2022 AMC 8 -
Part of AMC 8
Subcontests
(25)Mistake In Grading (2022 AMC 8 P19)
Mr. Ramos gave a test to his class of 20 students. The dot plot below shows the distribution of test scores.
[asy]
//diagram by pog . give me 1,000,000,000 dollars for this diagram
size(5cm);
defaultpen(0.7);
dot((0.5,1));
dot((0.5,1.5));
dot((1.5,1));
dot((1.5,1.5));
dot((2.5,1));
dot((2.5,1.5));
dot((2.5,2));
dot((2.5,2.5));
dot((3.5,1));
dot((3.5,1.5));
dot((3.5,2));
dot((3.5,2.5));
dot((3.5,3));
dot((4.5,1));
dot((4.5,1.5));
dot((5.5,1));
dot((5.5,1.5));
dot((5.5,2));
dot((6.5,1));
dot((7.5,1));
draw((0,0.5)--(8,0.5),linewidth(0.7));
defaultpen(fontsize(10.5pt));
label("65", (0.5,-0.1));
label("70", (1.5,-0.1));
label("75", (2.5,-0.1));
label("80", (3.5,-0.1));
label("85", (4.5,-0.1));
label("90", (5.5,-0.1));
label("95", (6.5,-0.1));
label("100", (7.5,-0.1));
[/asy]
Later Mr. Ramos discovered that there was a scoring error on one of the questions. He regraded the tests, awarding some of the students 5 extra points, which increased the median test score to 85. What is the minimum number of students who received extra points?(Note that the median test score equals the average of the 2 scores in the middle if the 20 test scores are arranged in increasing order.)<spanclass=′latex−bold′>(A)</span> 2<spanclass=′latex−bold′>(B)</span> 3<spanclass=′latex−bold′>(C)</span> 4<spanclass=′latex−bold′>(D)</span> 5<spanclass=′latex−bold′>(E)</span> 6 Reflection over 2 Lines (2022 AMC 8 P4)
The letter M in the figure below is first reflected over the line q and then reflected over the line p. What is the resulting image?
[asy]
// pog diagram
usepackage("newtxtext");
size(3cm);
draw((-1,0)--(1,0)); draw((0,-1)--(0,1)); label("<spanclass=′latex−bold′>M</span>",(0.25,0.6));
draw((-0.8,-0.8)--(0.8,0.8),linewidth(1.1)); label("p", (-1,0),NE); label("q", (-0.75,-0.75), N*1.5);
[/asy][asy]
// pog diagram
usepackage("newtxtext");
size(12.5cm);
draw((-1,0)--(1,0)); draw((0,-1)--(0,1)); label(rotate(90)*"<spanclass=′latex−bold′>M</span>",(0.6,-0.25));
draw((-0.8,-0.8)--(0.8,0.8),linewidth(1.1));
label("<spanclass=′latex−bold′>(A)</span>",(-1,1),W);
draw((2,0)--(4,0)); draw((3,-1)--(3,1)); label(rotate(270)*"<spanclass=′latex−bold′>M</span>",(2.8,0.7));
draw((2.2,-0.8)--(3.8,0.8),linewidth(1.1));
label("<spanclass=′latex−bold′>(B)</span>",(2,1),W);
draw((5,0)--(7,0)); draw((6,-1)--(6,1)); label(rotate(90)*"<spanclass=′latex−bold′>M</span>",(5.4,0.2));
draw((5.2,-0.8)--(6.8,0.8),linewidth(1.1));
label("<spanclass=′latex−bold′>(C)</span>",(5,1),W);
draw((-1,-2.5)--(1,-2.5)); draw((0,-3.5)--(0,-1.5)); label(rotate(180)*"<spanclass=′latex−bold′>M</span>",(-0.25,-3.1));
draw((-0.8,-3.3)--(0.8,-1.7),linewidth(1.1));
label("<spanclass=′latex−bold′>(D)</span>",(-1,-1.5),W);
draw((2,-2.5)--(4,-2.5)); draw((3,-3.5)--(3,-1.5)); label(rotate(270)*"<spanclass=′latex−bold′>M</span>",(3.6,-2.75));
draw((2.2,-3.3)--(3.8,-1.7),linewidth(1.1));
label("<spanclass=′latex−bold′>(E)</span>",(2,-1.5),W);
[/asy] Distance v.s. Time Graph (2022 AMC 8 P10)
One sunny day, Ling decided to take a hike in the mountains. She left her house at 8 \, \textsc{am}, drove at a constant speed of 45 miles per hour, and arrived at the hiking trail at 10 \, \textsc{am}. After hiking for 3 hours, Ling drove home at a constant speed of 60 miles per hour. Which of the following graphs best illustrates the distance between Ling’s car and her house over the course of her trip?
