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Difference between revisions of "2022 AMC 8 Problems"

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IMPORTANT: THESE ARE NOT THE 2022 AMC 8 PROBLEMS. THIS IS COPY PASTED FROM THE 2020 AMC 8 PROBLEMS WIKI PAGE.
 +
 
 +
==Problem 1==
 +
Luka is making lemonade to sell at a school fundraiser. His recipe requires <math>4</math> times as much water as sugar and twice as much sugar as lemon juice. He uses <math>3</math> cups of lemon juice. How many cups of water does he need?
 +
 
 +
<math>\textbf{(A) } 6\qquad\textbf{(B) } 8\qquad\textbf{(C) } 12\qquad\textbf{(D) } 18\qquad\textbf{(E) } 24\qquad</math>
 +
 
 +
[[2020 AMC 8 Problems/Problem 1|Solution]]
 +
 
 +
==Problem 2==
 +
Four friends do yardwork for their neighbors over the weekend, earning <math>\$15, \$20, \$25,</math> and <math>\$40,</math> respectively. They decide to split their earnings equally among themselves. In total how much will the friend who earned <math>\$40</math> give to the others?
 +
 
 +
<math>\textbf{(A) }\$5 \qquad \textbf{(B) }\$10 \qquad \textbf{(C) }\$15 \qquad \textbf{(D) }\$20 \qquad \textbf{(E) }\$25</math>
 +
 
 +
[[2020 AMC 8 Problems/Problem 2|Solution]]
 +
 
 +
==Problem 3==
 +
Carrie has a rectangular garden that measures <math>6</math> feet by <math>8</math> feet. She plants the entire garden with strawberry plants. Carrie is able to plant <math>4</math> strawberry plants per square foot, and she harvests an average of <math>10</math> strawberries per plant. How many strawberries can she expect to harvest?
 +
<math>\textbf{(A) }560 \qquad \textbf{(B) }960 \qquad \textbf{(C) }1120 \qquad \textbf{(D) }1920 \qquad \textbf{(E) }3840</math>
 +
 
 +
[[2020 AMC 8 Problems/Problem 3|Solution]]
 +
 
 +
==Problem 4==
 +
Three hexagons of increasing size are shown below. Suppose the dot pattern continues so that each successive hexagon contains one more band of dots. How many dots are in the next hexagon?
 +
 
 +
<asy>
 +
// diagram by SirCalcsALot, edited by MRENTHUSIASM
 +
size(250);
 +
path p = scale(0.8)*unitcircle;
 +
pair[] A;
 +
pen grey1 = rgb(100/256, 100/256, 100/256);
 +
pen grey2 = rgb(183/256, 183/256, 183/256);
 +
for (int i=0; i<7; ++i) { A[i] = rotate(60*i)*(1,0);}
 +
path hex = A[0]--A[1]--A[2]--A[3]--A[4]--A[5]--cycle;
 +
fill(p,grey1);
 +
draw(scale(1.25)*hex,black+linewidth(1.25));
 +
pair S = 6A[0]+2A[1];
 +
fill(shift(S)*p,grey1);
 +
for (int i=0; i<6; ++i) { fill(shift(S+2*A[i])*p,grey2);}
 +
draw(shift(S)*scale(3.25)*hex,black+linewidth(1.25));
 +
pair T = 16A[0]+4A[1];
 +
fill(shift(T)*p,grey1);
 +
for (int i=0; i<6; ++i) {
 +
fill(shift(T+2*A[i])*p,grey2);
 +
fill(shift(T+4*A[i])*p,grey1);
 +
fill(shift(T+2*A[i]+2*A[i+1])*p,grey1);
 +
}
 +
draw(shift(T)*scale(5.25)*hex,black+linewidth(1.25));
 +
</asy>
 +
 
 +
<math>\textbf{(A) }35 \qquad \textbf{(B) }37 \qquad \textbf{(C) }39 \qquad \textbf{(D) }43 \qquad \textbf{(E) }49</math>
 +
 
 +
[[2020 AMC 8 Problems/Problem 4|Solution]]
 +
 
 +
==Problem 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 <math>5</math> cups. What percent of the total capacity of the pitcher did each cup receive?
 +
 
 +
<math>\textbf{(A) }5 \qquad \textbf{(B) }10 \qquad \textbf{(C) }15 \qquad \textbf{(D) }20 \qquad \textbf{(E) }25</math>
 +
 
 +
[[2020 AMC 8 Problems/Problem 5|Solution]]
 +
 
 +
==Problem 6==
 +
Aaron, Darren, Karen, Maren, and Sharon rode on a small train that has five cars that seat one person each. Maren sat in the last car. Aaron sat directly behind Sharon. Darren sat in one of the cars in front of Aaron. At least one person sat between Karen and Darren. Who sat in the middle car?
 +
 
