2019 AMC 8

Part of AMC 8

Subcontests

(25)

1

Coloring a Cube

The faces of a cube are painted in six different colors: red (R), white (W), green (G), brown (B), aqua (A), and purple (P). Three views of the cube are shown below. What is the color of the face opposite the aqua face?

\text{<span class='latex-bold'>(A) </span>red}\qquad \text{<span class='latex-bold'>(B) </span> white}\qquad\text{<span class='latex-bold'>(C) </span>green}\qquad\text{<span class='latex-bold'>(D) </span>brown }\qquad\text{<span class='latex-bold'>(E)</span> purple}

[asy] unitsize(2 cm);
pair x, y, z, trans; int i;
x = dir(-5); y = (0.6,0.5); z = (0,1); trans = (2,0);
for (i = 0; i <= 2; ++i) { draw(shift(i*trans)*((0,0)--x--(x + y)--(x + y + z)--(y + z)--z--cycle)); draw(shift(i*trans)*((x + z)--x)); draw(shift(i*trans)*((x + z)--(x + y + z))); draw(shift(i*trans)*((x + z)--z)); }
label(rotate(-3)*"

R

", (x + z)/2); label(rotate(-5)*slant(0.5)*"

B

", ((x + z) + (y + z))/2); label(rotate(35)*slant(0.5)*"

G

", ((x + z) + (x + y))/2);
label(rotate(-3)*"

W

", (x + z)/2 + trans); label(rotate(50)*slant(-1)*"

B

", ((x + z) + (y + z))/2 + trans); label(rotate(35)*slant(0.5)*"

R

", ((x + z) + (x + y))/2 + trans);
label(rotate(-3)*"

P

", (x + z)/2 + 2*trans); label(rotate(-5)*slant(0.5)*"

R

", ((x + z) + (y + z))/2 + 2*trans); label(rotate(-85)*slant(-1)*"

G

", ((x + z) + (x + y))/2 + 2*trans); [/asy]

1

Soccer Practice

The diagram shows the number of students at soccer practice each weekday during last week. After computing the mean and median values, Coach discovers that there were actually

21

participants on Wednesday. Which of the following statements describes the change in the mean and median after the correction is made?
[asy] unitsize(1 cm);
real unitwidth, dayheight, barheight; int i;
unitwidth = 0.5; dayheight = 1; barheight = 0.3;
draw((unitwidth,0)--(unitwidth,5*dayheight),gray(0.7)); draw((2*unitwidth,0)--(2*unitwidth,5*dayheight),gray(0.7)); draw((3*unitwidth,0)--(3*unitwidth,5*dayheight),gray(0.7)); draw((4*unitwidth,0)--(4*unitwidth,5*dayheight),gray(0.7)); draw((5*unitwidth,0)--(5*unitwidth,5*dayheight),gray(0.7)); draw((6*unitwidth,0)--(6*unitwidth,5*dayheight),gray(0.7)); draw((7*unitwidth,0)--(7*unitwidth,5*dayheight),gray(0.7));
fill((0,1/2*dayheight - 1/2*barheight)--(8*unitwidth,1/2*dayheight - 1/2*barheight)--(8*unitwidth,1/2*dayheight + 1/2*barheight)--(0,1/2*dayheight + 1/2*barheight)--cycle,gray(0.5));
fill((0,5/2*dayheight - 1/2*barheight)--(8*unitwidth,5/2*dayheight - 1/2*barheight)--(8*unitwidth,5/2*dayheight + 1/2*barheight)--(0,5/2*dayheight + 1/2*barheight)--cycle,gray(0.5));
draw((8*unitwidth,0)--(8*unitwidth,5*dayheight),gray(0.7)); draw((9*unitwidth,0)--(9*unitwidth,5*dayheight),gray(0.7));
fill((0,9/2*dayheight - 1/2*barheight)--(10*unitwidth,9/2*dayheight - 1/2*barheight)--(10*unitwidth,9/2*dayheight + 1/2*barheight)--(0,9/2*dayheight + 1/2*barheight)--cycle,gray(0.5));
draw((10*unitwidth,0)--(10*unitwidth,5*dayheight),gray(0.7));
fill((0,3/2*dayheight - 1/2*barheight)--(11*unitwidth,3/2*dayheight - 1/2*barheight)--(11*unitwidth,3/2*dayheight + 1/2*barheight)--(0,3/2*dayheight + 1/2*barheight)--cycle,gray(0.5));
draw((11*unitwidth,0)--(11*unitwidth,5*dayheight),gray(0.7)); draw((12*unitwidth,0)--(12*unitwidth,5*dayheight),gray(0.7));
fill((0,7/2*dayheight - 1/2*barheight)--(13*unitwidth,7/2*dayheight - 1/2*barheight)--(13*unitwidth,7/2*dayheight + 1/2*barheight)--(0,7/2*dayheight + 1/2*barheight)--cycle,gray(0.5));
draw((0*unitwidth,0)--(0*unitwidth,5*dayheight),gray(0.7)); draw((13*unitwidth,0)--(13*unitwidth,5*dayheight),gray(0.7)); draw((14*unitwidth,0)--(14*unitwidth,5*dayheight),gray(0.7));
label("