[asy]
unitsize(12);
usepackage("mathptmx");
defaultpen(fontsize(8)+linewidth(.7));
int mod12(int i) {if (i<13) {return i;} else {return i-12;}}
void drawgraph(pair sh,string lab) {
for (int i=0;i<11;++i) {
for (int j=0;j<6;++j) {
draw(shift(sh+(i,j))*unitsquare,mediumgray);
}
}
draw(shift(sh)*((-1,0)--(11,0)),EndArrow(angle=20,size=8));
draw(shift(sh)*((0,-1)--(0,6)),EndArrow(angle=20,size=8));
for (int i=1;i<10;++i) {
draw(shift(sh)*((i,-.2)--(i,.2)));
}
label("8\tiny{\textsc{am}}",sh+(1,-.2),S);for (int i=2;i<9;++i) {
label(string(mod12(i+7)),sh+(i,-.2),S);
}
label("4\tiny{\textsc{pm}}",sh+(9,-.2),S);
for (int i=1;i<6;++i) {
label(string(30*i),sh+(0,i),2*W);
}
draw(rotate(90)*"Distance (miles)",sh+(-2.1,3),fontsize(10));
label("<spanclass=′latex−bold′>("+lab+")</span>",sh+(-2.1,6.8),fontsize(10));
}
drawgraph((0,0),"A");
drawgraph((15,0),"B");
drawgraph((0,-10),"C");
drawgraph((15,-10),"D");
drawgraph((0,-20),"E");
dotfactor=6;
draw((1,0)--(3,3)--(6,3)--(8,0),linewidth(.9));
dot((1,0)^^(3,3)^^(6,3)^^(8,0));
pair sh = (15,0);
draw(shift(sh)*((1,0)--(3,1.5)--(6,1.5)--(8,0)),linewidth(.9));
dot(sh+(1,0)^^sh+(3,1.5)^^sh+(6,1.5)^^sh+(8,0));
pair sh = (0,-10);
draw(shift(sh)*((1,0)--(3,1.5)--(6,1.5)--(7.5,0)),linewidth(.9));
dot(sh+(1,0)^^sh+(3,1.5)^^sh+(6,1.5)^^sh+(7.5,0));
pair sh = (15,-10);
draw(shift(sh)*((1,0)--(3,4)--(6,4)--(9.3,0)),linewidth(.9));
dot(sh+(1,0)^^sh+(3,4)^^sh+(6,4)^^sh+(9.3,0));
pair sh = (0,-20);
draw(shift(sh)*((1,0)--(3,3)--(6,3)--(7.5,0)),linewidth(.9));
dot(sh+(1,0)^^sh+(3,3)^^sh+(6,3)^^sh+(7.5,0));
[/asy] Scatter Plot of Black Pepper (2022 AMC 8 P15)
Laszlo went online to shop for black pepper and found thirty different black pepper options varying in weight and price, shown in the scatter plot below. In ounces, what is the weight of the pepper that offers the lowest price per ounce?[asy]
//diagram by pog
size(8.5cm);
usepackage("mathptmx");
defaultpen(mediumgray*0.5+gray*0.5+linewidth(0.63));
add(grid(6,6));
label(scale(0.7)*"1", (1,-0.3), black);
label(scale(0.7)*"2", (2,-0.3), black);
label(scale(0.7)*"3", (3,-0.3), black);
label(scale(0.7)*"4", (4,-0.3), black);
label(scale(0.7)*"5", (5,-0.3), black);
label(scale(0.7)*"1", (-0.3,1), black);
label(scale(0.7)*"2", (-0.3,2), black);
label(scale(0.7)*"3", (-0.3,3), black);
label(scale(0.7)*"4", (-0.3,4), black);
label(scale(0.7)*"5", (-0.3,5), black);
label(scale(0.75)*rotate(90)*"Price (dollars)", (-1,3.2), black);
label(scale(0.75)*"Weight (ounces)", (3.2,-1), black);
dot((1,1.2),black);
dot((1,1.7),black);
dot((1,2),black);
dot((1,2.8),black);dot((1.5,2.1),black);
dot((1.5,3),black);
dot((1.5,3.3),black);
dot((1.5,3.75),black);dot((2,2),black);
dot((2,2.9),black);
dot((2,3),black);
dot((2,4),black);
dot((2,4.35),black);
dot((2,4.8),black);dot((2.5,2.7),black);
dot((2.5,3.7),black);
dot((2.5,4.2),black);
dot((2.5,4.4),black);dot((3,2.5),black);
dot((3,3.4),black);
dot((3,4.2),black);dot((3.5,3.8),black);
dot((3.5,4.5),black);
dot((3.5,4.8),black);dot((4,3.9),black);
dot((4,5.1),black);dot((4.5,4.75),black);
dot((4.5,5),black);dot((5,4.5),black);