 +
<math>\textbf{(A) }\text{Aaron} \qquad \textbf{(B) }\text{Darren} \qquad \textbf{(C) }\text{Karen} \qquad \textbf{(D) }\text{Maren}\qquad \textbf{(E) }\text{Sharon}</math>
 +
 
 +
[[2020 AMC 8 Problems/Problem 6|Solution]]
 +
 
 +
==Problem 7==
 +
How many integers between <math>2020</math> and <math>2400</math> have four distinct digits arranged in increasing order? (For example, <math>2347</math> is one integer.)
 +
 
 +
<math>\textbf{(A) }\text{9} \qquad \textbf{(B) }\text{10} \qquad \textbf{(C) }\text{15} \qquad \textbf{(D) }\text{21}\qquad \textbf{(E) }\text{28}</math>
 +
 
 +
[[2020 AMC 8 Problems/Problem 7|Solution]]
 +
 
 +
==Problem 8==
 +
Ricardo has <math>2020</math> coins, some of which are pennies (<math>1</math>-cent coins) and the rest of which are nickels (<math>5</math>-cent coins). He has at least one penny and at least one nickel. What is the difference in cents between the greatest possible and least possible amounts of money that Ricardo can have?
 +
 
 +
<math>\textbf{(A) }\text{8062} \qquad \textbf{(B) }\text{8068} \qquad \textbf{(C) }\text{8072} \qquad \textbf{(D) }\text{8076}\qquad \textbf{(E) }\text{8082}</math>
 +
 
 +
[[2020 AMC 8 Problems/Problem 8|Solution]]
 +
 
 +
==Problem 9==
 +
Akash's birthday cake is in the form of a <math>4 \times 4 \times 4</math> 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 <math>64</math> smaller cubes, each measuring <math>1 \times 1 \times 1</math> inch, as shown below. How many small pieces will have icing on exactly two sides?
 +
 
 +
<asy>
 +
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);
 +
 
 +
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>
 +
 
 +
 
 +
<math>\textbf{(A) }\text{12} \qquad \textbf{(B) }\text{16} \qquad \textbf{(C) }\text{18} \qquad \textbf{(D) }\text{20}\qquad \textbf{(E) }\text{24}</math>
 +
 
 +
[[2020 AMC 8 Problems/Problem 9|Solution]]
 +
 
 +
==Problem 10==
 +
 
 +
Zara has a collection of <math>4</math> marbles: an Aggie, a Bumblebee, a Steelie, and a Tiger. She wants to display them in a row on a shelf, but does not want to put the Steelie and the Tiger next to one another. In how many ways can she do this?
 +
 
 +
<math>\textbf{(A) }6 \qquad \textbf{(B) }8 \qquad \textbf{(C) }12 \qquad \textbf{(D) }18 \qquad \textbf{(E) }24</math>
 +
 
 +
[[2020 AMC 8 Problems/Problem 10|Solution]]
 +
 
 +
==Problem 11==
 +
After school, Maya and Naomi headed to the beach, <math>6</math> miles away. Maya decided to bike while Naomi took a bus. The graph below shows their journeys, indicating the time and distance traveled. What was the difference, in miles per hour, between Naomi's and Maya's average speeds?
 +
 
 +
<asy>
 +
// diagram by SirCalcsALot
 +
unitsize(1.25cm);
 +
dotfactor = 10;
 +
pen shortdashed=linetype(new real[] {2.7,2.7});
 +
 
 +
for (int i = 0; i < 6; ++i) {
 +
    for (int j = 0; j < 6; ++j) {
 +
        draw((i,0)--(i,6), grey);
 +
        draw((0,j)--(6,j), grey);
 +
    }
 +
}
 +
 
 +
for (int i = 1; i <= 6; ++i) {
 +
    draw((-0.1,i)--(0.1,i),linewidth(1.25));
 +
    draw((i,-0.1)--(i,0.1),linewidth(1.25));
 +
    label(string(5*i), (i,0), 2*S);
 +
    label(string(i), (0, i), 2*W);
 +
}
 +
 
 +
draw((0,0)--(0,6)--(6,6)--(6,0)--(0,0)--cycle,linewidth(1.25));
 +
 
 +
label(rotate(90) * "Distance (miles)", (-0.5,3), W);
 +
label("Time (minutes)", (3,-0.5), S);
 +
 
 +
dot("Naomi", (2,6), 3*dir(305));
 +
dot((6,6));
 +
 
 +
label("Maya", (4.45,3.5));
 +
 
 +
draw((0,0)--(1.15,1.3)--(1.55,1.3)--(3.15,3.2)--(3.65,3.2)--(5.2,5.2)--(5.4,5.2)--(6,6),linewidth(1.35));
 +
draw((0,0)--(0.4,0.1)--(1.15,3.7)--(1.6,3.7)--(2,6),linewidth(1.35)+shortdashed);
 +
</asy>
 +
 