0

", (0,5*dayheight), N); label("

4

", (2*unitwidth,5*dayheight), N); label("

8

", (4*unitwidth,5*dayheight), N); label("

12

", (6*unitwidth,5*dayheight), N); label("

16

", (8*unitwidth,5*dayheight), N); label("

20

", (10*unitwidth,5*dayheight), N); label("

24

", (12*unitwidth,5*dayheight), N); label("

28

", (14*unitwidth,5*dayheight), N); label("Number of students at soccer practice", (7*unitwidth,6*dayheight)); label("Monday", (-0.5*unitwidth,9/2*dayheight), W); label("Tuesday", (-0.5*unitwidth,7/2*dayheight), W); label("Wednesday", (-0.5*unitwidth,5/2*dayheight), W); label("Thursday", (-0.5*unitwidth,3/2*dayheight), W); label("Friday", (-0.5*unitwidth,1/2*dayheight), W); [/asy]

<span class='latex-bold'>(A) </span> \text{The mean increases by 1 and the median does not change.}

<span class='latex-bold'>(B) </span> \text{The mean increases by 1 and the median increases by 1.}

<span class='latex-bold'>(C) </span> \text{The mean increases by 1 and the median increases by 5.}

<span class='latex-bold'>(D) </span> \text{The mean increases by 5 and the median increases by 1.}

<span class='latex-bold'>(E) </span> \text{The mean increases by 5 and the median increases by 5.}

1

Three Congruent Rectangles

Three identical rectangles are put together to form rectangle

ABCD

, as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles

5

feet, what is the area in square feet of rectangle

ABCD

? [asy]draw((0,0)--(0,10)--(15,10)--(15,0)--(0,0)); draw((0,5)--(10,5)); draw((10,0)--(10,10)); label("

A

",(0,0),SW); label("

B

",(15,0),SE); label("

C

",(15,10),NE); label("

D

",(0,10),NW); dot((0,10)); dot((15,0)); dot((15,10)); dot((0,0)); [/asy]

<span class='latex-bold'>(A) </span>45\qquad <span class='latex-bold'>(B) </span>75\qquad <span class='latex-bold'>(C) </span>100\qquad <span class='latex-bold'>(D) </span>125\qquad <span class='latex-bold'>(E) </span>150\qquad

1

Lincoln MS Schedules

The eighth grade class at Lincoln Middle School has

93

students. Each student takes a math class or a foreign language class or both. There are

70

eighth graders taking a math class, and there are

54

eighth graders taking a foreign language class. How many eighth graders take only a math class and not a foreign language class?

