Difference between revisions of "2019 AMC 8 Problems"
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− | == Problem 1 == | + | {{AMC8 Problems|year=2019|}} |
− | + | ==Problem 1== | |
+ | |||
+ | Ike and Mike go into a sandwich shop with a total of <math>\$30.00</math> to spend. Sandwiches cost <math>\$4.50</math> each and soft drinks cost <math>\$1.00</math> each. Ike and Mike plan to buy as many sandwiches as they can, and use any remaining money to buy soft drinks. Counting both sandwiches and soft drinks, how many items will they buy? | ||
+ | |||
<math>\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }10</math> | <math>\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }10</math> | ||
Line 7: | Line 10: | ||
== Problem 2 == | == Problem 2 == | ||
− | Three identical rectangles are | + | Three identical rectangles are put together to form rectangle <math>ABCD</math>, as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles is 5 feet, what is the area in square feet of rectangle <math>ABCD</math>? |
+ | |||
<asy> | <asy> | ||
draw((0,0)--(3,0)); | draw((0,0)--(3,0)); | ||
Line 29: | Line 33: | ||
[[2019 AMC 8 Problems/Problem 2|Solution]] | [[2019 AMC 8 Problems/Problem 2|Solution]] | ||
− | |||
− | |||
− | <math>\textbf{(A) }\frac{15}{11} < \frac{17}{13} < \frac{19}{15}\qquad\textbf{(B) }\frac{15}{11} < \frac{19}{15} < \frac{17}{13}\qquad\textbf{(C) }\frac{17}{13} < \frac{19}{15} < \frac{15}{11}\qquad\textbf{(D) }\frac{19}{15} < \frac{15}{11} < \frac{17}{13}\qquad\textbf{(E) }\frac{19}{15} < \frac{17}{13} < \frac{15}{11}</math> | + | ==Problem 3== |
+ | Which of the following is the correct order of the fractions <math>\frac{15}{11},\frac{19}{15},</math> and <math>\frac{17}{13},</math> from least to greatest? | ||
+ | |||
+ | <math>\textbf{(A) }\frac{15}{11}< \frac{17}{13}< \frac{19}{15} \qquad\textbf{(B) }\frac{15}{11}< \frac{19}{15}<\frac{17}{13} \qquad\textbf{(C) }\frac{17}{13}<\frac{19}{15}<\frac{15}{11} \qquad\textbf{(D) } \frac{19}{15}<\frac{15}{11}<\frac{17}{13} \qquad\textbf{(E) } \frac{19}{15}<\frac{17}{13}<\frac{15}{11}</math> | ||
[[2019 AMC 8 Problems/Problem 3|Solution]] | [[2019 AMC 8 Problems/Problem 3|Solution]] | ||
== Problem 4 == | == Problem 4 == | ||
− | + | ||
+ | Quadrilateral <math>ABCD</math> is a rhombus with perimeter <math>52</math> meters. The length of diagonal <math>\overline{AC}</math> is <math>24</math> meters. What is the area in square meters of rhombus <math>ABCD</math>? | ||
<asy> | <asy> | ||
Line 59: | Line 65: | ||
== Problem 5 == | == Problem 5 == | ||
− | the | + | 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 <math>d</math> traveled by the two animals over time <math>t</math> from start to finish? |
− | + | <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> | ||
[[2019 AMC 8 Problems/Problem 5|Solution]] | [[2019 AMC 8 Problems/Problem 5|Solution]] | ||
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== Problem 6 == | == Problem 6 == | ||
− | + | There are <math>81</math> grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point <math>P</math> is in the center of the square. Given that point <math>Q</math> is randomly chosen among the other <math>80</math> points, what is the probability that the line <math>PQ</math> is a line of symmetry for the square? | |
+ | |||
<asy> | <asy> | ||
draw((0,0)--(0,8)); | draw((0,0)--(0,8)); | ||
Line 171: | Line 218: | ||
== Problem 7 == | == Problem 7 == | ||