 +
<math>\textbf{(A) }6 \qquad \textbf{(B) }12 \qquad \textbf{(C) }18 \qquad \textbf{(D) }20 \qquad \textbf{(E) }24</math>
 +
 
 +
[[2020 AMC 8 Problems/Problem 11|Solution]]
 +
 
 +
==Problem 12==
 +
For a positive integer <math>n,</math> the factorial notation <math>n!</math> represents the product of the integers
 +
from <math>n</math> to <math>1</math>. (For example, <math>6! = 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1</math>.) What value of <math>N</math> satisfies the following equation?
 +
<cmath>5! \cdot 9! = 12 \cdot N!</cmath>
 +
<math>\textbf{(A) }10 \qquad \textbf{(B) }11 \qquad \textbf{(C) }12 \qquad \textbf{(D) }13 \qquad \textbf{(E) }14</math>
 +
 
 +
[[2020 AMC 8 Problems/Problem 12|Solution]]
 +
 
 +
==Problem 13==
 +
Jamal has a drawer containing <math>6</math> green socks, <math>18</math> purple socks, and <math>12</math> orange socks. After adding more purple socks, Jamal noticed that there is now a <math>60\%</math> chance that a sock randomly selected from the drawer is purple. How many purple socks did Jamal add?
 +
 
 +
<math>\textbf{(A) }6 \qquad \textbf{(B) }9 \qquad \textbf{(C) }12 \qquad \textbf{(D) }18 \qquad \textbf{(E) }24</math>
 +
 
 +
[[2020 AMC 8 Problems/Problem 13|Solution]]
 +
 
 +
==Problem 14==
 +
There are <math>20</math> 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 <math>20</math> 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>
 +
 
 +
<math>\textbf{(A) }65{,}000 \qquad \textbf{(B) }75{,}000 \qquad \textbf{(C) }85{,}000 \qquad \textbf{(D) }95{,}000 \qquad \textbf{(E) }105{,}000</math>
 +
 
 +
[[2020 AMC 8 Problems/Problem 14|Solution]]
 +
 
 +
==Problem 15==
 +
Suppose <math>15\%</math> of <math>x</math> equals <math>20\%</math> of <math>y.</math> What percentage of <math>x</math> is <math>y?</math>
 +
 
 +
<math>\textbf{(A) }5 \qquad \textbf{(B) }35 \qquad \textbf{(C) }75 \qquad \textbf{(D) }133 \frac13 \qquad \textbf{(E) }300</math>
 +
 
 +
[[2020 AMC 8 Problems/Problem 15|Solution]]
 +
 
 +
==Problem 16==
 +
Each of the points <math>A,B,C,D,E,</math> and <math>F</math> in the figure below represents a different digit from <math>1</math> to <math>6.</math> Each of the five lines shown passes through some of these points. The digits along each line are added to produce five sums, one for each line. The total of the five sums is <math>47.</math> What is the digit represented by B?
 +
 
 +
<asy>
 +
// made by SirCalcsALot
 +
 
 +
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>
 +
 
 +
<math>\textbf{(A) }1 \qquad \textbf{(B) }2 \qquad \textbf{(C) }3 \qquad \textbf{(D) }4 \qquad \textbf{(E) }5</math>
 +
 
 +
[[2020 AMC 8 Problems/Problem 16|Solution]]
 +
 
 +
==Problem 17==
 +
How many factors of <math>2020</math> have more than <math>3</math> factors? (As an example, <math>12</math> has <math>6</math> factors, namely <math>1, 2, 3, 4, 6,</math> and <math>12.</math>)
 +
 
 +
<math>\textbf{(A) }6 \qquad \textbf{(B) }7 \qquad \textbf{(C) }8 \qquad \textbf{(D) }9 \qquad \textbf{(E) }10</math>
 +
 
 +
[[2020 AMC 8 Problems/Problem 17|Solution]]
 +
 
 +
==Problem 18==
 +
Rectangle <math>ABCD</math> is inscribed in a semicircle with diameter <math>\overline{FE},</math> as shown in the figure. Let <math>DA=16,</math> and let <math>FD=AE=9.</math> What is the area of <math>ABCD?</math>
 +
 
 +
<asy>
 +
// diagram by SirCalcsALot
 +
draw(arc((0,0),17,180,0));
 +
draw((-17,0)--(17,0));
 +
fill((-8,0)--(-8,15)--(8,15)--(8,0)--cycle, 1.5*grey);
 +
draw((-8,0)--(-8,15)--(8,15)--(8,0)--cycle);
 +
dot("$A$",(8,0), 1.25*S);
 +
dot("$B$",(8,15), 1.25*N);
 +
dot("$C$",(-8,15), 1.25*N);
 +
dot("$D$",(-8,0), 1.25*S);
 +
dot("$E$",(17,0), 1.25*S);
 +
dot("$F$",(-17,0), 1.25*S);
 +
label("$16$",(0,0),N);
 +
label("$9$",(12.5,0),N);
 +
label("$9$",(-12.5,0),N);
 +
</asy>
 +
 