<span class='latex-bold'>(A) </span>16\qquad <span class='latex-bold'>(B) </span>23\qquad <span class='latex-bold'>(C) </span>31\qquad <span class='latex-bold'>(D) </span>39\qquad <span class='latex-bold'>(E) </span>70\qquad

1

Tortoise and hare

A tortoise challenges a hare to a race. The hare eagerly agrees and quickly runs ahead, leaving the slow-moving tortoise behind. Confident that he will win, the hare stops to take a nap. Meanwhile, the tortoise walks at a slow steady pace for the entire race. The hare awakes and runs to the finish line, only to find the tortoise already there. Which of the following graphs matches the description of the race, showing the distance

d

traveled by the two animals over time

t

from start to finish?

\phantom{h}

[asy] unitsize(0.4 cm);
pair transx, transy; int i, j; int x, y;
transx = (13,0); transy = (0,-9);
for (i = 0; i <= 2; ++i) { for (j = 0; j <= 1; ++j) { if (i <= 1 || j <= 0) { for (x = 1; x <= 10; ++x) { draw(shift(i*transx + j*transy)*((x,0)--(x,5)),gray(0.7) + dashed); } for (y = 1; y <= 5; ++y) { draw(shift(i*transx + j*transy)*((0,y)--(10,y)),gray(0.7) + dashed); } draw(shift(i*transx + j*transy)*((0,0)--(11,0)),Arrow(6)); draw(shift(i*transx + j*transy)*((0,0)--(0,6)),Arrow(6)); label("time", (5,-0.5) + i*transx + j*transy); label(rotate(90)*"distance", (-0.5,2.5) + i*transx + j*transy); } }}
draw((0,0)--(1.5,2.5)--(7.5,2.5)--(9,5),linewidth(1.5*bp)); draw((0,0)--(10,5),linewidth(1.5*bp)); draw(shift(transx)*((0,0)--(2.5,2.5)--(7.5,2.5)--(10,5)),linewidth(1.5*bp)); draw(shift(transx)*((0,0)--(9,5)),linewidth(1.5*bp)); draw(shift(2*transx)*((0,0)--(2.5,3)--(7,2)--(10,5)),linewidth(1.5*bp)); draw(shift(2*transx)*((0,0)--(9,5)),linewidth(1.5*bp)); draw(shift(transy)*((0,0)--(2.5,2.5)--(6.5,2.5)--(9,5)),linewidth(1.5*bp)); draw(shift(transy)*((0,0)--(7.5,2)--(10,5)),linewidth(1.5*bp)); draw(shift(transx + transy)*((0,0)--(2.5,2)--(7.5,3)--(10,5)),linewidth(1.5*bp)); draw(shift(transx + transy)*((0,0)--(9,5)),linewidth(1.5*bp));
label("(A)", (-1,6)); label("(B)", (-1,6) + transx); label("(C)", (-1,6) + 2*transx); label("(D)", (-1,6) + transy); label("(E)", (-1,6) + transx + transy); [/asy]

1

Many dots

There are 81 grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point

P

is the center of the square. Given that point

Q

is randomly chosen from among the other 80 points, what is the probability that line

PQ

is a line of symmetry for the square? [asy] size(130); defaultpen(fontsize(11)); int i, j; for(i=0; i<9; i=i+1) { for(j=0; j<9; j=j+1) if((i==4) && (j==4)) { dot((i,j),linewidth(5)); } else { dot((i,j),linewidth(3)); } } dot("

P

",(4,4),NE); draw((0,0)--(0,8)--(8,8)--(8,0)--cycle); [/asy]

<span class='latex-bold'>(A) </span> \frac{1}{5} \qquad<span class='latex-bold'>(B) </span> \frac{1}{4} \qquad<span class='latex-bold'>(C) </span> \frac{2}{5} \qquad<span class='latex-bold'>(D) </span> \frac{9}{20} \qquad<span class='latex-bold'>(E) </span> \frac{1}{2}

1

Rhombus on #4?!?