dot((5,5),black);
[/asy]<spanclass=′latex−bold′>(A)</span> 1<spanclass=′latex−bold′>(B)</span> 2<spanclass=′latex−bold′>(C)</span> 3<spanclass=′latex−bold′>(D)</span> 4<spanclass=′latex−bold′>(E)</span> 5 A Math Team's Lame Logo (2022 AMC 8 #1)
The Math Team designed a logo shaped like a multiplication symbol, shown below on a grid of 1-inch squares. What is the area of the logo in square inches?[asy]
usepackage("mathptmx");
defaultpen(linewidth(0.5));
size(5cm);
defaultpen(fontsize(14pt));
label("<spanclass=′latex−bold′>Math</span>", (2.1,3.7)--(3.9,3.7));
label("<spanclass=′latex−bold′>Team</span>", (2.1,3)--(3.9,3));
filldraw((1,2)--(2,1)--(3,2)--(4,1)--(5,2)--(4,3)--(5,4)--(4,5)--(3,4)--(2,5)--(1,4)--(2,3)--(1,2)--cycle, mediumgray*0.5 + lightgray*0.5);
draw((0,0)--(6,0), gray);
draw((0,1)--(6,1), gray);
draw((0,2)--(6,2), gray);
draw((0,3)--(6,3), gray);
draw((0,4)--(6,4), gray);
draw((0,5)--(6,5), gray);
draw((0,6)--(6,6), gray);draw((0,0)--(0,6), gray);
draw((1,0)--(1,6), gray);
draw((2,0)--(2,6), gray);
draw((3,0)--(3,6), gray);
draw((4,0)--(4,6), gray);
draw((5,0)--(5,6), gray);
draw((6,0)--(6,6), gray);[/asy]<spanclass=′latex−bold′>(A)</span>10<spanclass=′latex−bold′>(B)</span>12<spanclass=′latex−bold′>(C)</span>13<spanclass=′latex−bold′>(D)</span>14<spanclass=′latex−bold′>(E)</span>15 Two Spinners (2022 AMC 8 P12)
The arrows on the two spinners shown below are spun. Let the number N equal 10 times the number on Spinner A, added to the number on Spinner B. What is the probability that N is a perfect square number?<spanclass=′latex−bold′>(A)</span> 161<spanclass=′latex−bold′>(B)</span> 81<spanclass=′latex−bold′>(C)</span> 41<spanclass=′latex−bold′>(D)</span> 83<spanclass=′latex−bold′>(E)</span> 21
[asy]
//diagram by pog give me 1 billion dollars for this
size(6cm);
usepackage("mathptmx");
filldraw(arc((0,0), r=4, angle1=0, angle2=90)--(0,0)--cycle,mediumgray*0.5+gray*0.5);
filldraw(arc((0,0), r=4, angle1=90, angle2=180)--(0,0)--cycle,lightgray);
filldraw(arc((0,0), r=4, angle1=180, angle2=270)--(0,0)--cycle,mediumgray);
filldraw(arc((0,0), r=4, angle1=270, angle2=360)--(0,0)--cycle,lightgray*0.5+mediumgray*0.5);
label("5", (-1.5,1.7));
label("6", (1.5,1.7));
label("7", (1.5,-1.7));
label("8", (-1.5,-1.7));
label("Spinner A", (0, -5.5));
filldraw(arc((12,0), r=4, angle1=0, angle2=90)--(12,0)--cycle,mediumgray*0.5+gray*0.5);
filldraw(arc((12,0), r=4, angle1=90, angle2=180)--(12,0)--cycle,lightgray);
filldraw(arc((12,0), r=4, angle1=180, angle2=270)--(12,0)--cycle,mediumgray);
filldraw(arc((12,0), r=4, angle1=270, angle2=360)--(12,0)--cycle,lightgray*0.5+mediumgray*0.5);
label("1", (10.5,1.7));
label("2", (13.5,1.7));
label("3", (13.5,-1.7));
label("4", (10.5,-1.7));
label("Spinner B", (12, -5.5));
[/asy]
Pigeon(hole) orz
The grid below is to be filled with integers in such a way that the sum of the numbers in each row and the sum of the numbers in each column are the same. Four numbers are missing. The number x in the lower left corner is larger than the other three missing numbers. What is the smallest possible value of x?