 +
<math>\textbf{(A) }240 \qquad \textbf{(B) }248 \qquad \textbf{(C) }256 \qquad \textbf{(D) }264 \qquad \textbf{(E) }272</math>
 +
 
 +
[[2020 AMC 8 Problems/Problem 18|Solution]]
 +
 
 +
==Problem 19==
 +
A number is called flippy if its digits alternate between two distinct digits. For example, <math>2020</math> and <math>37373</math> are flippy, but <math>3883</math> and <math>123123</math> are not. How many five-digit flippy numbers are divisible by <math>15?</math>
 +
 
 +
<math>\textbf{(A) }3 \qquad \textbf{(B) }4 \qquad \textbf{(C) }5 \qquad \textbf{(D) }6 \qquad \textbf{(E) }8</math>
 +
 
 +
[[2020 AMC 8 Problems/Problem 19|Solution]]
 +
 
 +
==Problem 20==
 +
A scientist walking through a forest recorded as integers the heights of <math>5</math> trees standing in a row. She observed that each tree was either twice as tall or half as tall as the one to its right. Unfortunately some of her data was lost when rain fell on her notebook. Her notes are shown below, with blanks indicating the missing numbers. Based on her observations, the scientist was able to reconstruct the lost data. What was the average height of the trees, in meters?
 +
 
 +
<cmath>
 +
\begingroup
 +
\setlength{\tabcolsep}{10pt}
 +
\renewcommand{\arraystretch}{1.5}
 +
\begin{tabular}{|c|c|}
 +
\hline Tree 1 & \rule{0.4cm}{0.15mm} meters \\
 +
Tree 2 & 11 meters \\
 +
Tree 3 & \rule{0.5cm}{0.15mm} meters \\
 +
Tree 4 & \rule{0.5cm}{0.15mm} meters \\
 +
Tree 5 & \rule{0.5cm}{0.15mm} meters \\ \hline
 +
Average height & \rule{0.5cm}{0.15mm}\text{ .}2 meters \\
 +
\hline
 +
\end{tabular}
 +
\endgroup</cmath>
 +
<math>\newline \textbf{(A) }22.2 \qquad \textbf{(B) }24.2 \qquad \textbf{(C) }33.2 \qquad \textbf{(D) }35.2 \qquad \textbf{(E) }37.2</math>
 +
 
 +
[[2020 AMC 8 Problems/Problem 20|Solution]]
 +
 
 +
==Problem 21==
 +
A game board consists of <math>64</math> squares that alternate in color between black and white. The figure below shows square <math>P</math> in the bottom row and square <math>Q</math> in the top row. A marker is placed at <math>P.</math> A step consists of moving the marker onto one of the adjoining white squares in the row above. How many <math>7</math>-step paths are there from <math>P</math> to <math>Q?</math> (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),grey);
 +
}
 +
label("$P$", (5.5, 0.5));
 +
label("$Q$", (6.5, 7.5));
 +
</asy>
 +
 
 +
<math>\textbf{(A) }28 \qquad \textbf{(B) }30 \qquad \textbf{(C) }32 \qquad \textbf{(D) }33 \qquad \textbf{(E) }35</math>
 +
 
 +
[[2020 AMC 8 Problems/Problem 21|Solution]]
 +
 
 +
==Problem 22==
 +
When a positive integer <math>N</math> is fed into a machine, the output is a number calculated according to the rule shown below.
 +
 
 +
<asy>
 +
size(300);
 +
defaultpen(linewidth(0.8)+fontsize(13));
 +
real r = 0.05;
 +
draw((0.9,0)--(3.5,0),EndArrow(size=7));
 +
filldraw((4,2.5)--(7,2.5)--(7,-2.5)--(4,-2.5)--cycle,gray(0.65));
 +
fill(circle((5.5,1.25),0.8),white);
 +
fill(circle((5.5,1.25),0.5),gray(0.65));
 +
fill((4.3,-r)--(6.7,-r)--(6.7,-1-r)--(4.3,-1-r)--cycle,white);
 +
fill((4.3,-1.25+r)--(6.7,-1.25+r)--(6.7,-2.25+r)--(4.3,-2.25+r)--cycle,white);
 +
fill((4.6,-0.25-r)--(6.4,-0.25-r)--(6.4,-0.75-r)--(4.6,-0.75-r)--cycle,gray(0.65));
 +
fill((4.6,-1.5+r)--(6.4,-1.5+r)--(6.4,-2+r)--(4.6,-2+r)--cycle,gray(0.65));
 +
label("$N$",(0.45,0));
 +
draw((7.5,1.25)--(11.25,1.25),EndArrow(size=7));
 +
draw((7.5,-1.25)--(11.25,-1.25),EndArrow(size=7));
 +
label("if $N$ is even",(9.25,1.25),N);
 +
label("if $N$ is odd",(9.25,-1.25),N);
 +
label("$\frac N2$",(12,1.25));
 +
label("$3N+1$",(12.6,-1.25));
 +
</asy>
 +
 