ABCD

is a rhombus with perimeter

52

meters. The length of diagonal

\overline{AC}

24

meters. What is the area in square meters of rhombus

ABCD

? [asy] unitsize(1cm); draw((0,1)--(2,2)--(4,1)--(2,0)--cycle); dot("

A

",(0,1),W); dot("

D

",(2,2),N); dot("

C

",(4,1),E); dot("

B

",(2,0),S); [/asy]

<span class='latex-bold'>(A) </span> 60 \qquad<span class='latex-bold'>(B) </span> 90 \qquad<span class='latex-bold'>(C) </span> 105 \qquad<span class='latex-bold'>(D) </span> 120 \qquad<span class='latex-bold'>(E) </span> 144

1

Sunglasses and Caps

On a beach 50 people are wearing sunglasses and 35 people are wearing caps. Some people are wearing both sunglasses and caps. If one of the people wearing a cap is selected at random, the probability that this person is also wearing sunglasses is

\frac{2}{5}

. If instead, someone wearing sunglasses is selected at random, what is the probability that this person is also wearing a cap?

<span class='latex-bold'>(A) </span>\frac{14}{85}\qquad<span class='latex-bold'>(B) </span>\frac{7}{25}\qquad<span class='latex-bold'>(C) </span>\frac{2}{5}\qquad<span class='latex-bold'>(D) </span>\frac{4}{7}\qquad<span class='latex-bold'>(E) </span>\frac{7}{10}

1

Ice Cream for Isabella!

6

coupons that can be redeemed for free ice cream cones at Pete's Sweet Treats. In order to make the coupons last, she decides that she will redeem one every 10 days until she has used them all. She knows that Pete's is closed on Sundays, but as she circles the 6 dates on her calender, she realizes that no circled date falls on a Sunday. On what day of the week does Isabella redeem her first coupon?

<span class='latex-bold'>(A) </span>\text{Monday}\qquad<span class='latex-bold'>(B) </span>\text{Tuesday}\qquad<span class='latex-bold'>(C) </span>\text{Wednesday}\qquad<span class='latex-bold'>(D) </span>\text{Thursday}\qquad<span class='latex-bold'>(E) </span>\text{Friday}

1

Three-Digit Sum of Palindromes

A palindrome is a number that has the same value when read from left to right or from right to left. (For example, 12321 is a palindrome). Let

N

be the least three-digit integer which is not a palindrome but which is the sum of three distinct two-digit palindromes. What is the sum of the digits of

N

?

<span class='latex-bold'>(A) </span>2\qquad<span class='latex-bold'>(B) </span>3\qquad<span class='latex-bold'>(C) </span>4\qquad<span class='latex-bold'>(D) </span>5\qquad<span class='latex-bold'>(E) </span>6

1

Cat Food Party

Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are 6 cm in diameter and 12 cm high. Felicia buys cat food in cylindrical cans that are 12 cm in diameter and 6 cm high. What is the ratio of the volume of one of Alex's cans to the volume of one of Felicia's cans?

<span class='latex-bold'>(A) </span> 1:4 \qquad<span class='latex-bold'>(B) </span>1:2\qquad<span class='latex-bold'>(C) </span>1:1\qquad<span class='latex-bold'>(D) </span>2:1\qquad<span class='latex-bold'>(E) </span>4:1

1

Shirt Price Magic

A store increased the original price of a shirt by a certain percent and then decreased the new price by the same percent. Given that the resulting price was 84% of the original price, by what percent was the price increased and decreased?

