Difference between revisions of "2020 AMC 8 Problems"
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+ | {{AMC8 Problems|year=2020|}} | ||
==Problem 1== | ==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? | + | 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> | <math>\textbf{(A) } 6\qquad\textbf{(B) } 8\qquad\textbf{(C) } 12\qquad\textbf{(D) } 18\qquad\textbf{(E) } 24\qquad</math> | ||
Line 8: | Line 9: | ||
==Problem 2== | ==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? | 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]] | [[2020 AMC 8 Problems/Problem 2|Solution]] | ||
− | |||
− | |||
==Problem 3== | ==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? | 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> | <math>\textbf{(A) }560 \qquad \textbf{(B) }960 \qquad \textbf{(C) }1120 \qquad \textbf{(D) }1920 \qquad \textbf{(E) }3840</math> | ||
Line 22: | Line 22: | ||
==Problem 4== | ==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? | 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> | <asy> | ||
+ | // diagram by SirCalcsALot, edited by MRENTHUSIASM | ||
size(250); | size(250); | ||
− | + | path p = scale(0.8)*unitcircle; | |
− | + | pair[] A; | |
− | |||
− | |||
− | |||
− | pair | ||
pen grey1 = rgb(100/256, 100/256, 100/256); | pen grey1 = rgb(100/256, 100/256, 100/256); | ||
pen grey2 = rgb(183/256, 183/256, 183/256); | pen grey2 = rgb(183/256, 183/256, 183/256); | ||
− | + | for (int i=0; i<7; ++i) { A[i] = rotate(60*i)*(1,0);} | |
− | for (int i = 0; i < | + | 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)); | |
− | fill( | + | pair S = 6A[0]+2A[1]; |
− | for (int i = 0; i < 6; ++i) { | + | fill(shift(S)*p,grey1); |
− | fill( | + | for (int i=0; i<6; ++i) { fill(shift(S+2*A[i])*p,grey2);} |
− | draw( | + | draw(shift(S)*scale(3.25)*hex,black+linewidth(1.25)); |
− | + | pair T = 16A[0]+4A[1]; | |
− | fill( | + | fill(shift(T)*p,grey1); |
− | for (int i = 0; i < 6; ++i) { | + | for (int i=0; i<6; ++i) { |
− | fill( | + | fill(shift(T+2*A[i])*p,grey2); |
− | fill( | + | fill(shift(T+4*A[i])*p,grey1); |
− | fill( | + | fill(shift(T+2*A[i]+2*A[i+1])*p,grey1); |
− | |||
} | } | ||
+ | draw(shift(T)*scale(5.25)*hex,black+linewidth(1.25)); | ||
</asy> | </asy> | ||
− | |||
− | |||
<math>\textbf{(A) }35 \qquad \textbf{(B) }37 \qquad \textbf{(C) }39 \qquad \textbf{(D) }43 \qquad \textbf{(E) }49</math> | <math>\textbf{(A) }35 \qquad \textbf{(B) }37 \qquad \textbf{(C) }39 \qquad \textbf{(D) }43 \qquad \textbf{(E) }49</math> | ||
Line 57: | Line 53: | ||
==Problem 5== | ==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? | + | 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> | <math>\textbf{(A) }5 \qquad \textbf{(B) }10 \qquad \textbf{(C) }15 \qquad \textbf{(D) }20 \qquad \textbf{(E) }25</math> | ||
Line 87: | Line 83: | ||
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? | 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> | <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]] | [[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}); | ||
+ | |||
+ | 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); | ||
+ | |||
+ | // axis | ||
+ | draw((0,0)--(0,9300), linewidth(1.25)); | ||
+ | draw((0,0)--(11550,0), linewidth(1.25)); | ||
+ | </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>\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]] | ||
+ | |||
+ | ==See Also== | ||
+ | {{AMC8 box|year=2020|before=[[2019 AMC 8 Problems|2019 AMC 8]]|after=[[2022 AMC 8 Problems|2022 AMC 8]]}} | ||
+ | * [[AMC 8]] | ||
+ | * [[AMC 8 Problems and Solutions]] | ||
+ | * [[Mathematics competition resources|Mathematics Competition Resources]] |
Latest revision as of 23:06, 23 August 2024
2020 AMC 8 (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 |
Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 See Also
Problem 1
Luka is making lemonade to sell at a school fundraiser. His recipe requires times as much water as sugar and twice as much sugar as lemon juice. He uses
cups of lemon juice. How many cups of water does he need?
Problem 2
Four friends do yardwork for their neighbors over the weekend, earning and
respectively. They decide to split their earnings equally among themselves. In total how much will the friend who earned
give to the others?
Problem 3
Carrie has a rectangular garden that measures feet by
feet. She plants the entire garden with strawberry plants. Carrie is able to plant
strawberry plants per square foot, and she harvests an average of
strawberries per plant. How many strawberries can she expect to harvest?
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?
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 cups. What percent of the total capacity of the pitcher did each cup receive?
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?
Problem 7
How many integers between and
have four distinct digits arranged in increasing order? (For example,
is one integer.)
Problem 8
Ricardo has coins, some of which are pennies (
-cent coins) and the rest of which are nickels (
-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?
Problem 9
Akash's birthday cake is in the form of a 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
smaller cubes, each measuring
inch, as shown below. How many small pieces will have icing on exactly two sides?
Problem 10
Zara has a collection of 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?
Problem 11
After school, Maya and Naomi headed to the beach, 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?
Problem 12
For a positive integer the factorial notation
represents the product of the integers
from
to
. (For example,
.) What value of
satisfies the following equation?
Problem 13
Jamal has a drawer containing green socks,
purple socks, and
orange socks. After adding more purple socks, Jamal noticed that there is now a
chance that a sock randomly selected from the drawer is purple. How many purple socks did Jamal add?
Problem 14
There are 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
cities?
Problem 15
Suppose of
equals
of
What percentage of
is
Problem 16
Each of the points and
in the figure below represents a different digit from
to
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
What is the digit represented by B?
Problem 17
How many factors of have more than
factors? (As an example,
has
factors, namely
and
)
Problem 18
Rectangle is inscribed in a semicircle with diameter
as shown in the figure. Let
and let
What is the area of
Problem 19
A number is called flippy if its digits alternate between two distinct digits. For example, and
are flippy, but
and
are not. How many five-digit flippy numbers are divisible by
Problem 20
A scientist walking through a forest recorded as integers the heights of 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?
Problem 21
A game board consists of squares that alternate in color between black and white. The figure below shows square
in the bottom row and square
in the top row. A marker is placed at
A step consists of moving the marker onto one of the adjoining white squares in the row above. How many
-step paths are there from
to
(The figure shows a sample path.)
Problem 22
When a positive integer is fed into a machine, the output is a number calculated according to the rule shown below.
For example, starting with an input of the machine will output
Then if the output is repeatedly inserted into the machine five more times, the final output is
When the same
-step process is applied to a different starting value of
the final output is
What is the sum of all such integers
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?
Problem 24
A large square region is paved with gray square tiles, each measuring
inches on a side. A border
inches wide surrounds each tile. The figure below shows the case for
. When
, the
gray tiles cover
of the area of the large square region. What is the ratio
for this larger value of
Problem 25
Rectangles and
and squares
and
shown below, combine to form a rectangle that is 3322 units wide and 2020 units high. What is the side length of
in units?
See Also
2020 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by 2019 AMC 8 |
Followed by 2022 AMC 8 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AJHSME/AMC 8 Problems and Solutions |