 +
For example, starting with an input of <math>N=7,</math> the machine will output <math>3 \cdot 7 +1 = 22.</math> Then if the output is repeatedly inserted into the machine five more times, the final output is <math>26.</math>
 +
<cmath>7 \to 22 \to 11 \to 34 \to 17 \to 52 \to 26</cmath>When the same <math>6</math>-step process is applied to a different starting value of <math>N,</math> the final output is <math>1.</math> What is the sum of all such integers <math>N?</math>
 +
<cmath>N \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to 1</cmath>
 +
 
 +
<math>\textbf{(A) }73 \qquad \textbf{(B) }74 \qquad \textbf{(C) }75 \qquad \textbf{(D) }82 \qquad \textbf{(E) }83</math>
 +
 
 +
[[2020 AMC 8 Problems/Problem 22|Solution]]
 +
 
 +
==Problem 23==
 +
Five different awards are to be given to three students. Each student will receive at least one award. In how many different ways can the awards be distributed?
 +
 
 +
<math>\textbf{(A) }120 \qquad \textbf{(B) }150 \qquad \textbf{(C) }180 \qquad \textbf{(D) }210 \qquad \textbf{(E) }240</math>
 +
 
 +
[[2020 AMC 8 Problems/Problem 23|Solution]]
 +
 
 +
==Problem 24==
 +
A large square region is paved with <math>n^2</math> gray square tiles, each measuring <math>s</math> inches on a side. A border <math>d</math> inches wide surrounds each tile. The figure below shows the case for <math>n=3</math>. When <math>n=24</math>
 +
, the <math>576</math> gray tiles cover <math>64\%</math> of the area of the large square region. What is the ratio <math>\frac{d}{s}</math> for this larger value of <math>n?</math>
 +
 
 +
<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>
 +
 
 +
<math>\textbf{(A) }\frac6{25} \qquad \textbf{(B) }\frac14 \qquad \textbf{(C) }\frac9{25} \qquad \textbf{(D) }\frac7{16} \qquad \textbf{(E) }\frac9{16}</math>
 +
 
 +
[[2020 AMC 8 Problems/Problem 24|Solution]]
 +
 
 +
==Problem 25==
 +
Rectangles <math>R_1</math> and <math>R_2,</math> and squares <math>S_1,\,S_2,\,</math> and <math>S_3,</math> shown below, combine to form a rectangle that is 3322 units wide and 2020 units high. What is the side length of <math>S_2</math> in units?
 +
 
 +
<asy>
 +
draw((0,0)--(5,0)--(5,3)--(0,3)--(0,0));
 +
draw((3,0)--(3,1)--(0,1));
 +
draw((3,1)--(3,2)--(5,2));
 +
draw((3,2)--(2,2)--(2,1)--(2,3));
 +
label("$R_1$",(3/2,1/2));
 +
label("$S_3$",(4,1));
 +
label("$S_2$",(5/2,3/2));
 +
label("$S_1$",(1,2));
 +
label("$R_2$",(7/2,5/2));
 +
</asy>
 +
 
 +
<math>\textbf{(A) }651 \qquad \textbf{(B) }655 \qquad \textbf{(C) }656 \qquad \textbf{(D) }662 \qquad \textbf{(E) }666</math>
 +
 
 +
[[2020 AMC 8 Problems/Problem 25|Solution]]

Revision as of 09:19, 28 January 2022

IMPORTANT: THESE ARE NOT THE 2022 AMC 8 PROBLEMS. THIS IS COPY PASTED FROM THE 2020 AMC 8 PROBLEMS WIKI PAGE.

Problem 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?

$\textbf{(A) } 6\qquad\textbf{(B) } 8\qquad\textbf{(C) } 12\qquad\textbf{(D) } 18\qquad\textbf{(E) } 24\qquad$

Solution

Problem 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) }$25$

Solution

Problem 3

Carrie has a rectangular garden that measures $6$ feet by $8$ feet. She plants the entire garden with strawberry plants. Carrie is able to plant $4$ strawberry plants per square foot, and she harvests an average of $10$ strawberries per plant. How many strawberries can she expect to harvest? $\textbf{(A) }560 \qquad \textbf{(B) }960 \qquad \textbf{(C) }1120 \qquad \textbf{(D) }1920 \qquad \textbf{(E) }3840$

Solution

Problem 4

Three hexagons of increasing size are shown below. Suppose the dot pattern continues so that each successive hexagon contains one more band of dots. How many dots are in the next hexagon?