[asy]
unitsize(0.5cm);
draw((3,3)--(-3,3));
draw((3,1)--(-3,1));
draw((3,-3)--(-3,-3));
draw((3,-1)--(-3,-1));
draw((3,3)--(3,-3));
draw((1,3)--(1,-3));
draw((-3,3)--(-3,-3));
draw((-1,3)--(-1,-3));
label((-2,2),"−2");
label((0,2),"9");
label((2,2),"5");
label((2,0),"−1");
label((2,-2),"8");
label((-2,-2),"x");
[/asy]
<spanclass=′latex−bold′>(A)</span>−1<spanclass=′latex−bold′>(B)</span>5<spanclass=′latex−bold′>(C)</span>6<spanclass=′latex−bold′>(D)</span>8<spanclass=′latex−bold′>(E)</span>9 Candice.
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]<spanclass=′latex−bold′>(A)</span> 7<spanclass=′latex−bold′>(B)</span> 8<spanclass=′latex−bold′>(C)</span> 9<spanclass=′latex−bold′>(D)</span> 10<spanclass=′latex−bold′>(E)</span> 11 Volume of Prism Given Net (2022 AMC 8 P24)
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]
<spanclass=′latex−bold′>(A)</span> 112<spanclass=′latex−bold′>(B)</span> 128<spanclass=′latex−bold′>(C)</span> 192<spanclass=′latex−bold′>(D)</span> 240<spanclass=′latex−bold′>(E)</span> 288 Tic-tac-toe but with triangles! (2022 AMC 8 Problem #23)
A △ or ◯ is placed in each of the nine squares in a 3-by-3 grid. Shown below is a sample configuration with three △s in a line.[asy]
//diagram by kante314
size(3.3cm);
defaultpen(linewidth(1));
real r = 0.37;
path equi = r * dir(-30) -- (r+0.03) * dir(90) -- r * dir(210) -- cycle;
draw((0,0)--(0,3)--(3,3)--(3,0)--cycle);
draw((0,1)--(3,1)--(3,2)--(0,2)--cycle);
draw((1,0)--(1,3)--(2,3)--(2,0)--cycle);
draw(circle((3/2,5/2),1/3));
draw(circle((5/2,1/2),1/3));
draw(circle((3/2,3/2),1/3));
draw(shift(0.5,0.38) * equi);
draw(shift(1.5,0.38) * equi);
draw(shift(0.5,1.38) * equi);
draw(shift(2.5,1.38) * equi);
draw(shift(0.5,2.38) * equi);
draw(shift(2.5,2.38) * equi);
[/asy]How many configurations will have three △s in a line and three ◯s in a line?<spanclass=′latex−bold′>(A)</span>39<spanclass=′latex−bold′>(B)</span>42<spanclass=′latex−bold′>(C)</span>78<spanclass=′latex−bold′>(D)</span>84<spanclass=′latex−bold′>(E)</span>96 Midpoint Rectangle
The midpoints of the four sides of a rectangle are (−3,0),(2,0),(5,4) and (0,4). What is the area of the rectangle?<spanclass=′latex−bold′>(A)</span> 20<spanclass=′latex−bold′>(B)</span> 25<spanclass=′latex−bold′>(C)</span> 40<spanclass=′latex−bold′>(D)</span> 50<spanclass=′latex−bold′>(E)</span> 80