad\textbf{(B)}\mbox{ }560\qquad\textbf{(C)}\mbox{ }595\qquad\textbf{(D)}\mbox{ }665\qquad\textbf{(E)}\mbox{ }735$
2020 AMC 8 -, 25
I was wondering if anyone had a sol for this. I am probably just going to bash it out.
2017 AMC 8, 7
Let $Z$ be a 6-digit positive integer, such as 247247, whose first three digits are the same as its last three digits taken in the same order. Which of the following numbers must also be a factor of $Z$?
$\textbf{(A) }11\qquad\textbf{(B) }19\qquad\textbf{(C) }101\qquad\textbf{(D) }111\qquad\textbf{(E) }1111$
2018 AMC 8, 16
Professor Chang has nine different language books lined up on a bookshelf: two Arabic, three German, and four Spanish. How many ways are there to arrange the nine books on the shelf keeping the Arabic books together and keeping the Spanish books together?
$\textbf{(A) }1440\qquad\textbf{(B) }2880\qquad\textbf{(C) }5760\qquad\textbf{(D) }182,440\qquad \textbf{(E) }362,880$
2018 AMC 8, 9
Tyler is tiling the floor of his 12 foot by 16 foot living room. He plans to place one-foot by one-foot square tiles to form a border along the edges of the room and to fill in the rest of the floor with two-foot by two-foot square tiles. How many tiles will he use?
$\textbf{(A) }48\qquad\textbf{(B) }87\qquad\textbf{(C) }91\qquad\textbf{(D) }96\qquad \textbf{(E) }120$
2015 AMC 8, 15
At Euler Middle School, $198$ students voted on two issues in a school referendum with the following results: $149$ voted in favor of the first issue and $119$ voted in favor of the second issue. If there were exactly $29$ students who voted against both issues, how many students voted in favor of both issues?
$\textbf{(A) }49\qquad\textbf{(B) }70\qquad\textbf{(C) }79\qquad\textbf{(D) }99\qquad \textbf{(E) }149$
2020 AMC 8 -, 1
Luka is making lemonade to sell at a school fundraiser. His recipe requires $4$ times as much water as sugar and twice as much sugar as lemon juice. He uses $3$ cups of lemon juice. How many cups of water does he need?
$\textbf{(A)}\ 6 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 12\qquad \textbf{(D)}\ 18 \qquad \textbf{(E)}\ 24$
2024 AMC 8 -, 1
What is the ones digit of \[222{,}222-22{,}222-2{,}222-222-22-2?\]
$\textbf{(A) }0\qquad\textbf{(B) }2\qquad\textbf{(C) }4\qquad\textbf{(D) }6\qquad\textbf{(E) }8$
2020 AMC 8 -, 23
Five different awards are to be given to three students. Each student will receive at least one award. In how many ways can the awards be distributed?
$\textbf{(A)}\ 120 \qquad \textbf{(B)}\ 150 \qquad \textbf{(C)}\ 180 \qquad \textbf{(D)}\ 210 \qquad \textbf{(E)}\ 240$
2010 AMC 8, 22
The hundreds digit of a three-digit number is $2$ more than the units digit. The digits of the three-digit number are reversed, and the result is subtracted from the original three-digit number. What is the units digit of the result?
$ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 8 $
2013 AMC 8, 25
A ball with diameter 4 inches starts at point A to roll along the track shown. The track is comprised of 3 semicircular arcs whose radii are $R_1 = 100$ inches, $R_2 = 60$ inches, and $R_3 = 80$ inches, respectively. The ball always remains in contact with the track and does not slip. What is the distance the center of the ball travels over the course from A to B?
[asy]
size(8cm);
draw((0,0)--(480,0),linetype("3 4"));
filldraw(circle((8,0),8),black);
draw((0,0)..(100,-100)..(200,0));
draw((200,0)..(260,60)..(320,0));
draw((320,0)..(400,-80)..(480,0));
draw((100,0)--(150,-50sqrt(3)),Arrow(size=4));
draw((260,0)--(290,30sqrt(3)),Arrow(size=4));
draw((400,0)--(440,-40sqrt(3)),Arrow(size=4));
label("$R_1$",(100,0)--(150,-50sqrt(3)), W, fontsize(10));
label("$R_2$",(260,0)--(290,30sqrt(3)), W, fontsize(10));
label("$R_3$",(400,0)--(440,-40sqrt(3)), W, fontsize(10));
filldraw(circle((8,0),8),black);
label("$A$",(0,0),W,fontsize(10));[/asy]
$\textbf{(A)}\ 238\pi \qquad \textbf{(B)}\ 240\pi \qquad \textbf{(C)}\ 260\pi \qquad \textbf{(D)}\ 280\pi \qquad \textbf{(E)}\ 500\pi$
2018 AMC 8, 12
The clock in Sri's car, which is not accurate, gains time at a constant rate. One day as he begins shopping he notes that his car clock and his watch (which is accurate) both say 12:00 noon. When he is done shopping, his watch says 12:30 and his car clock says 12:35. Later that day, Sri loses his watch. He looks at his car clock and it says 7:00. What is the actual time?
$\textbf{(A) }5:50\qquad\textbf{(B) }6:00\qquad\textbf{(C) }6:30\qquad\textbf{(D) }6:55\qquad \textbf{(E) }8:10$
2009 AMC 10, 14
On Monday, Millie puts a quart of seeds, $ 25\%$ of which are millet, into a bird feeder. On each successive day she adds another quart of the same mix of seeds without removing any seeds that are left. Each day the birds eat only $ 25\%$ of the millet in the feeder, but they eat all of the other seeds. On which day, just after Millie has placed the seeds, will the birds find that more than half the seeds in the feeder are millet?