[asy] // diagram by SirCalcsALot, edited by MRENTHUSIASM size(250); path p = scale(0.8)*unitcircle; pair[] A; pen grey1 = rgb(100/256, 100/256, 100/256); pen grey2 = rgb(183/256, 183/256, 183/256); for (int i=0; i<7; ++i) { A[i] = rotate(60*i)*(1,0);} path hex = A[0]--A[1]--A[2]--A[3]--A[4]--A[5]--cycle; fill(p,grey1); draw(scale(1.25)*hex,black+linewidth(1.25)); pair S = 6A[0]+2A[1]; fill(shift(S)*p,grey1); for (int i=0; i<6; ++i) { fill(shift(S+2*A[i])*p,grey2);} draw(shift(S)*scale(3.25)*hex,black+linewidth(1.25)); pair T = 16A[0]+4A[1]; fill(shift(T)*p,grey1); for (int i=0; i<6; ++i) { fill(shift(T+2*A[i])*p,grey2); fill(shift(T+4*A[i])*p,grey1); fill(shift(T+2*A[i]+2*A[i+1])*p,grey1); } draw(shift(T)*scale(5.25)*hex,black+linewidth(1.25)); [/asy]

$\textbf{(A) }35 \qquad \textbf{(B) }37 \qquad \textbf{(C) }39 \qquad \textbf{(D) }43 \qquad \textbf{(E) }49$

Solution

Problem 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?

$\textbf{(A) }5 \qquad \textbf{(B) }10 \qquad \textbf{(C) }15 \qquad \textbf{(D) }20 \qquad \textbf{(E) }25$

Solution

Problem 6

Aaron, Darren, Karen, Maren, and Sharon rode on a small train that has five cars that seat one person each. Maren sat in the last car. Aaron sat directly behind Sharon. Darren sat in one of the cars in front of Aaron. At least one person sat between Karen and Darren. Who sat in the middle car?

$\textbf{(A) }\text{Aaron} \qquad \textbf{(B) }\text{Darren} \qquad \textbf{(C) }\text{Karen} \qquad \textbf{(D) }\text{Maren}\qquad \textbf{(E) }\text{Sharon}$

Solution

Problem 7

How many integers between $2020$ and $2400$ have four distinct digits arranged in increasing order? (For example, $2347$ is one integer.)

$\textbf{(A) }\text{9} \qquad \textbf{(B) }\text{10} \qquad \textbf{(C) }\text{15} \qquad \textbf{(D) }\text{21}\qquad \textbf{(E) }\text{28}$

Solution

Problem 8

Ricardo has $2020$ coins, some of which are pennies ($1$-cent coins) and the rest of which are nickels ($5$-cent coins). He has at least one penny and at least one nickel. What is the difference in cents between the greatest possible and least possible amounts of money that Ricardo can have?

$\textbf{(A) }\text{8062} \qquad \textbf{(B) }\text{8068} \qquad \textbf{(C) }\text{8072} \qquad \textbf{(D) }\text{8076}\qquad \textbf{(E) }\text{8082}$

Solution

Problem 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 small pieces will have icing on exactly two sides?

[asy] 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); 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]


$\textbf{(A) }\text{12} \qquad \textbf{(B) }\text{16} \qquad \textbf{(C) }\text{18} \qquad \textbf{(D) }\text{20}\qquad \textbf{(E) }\text{24}$

Solution

Problem 10

Zara has a collection of $4$ marbles: an Aggie, a Bumblebee, a Steelie, and a Tiger. She wants to display them in a row on a shelf, but does not want to put the Steelie and the Tiger next to one another. In how many ways can she do this?

$\textbf{(A) }6 \qquad \textbf{(B) }8 \qquad \textbf{(C) }12 \qquad \textbf{(D) }18 \qquad \textbf{(E) }24$

Solution

Problem 11

After school, Maya and Naomi headed to the beach, $6$ miles away. Maya decided to bike while Naomi took a bus. The graph below shows their journeys, indicating the time and distance traveled. What was the difference, in miles per hour, between Naomi's and Maya's average speeds?

[asy] // diagram by SirCalcsALot unitsize(1.25cm); dotfactor = 10; pen shortdashed=linetype(new real[] {2.7,2.7}); for (int i = 0; i < 6; ++i) { for (int j = 0; j < 6; ++j) { draw((i,0)--(i,6), grey); draw((0,j)--(6,j), grey); } } for (int i = 1; i <= 6; ++i) { draw((-0.1,i)--(0.1,i),linewidth(1.25)); draw((i,-0.1)--(i,0.1),linewidth(1.25)); label(string(5*i), (i,0), 2*S); label(string(i), (0, i), 2*W); } draw((0,0)--(0,6)--(6,6)--(6,0)--(0,0)--cycle,linewidth(1.25)); label(rotate(90) * "Distance (miles)", (-0.5,3), W); label("Time (minutes)", (3,-0.5), S); dot("Naomi", (2,6), 3*dir(305)); dot((6,6)); label("Maya", (4.45,3.5)); draw((0,0)--(1.15,1.3)--(1.55,1.3)--(3.15,3.2)--(3.65,3.2)--(5.2,5.2)--(5.4,5.2)--(6,6),linewidth(1.35)); draw((0,0)--(0.4,0.1)--(1.15,3.7)--(1.6,3.7)--(2,6),linewidth(1.35)+shortdashed); [/asy]