<span class='latex-bold'>(A) </span>16\qquad<span class='latex-bold'>(B) </span>20\qquad<span class='latex-bold'>(C) </span>28\qquad<span class='latex-bold'>(D) </span>36\qquad<span class='latex-bold'>(E) </span>40

1

Alexa, Score Some Points

After Euclid High School's last basketball game, it was determined that

\frac{1}{4}

of the team's points were scored by Alexa and

\frac{2}{7}

were scored by Brittany. Chelsea scored 15 points. None of the other 7 team members scored more than 2 points. What was the total number of points scored by the other 7 team members?

<span class='latex-bold'>(A) </span>10\qquad<span class='latex-bold'>(B) </span>11\qquad<span class='latex-bold'>(C) </span>12\qquad<span class='latex-bold'>(D) </span>13\qquad<span class='latex-bold'>(E) </span>14

1

The Team Tournament

In a tournament there are six teams that play each other twice. A team earns 3 points for a win, 1 point for a draw, and 0 points for a loss. After all the games have been played it turns out that the top three teams earned the same number of total points. What is the greatest possible number of total points for each of the top three teams?

<span class='latex-bold'>(A) </span>22\qquad<span class='latex-bold'>(B) </span>23\qquad<span class='latex-bold'>(C) </span>24\qquad<span class='latex-bold'>(D) </span>26\qquad<span class='latex-bold'>(E) </span>30

1

Telescoping Product

What is the value of the product

\left(\frac{1\cdot3}{2\cdot2}\right)\left(\frac{2\cdot4}{3\cdot3}\right)\left(\frac{3\cdot5}{4\cdot4}\right)\cdots\left(\frac{97\cdot99}{98\cdot98}\right)\left(\frac{98\cdot100}{99\cdot99}\right)?

<span class='latex-bold'>(A) </span>\frac{1}{2}\qquad<span class='latex-bold'>(B) </span>\frac{50}{99}\qquad<span class='latex-bold'>(C) </span>\frac{9800}{9801}\qquad<span class='latex-bold'>(D) </span>\frac{100}{99}\qquad<span class='latex-bold'>(E) </span> 50

1

Squares of a Square

How many different real numbers

x

satisfy the equation

(x^2-5)^2=16?

<span class='latex-bold'>(A) </span>0\qquad<span class='latex-bold'>(B) </span>1\qquad<span class='latex-bold'>(C) </span>2\qquad<span class='latex-bold'>(D) </span>4\qquad<span class='latex-bold'>(E) </span>8

1

Triangle Area in the Plane

What is the area of the triangle formed by the lines

y=5

y=1+x

y=1-x

?

<span class='latex-bold'>(A) </span>4\qquad<span class='latex-bold'>(B) </span>8\qquad<span class='latex-bold'>(C) </span>10\qquad<span class='latex-bold'>(D) </span>12\qquad<span class='latex-bold'>(E) </span>16

1

The Trip

Qiang drives 15 miles at an average speed of 30 miles per hour. How many additional miles will he have to drive at 55 miles per hour to average 50 miles per hour for the entire trip?

<span class='latex-bold'>(A) </span>45\qquad<span class='latex-bold'>(B) </span>62\qquad<span class='latex-bold'>(C) </span>90\qquad<span class='latex-bold'>(D) </span>110\qquad<span class='latex-bold'>(E) </span>135

1

The Dice of MAA

The faces on each of two fair dice are numbered 1, 2, 3, 5, 7, and 8. When the two dice are tossed, what is the probability that their sum will be an even number?

<span class='latex-bold'>(A) </span>\frac{4}{9}\qquad<span class='latex-bold'>(B) </span>\frac{1}{2}\qquad<span class='latex-bold'>(C) </span>\frac{5}{9}\qquad<span class='latex-bold'>(D) </span>\frac{3}{5}\qquad<span class='latex-bold'>(E) </span>\frac{2}{3}

1

Fractions......

Which of the following is the correct order of the fractions

\frac{15}{11}, \frac{19}{15}

\frac{17}{13}

, from least to greatest?