$ \textbf{(A)}\ \text{Tuesday}\qquad \textbf{(B)}\ \text{Wednesday}\qquad \textbf{(C)}\ \text{Thursday} \qquad \textbf{(D)}\ \text{Friday}\qquad \textbf{(E)}\ \text{Saturday}$
2024 AMC 8 -, 2
What is the value of this expression in decimal form?
\[\dfrac{44}{11}+\dfrac{110}{44}+\dfrac{44}{1100}\]
$\textbf{(A) }6.4\qquad\textbf{(B) }6.504\qquad\textbf{(C) }6.54\qquad\textbf{(D) }6.9\qquad\textbf{(E) }6.94$
2022 AMC 8 -, 24
The figure below shows a polygon $ABCDEFGH$, consisting of rectangles and right triangles. When cut out and folded on the dotted lines, the polygon forms a triangular prism. Suppose that $AH = EF = 8$ and $GH = 14$. What is the volume of the prism?
[asy]
// djmathman diagram
unitsize(1cm);
defaultpen(linewidth(0.7)+fontsize(11));
real r = 2, s = 2.5, theta = 14;
pair G = (0,0), F = (r,0), C = (r,s), B = (0,s), M = (C+F)/2, I = M + s/2 * dir(-theta);
pair N = (B+G)/2, J = N + s/2 * dir(180+theta);
pair E = F + r * dir(- 45 - theta/2), D = I+E-F;
pair H = J + r * dir(135 + theta/2), A = B+H-J;
draw(A--B--C--I--D--E--F--G--J--H--cycle^^rightanglemark(F,I,C)^^rightanglemark(G,J,B));
draw(J--B--G^^C--F--I,linetype ("4 4"));
dot("$A$",A,N);
dot("$B$",B,1.2*N);
dot("$C$",C,N);
dot("$D$",D,dir(0));
dot("$E$",E,S);
dot("$F$",F,1.5*S);
dot("$G$",G,S);
dot("$H$",H,W);
dot("$I$",I,NE);
dot("$J$",J,1.5*S);
[/asy]
$\textbf{(A)} ~112\qquad\textbf{(B)} ~128\qquad\textbf{(C)} ~192\qquad\textbf{(D)} ~240\qquad\textbf{(E)} ~288\qquad$
2024 AMC 8 -, 6
Sergei skated around an ice rink, gliding along different paths. The gray lines in the figures below show four of the paths labeled $P$, $Q$, $R$, and $S$. What is the sorted order of the four paths from shortest to longest?
[center][img]https://wiki-images.artofproblemsolving.com/9/94/2024_AMC_8_Problem_6.png[/img][/center]
$\textbf{(A) }\text{P, Q, R, S}\qquad\textbf{(B) }\text{P, R, S, Q}\qquad\textbf{(C) }\text{Q, S, P, R}\qquad\textbf{(D) }\text{R, P, S, Q}\qquad\textbf{(E) }\text{R, S, P, Q}$
2020 AMC 8 -, 21
A game board consists of $64$ squares that alternate in color between black and white. The figure below shows square $P$ in the bottom and square $Q$ in the top row. A marker is placed at $P$. A [i]step[/i] consists of moving the marker onto one of the adjoining white squares in the row above. How many $7$-step paths are there from $P$ to $Q$? (The figure shows a sample path.)
[asy]//diagram by SirCalcsALot
size(200); int[] x = {6, 5, 4, 5, 6, 5, 6}; int[] y = {1, 2, 3, 4, 5, 6, 7}; int N = 7; for (int i = 0; i < 8; ++i) { for (int j = 0; j < 8; ++j) { draw((i,j)--(i+1,j)--(i+1,j+1)--(i,j+1)--(i,j)); if ((i+j) % 2 == 0) { filldraw((i,j)--(i+1,j)--(i+1,j+1)--(i,j+1)--(i,j)--cycle,black); } } } for (int i = 0; i < N; ++i) { draw(circle((x[i],y[i])+(0.5,0.5),0.35)); } label("$P$", (5.5, 0.5)); label("$Q$", (6.5, 7.5)); [/asy]
$\textbf{(A)}\ 28 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\ 33 \qquad \textbf{(E)}\ 35$
2015 AMC 8, 10
How many integers between $1000$ and $9999$ have four distinct digits?
$\textbf{(A) }3024\qquad\textbf{(B) }4536\qquad\textbf{(C) }5040\qquad\textbf{(D) }6480\qquad \textbf{(E) }6561$
2017 AMC 8, 23
Each day for four days, Linda traveled for one hour at a speed that resulted in her traveling one mile in an integer number of minutes. Each day after the first, her speed decreased so that the number of minutes to travel one mile increased by 5 minutes over the preceding day. Each of the four days, her distance traveled was also an integer number of miles. What was the total number of miles for the four trips?
$\textbf{(A) }10\qquad\textbf{(B) }15\qquad\textbf{(C) }25\qquad\textbf{(D) }50\qquad\textbf{(E) }82$
2016 AMC 8, 12
Jefferson Middle School has the same number of boys and girls. Three-fourths of the girls and two-thirds of the boys went on a field trip. What fraction of the students were girls?
$\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{9}{17}\qquad\textbf{(C) }\frac{7}{13}\qquad\textbf{(D) }\frac{2}{3}\qquad \textbf{(E) }\frac{14}{15}$