$\textbf{(A) }6 \qquad \textbf{(B) }12 \qquad \textbf{(C) }18 \qquad \textbf{(D) }20 \qquad \textbf{(E) }24$

Solution

Problem 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? \[5! \cdot 9! = 12 \cdot N!\] $\textbf{(A) }10 \qquad \textbf{(B) }11 \qquad \textbf{(C) }12 \qquad \textbf{(D) }13 \qquad \textbf{(E) }14$

Solution

Problem 13

Jamal has a drawer containing $6$ green socks, $18$ purple socks, and $12$ orange socks. After adding more purple socks, Jamal noticed that there is now a $60\%$ chance that a sock randomly selected from the drawer is purple. How many purple socks did Jamal add?

$\textbf{(A) }6 \qquad \textbf{(B) }9 \qquad \textbf{(C) }12 \qquad \textbf{(D) }18 \qquad \textbf{(E) }24$

Solution

Problem 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]

$\textbf{(A) }65{,}000 \qquad \textbf{(B) }75{,}000 \qquad \textbf{(C) }85{,}000 \qquad \textbf{(D) }95{,}000 \qquad \textbf{(E) }105{,}000$

Solution

Problem 15

Suppose $15\%$ of $x$ equals $20\%$ of $y.$ What percentage of $x$ is $y?$

$\textbf{(A) }5 \qquad \textbf{(B) }35 \qquad \textbf{(C) }75 \qquad \textbf{(D) }133 \frac13 \qquad \textbf{(E) }300$

Solution

Problem 16

Each of the points $A,B,C,D,E,$ and $F$ in the figure below represents a different digit from $1$ to $6.$ Each of the five lines shown passes through some of these points. The digits along each line are added to produce five sums, one for each line. The total of the five sums is $47.$ What is the digit represented by B?

[asy] // made by SirCalcsALot 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]

$\textbf{(A) }1 \qquad \textbf{(B) }2 \qquad \textbf{(C) }3 \qquad \textbf{(D) }4 \qquad \textbf{(E) }5$

Solution

Problem 17

How many factors of $2020$ have more than $3$ factors? (As an example, $12$ has $6$ factors, namely $1, 2, 3, 4, 6,$ and $12.$)

$\textbf{(A) }6 \qquad \textbf{(B) }7 \qquad \textbf{(C) }8 \qquad \textbf{(D) }9 \qquad \textbf{(E) }10$

Solution

Problem 18

Rectangle $ABCD$ is inscribed in a semicircle with diameter $\overline{FE},$ as shown in the figure. Let $DA=16,$ and let $FD=AE=9.$ What is the area of $ABCD?$

[asy] // diagram by SirCalcsALot draw(arc((0,0),17,180,0)); draw((-17,0)--(17,0)); fill((-8,0)--(-8,15)--(8,15)--(8,0)--cycle, 1.5*grey); draw((-8,0)--(-8,15)--(8,15)--(8,0)--cycle); dot("$A$",(8,0), 1.25*S); dot("$B$",(8,15), 1.25*N); dot("$C$",(-8,15), 1.25*N); dot("$D$",(-8,0), 1.25*S); dot("$E$",(17,0), 1.25*S); dot("$F$",(-17,0), 1.25*S); label("$16$",(0,0),N); label("$9$",(12.5,0),N); label("$9$",(-12.5,0),N); [/asy]

$\textbf{(A) }240 \qquad \textbf{(B) }248 \qquad \textbf{(C) }256 \qquad \textbf{(D) }264 \qquad \textbf{(E) }272$

Solution

Problem 19

A number is called flippy if its digits alternate between two distinct digits. For example, $2020$ and $37373$ are flippy, but $3883$ and $123123$ are not. How many five-digit flippy numbers are divisible by $15?$

$\textbf{(A) }3 \qquad \textbf{(B) }4 \qquad \textbf{(C) }5 \qquad \textbf{(D) }6 \qquad \textbf{(E) }8$

Solution

Problem 20

A scientist walking through a forest recorded as integers the heights of $5$ trees standing in a row. She observed that each tree was either twice as tall or half as tall as the one to its right. Unfortunately some of her data was lost when rain fell on her notebook. Her notes are shown below, with blanks indicating the missing numbers. Based on her observations, the scientist was able to reconstruct the lost data. What was the average height of the trees, in meters?