<span class='latex-bold'>(A) </span> \frac{15}{11} < \frac{17}{13} < \frac{19}{15} \qquad<span class='latex-bold'>(B) </span> \frac{15}{11} < \frac{19}{15} < \frac{17}{13} \qquad<span class='latex-bold'>(C) </span> \frac{17}{13} < \frac{19}{15} < \frac{15}{11} \newline\newline \qquad<span class='latex-bold'>(D) </span> \frac{19}{15} < \frac{15}{11} < \frac{17}{13} \qquad<span class='latex-bold'>(E) </span> \frac{19}{15} < \frac{17}{13} < \frac{15}{11}

1

Marbles?

Gilda has a bag of marbles. She gives

20 \%

of them to her friend Pedro. The, Gilda gives

10 \%

of what is left to her other friend, Ebony. Finally, Gilda gives

25 \%

of what is left in the bag to her brother. What percentage of her original bag does she have left?

<span class='latex-bold'>(A) </span> 20 \qquad<span class='latex-bold'>(B) </span> 33\tfrac{1}{3} \qquad<span class='latex-bold'>(C) </span> 38 \qquad<span class='latex-bold'>(D) </span> 45 \qquad<span class='latex-bold'>(E) </span> 54

1

Apples and Bars

Alice has 24 apples. In how many ways can she share them with Becky and Chris so that each of the people has at least 2 apples?

<span class='latex-bold'>(A) </span>105\qquad<span class='latex-bold'>(B) </span>114\qquad<span class='latex-bold'>(C) </span>190\qquad<span class='latex-bold'>(D) </span>210\qquad<span class='latex-bold'>(E) </span>380

1

Averaging........

5

tests, each worth a maximum of a

100

points. Her scores on the first three tests were

76

94

87

. In order to average an

81

on all five tests, what is the lowest score she could earn on one of the two tests?

<span class='latex-bold'>(A) </span> 48 \qquad<span class='latex-bold'>(B) </span> 52 \qquad<span class='latex-bold'>(C) </span> 66 \qquad<span class='latex-bold'>(D) </span> 70 \qquad<span class='latex-bold'>(E) </span> 74

1

Medians.......

ABC

D

\overline{AC}

AD:DC=1:2

E

be the midpoint of

\overline{BD}

F

be the point of intersection of line

BC

AE

. Given that the area of

\triangle ABC

360

, what is the area of

\triangle EBF

? [asy] unitsize(1.5cm); pair A,B,C,DD,EE,FF; B = (0,0); C = (3,0); A = (1.2,1.7); DD = (2/3)*A+(1/3)*C; EE = (B+DD)/2; FF = intersectionpoint(B--C,A--A+2*(EE-A)); draw(A--B--C--cycle); draw(A--FF); draw(B--DD);dot(A); label("

A

",A,N); dot(B); label("

B

", B,SW);dot(C); label("

C

",C,SE); dot(DD); label("

D

",DD,NE); dot(EE); label("

E

",EE,NW); dot(FF); label("

F

",FF,S); [/asy]

<span class='latex-bold'>(A) </span>24\qquad<span class='latex-bold'>(B) </span>30\qquad<span class='latex-bold'>(C) </span>32\qquad<span class='latex-bold'>(D) </span>36\qquad<span class='latex-bold'>(E) </span>40

1

Ike and Mike's Sandwich Adventure

Ike and Mike go into a sandwich shop with a total of

\$30.00

to spend. Sandwiches cost

\$4.50

each and soft drinks cost

\$1.00

each. Ike and Mike plan to buy as many sandwiches as they can and use any remaining money to buy soft drinks. Counting both soft drinks and sandwiches, how many items will they buy?

<span class='latex-bold'>(A) </span> 6 \qquad<span class='latex-bold'>(B) </span>7\qquad<span class='latex-bold'>(C) </span>8\qquad<span class='latex-bold'>(D) </span>9\qquad<span class='latex-bold'>(E) </span>10