\[\begingroup \setlength{\tabcolsep}{10pt} \renewcommand{\arraystretch}{1.5} \begin{tabular}{|c|c|} \hline Tree 1 & \rule{0.4cm}{0.15mm} meters \\ Tree 2 & 11 meters \\ Tree 3 & \rule{0.5cm}{0.15mm} meters \\ Tree 4 & \rule{0.5cm}{0.15mm} meters \\ Tree 5 & \rule{0.5cm}{0.15mm} meters \\ \hline Average height & \rule{0.5cm}{0.15mm}\text{ .}2 meters \\ \hline \end{tabular} \endgroup\] $\newline \textbf{(A) }22.2 \qquad \textbf{(B) }24.2 \qquad \textbf{(C) }33.2 \qquad \textbf{(D) }35.2 \qquad \textbf{(E) }37.2$

Solution

Problem 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 row and square $Q$ in the top row. A marker is placed at $P.$ A step 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),grey); } label("$P$", (5.5, 0.5)); label("$Q$", (6.5, 7.5)); [/asy]

$\textbf{(A) }28 \qquad \textbf{(B) }30 \qquad \textbf{(C) }32 \qquad \textbf{(D) }33 \qquad \textbf{(E) }35$

Solution

Problem 22

When a positive integer $N$ is fed into a machine, the output is a number calculated according to the rule shown below.

[asy] size(300); defaultpen(linewidth(0.8)+fontsize(13)); real r = 0.05; draw((0.9,0)--(3.5,0),EndArrow(size=7)); filldraw((4,2.5)--(7,2.5)--(7,-2.5)--(4,-2.5)--cycle,gray(0.65)); fill(circle((5.5,1.25),0.8),white); fill(circle((5.5,1.25),0.5),gray(0.65)); fill((4.3,-r)--(6.7,-r)--(6.7,-1-r)--(4.3,-1-r)--cycle,white); fill((4.3,-1.25+r)--(6.7,-1.25+r)--(6.7,-2.25+r)--(4.3,-2.25+r)--cycle,white); fill((4.6,-0.25-r)--(6.4,-0.25-r)--(6.4,-0.75-r)--(4.6,-0.75-r)--cycle,gray(0.65)); fill((4.6,-1.5+r)--(6.4,-1.5+r)--(6.4,-2+r)--(4.6,-2+r)--cycle,gray(0.65)); label("$N$",(0.45,0)); draw((7.5,1.25)--(11.25,1.25),EndArrow(size=7)); draw((7.5,-1.25)--(11.25,-1.25),EndArrow(size=7)); label("if $N$ is even",(9.25,1.25),N); label("if $N$ is odd",(9.25,-1.25),N); label("$\frac N2$",(12,1.25)); label("$3N+1$",(12.6,-1.25)); [/asy]

For example, starting with an input of $N=7,$ the machine will output $3 \cdot 7 +1 = 22.$ Then if the output is repeatedly inserted into the machine five more times, the final output is $26.$ \[7 \to 22 \to 11 \to 34 \to 17 \to 52 \to 26\]When the same $6$-step process is applied to a different starting value of $N,$ the final output is $1.$ What is the sum of all such integers $N?$ \[N \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to 1\]

$\textbf{(A) }73 \qquad \textbf{(B) }74 \qquad \textbf{(C) }75 \qquad \textbf{(D) }82 \qquad \textbf{(E) }83$

Solution

Problem 23

Five different awards are to be given to three students. Each student will receive at least one award. In how many different ways can the awards be distributed?

$\textbf{(A) }120 \qquad \textbf{(B) }150 \qquad \textbf{(C) }180 \qquad \textbf{(D) }210 \qquad \textbf{(E) }240$

Solution

Problem 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]

$\textbf{(A) }\frac6{25} \qquad \textbf{(B) }\frac14 \qquad \textbf{(C) }\frac9{25} \qquad \textbf{(D) }\frac7{16} \qquad \textbf{(E) }\frac9{16}$

Solution

Problem 25

Rectangles $R_1$ and $R_2,$ and squares $S_1,\,S_2,\,$ and $S_3,$ shown below, combine to form a rectangle that is 3322 units wide and 2020 units high. What is the side length of $S_2$ in units?

[asy] draw((0,0)--(5,0)--(5,3)--(0,3)--(0,0)); draw((3,0)--(3,1)--(0,1)); draw((3,1)--(3,2)--(5,2)); draw((3,2)--(2,2)--(2,1)--(2,3)); label("$R_1$",(3/2,1/2)); label("$S_3$",(4,1)); label("$S_2$",(5/2,3/2)); label("$S_1$",(1,2)); label("$R_2$",(7/2,5/2)); [/asy]

$\textbf{(A) }651 \qquad \textbf{(B) }655 \qquad \textbf{(C) }656 \qquad \textbf{(D) }662 \qquad \textbf{(E) }666$

Solution

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