− | + | Shauna takes five tests, each worth a maximum of <math>100</math> points. Her scores on the first three tests are <math>76</math>, <math>94</math>, and <math>87</math>. In order to average <math>81</math> for all five tests, what is the lowest score she could earn on one of the other two tests? | |
− | + | ||
− | |||
− | |||
− | |||
− | |||
<math>\textbf{(A) }48\qquad\textbf{(B) }52\qquad\textbf{(C) }66\qquad\textbf{(D) }70\qquad\textbf{(E) }74</math> | <math>\textbf{(A) }48\qquad\textbf{(B) }52\qquad\textbf{(C) }66\qquad\textbf{(D) }70\qquad\textbf{(E) }74</math> | ||
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== Problem 8 == | == Problem 8 == | ||
− | + | Gilda has a bag of marbles. She gives <math>20\%</math> of them to her friend Pedro. Then Gilda gives <math>10\%</math> of her original to another friend, Ebony. Finally, Gilda gives <math>25\%</math> of her original in the bag to her brother Jimmy. What percentage of her original bag of marbles does Gilda have left for herself? | |
− | + | ||
− | |||
<math>\textbf{(A) }20\qquad\textbf{(B) }33\frac{1}{3}\qquad\textbf{(C) }38\qquad\textbf{(D) }45\qquad\textbf{(E) }54</math> | <math>\textbf{(A) }20\qquad\textbf{(B) }33\frac{1}{3}\qquad\textbf{(C) }38\qquad\textbf{(D) }45\qquad\textbf{(E) }54</math> | ||
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== Problem 9 == | == Problem 9 == | ||
− | + | Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are <math>6</math> cm in diameter and <math>12</math> cm high. Felicia buys cat food in cylindrical cans that are <math>12</math> cm in diameter and <math>6</math> cm high. What is the ratio of the volume of one of Alex's cans to the volume one of Felicia's cans? | |
+ | |||
<math>\textbf{(A) }1:4\qquad\textbf{(B) }1:2\qquad\textbf{(C) }1:1\qquad\textbf{(D) }2:1\qquad\textbf{(E) }4:1</math> | <math>\textbf{(A) }1:4\qquad\textbf{(B) }1:2\qquad\textbf{(C) }1:1\qquad\textbf{(D) }2:1\qquad\textbf{(E) }4:1</math> | ||
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== Problem 11 == | == Problem 11 == | ||
− | The | + | The third-grade class at Lincoln Elementary School has <math>93</math> students. Each student takes a math class or a foreign language class or both. There are <math>70</math> third graders taking a math class, and there are <math>54</math> third graders taking a foreign language class. How many third graders take ''only'' a math class and ''not'' a foreign language class? |
<math>\textbf{(A) }16\qquad\textbf{(B) }23\qquad\textbf{(C) }31\qquad\textbf{(D) }39\qquad\textbf{(E) }70</math> | <math>\textbf{(A) }16\qquad\textbf{(B) }23\qquad\textbf{(C) }31\qquad\textbf{(D) }39\qquad\textbf{(E) }70</math> | ||
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The faces of a cube are painted in six different colors: red <math>(R)</math>, white <math>(W)</math>, green <math>(G)</math>, brown <math>(B)</math>, aqua <math>(A)</math>, and purple <math>(P)</math>. Three views of the cube are shown below. What is the color of the face opposite the aqua face? | The faces of a cube are painted in six different colors: red <math>(R)</math>, white <math>(W)</math>, green <math>(G)</math>, brown <math>(B)</math>, aqua <math>(A)</math>, and purple <math>(P)</math>. Three views of the cube are shown below. What is the color of the face opposite the aqua face? | ||
− | + | <asy> | |
+ | unitsize(2cm); | ||
+ | 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> | ||
<math>\textbf{(A) }\text{red}\qquad\textbf{(B) }\text{white}\qquad\textbf{(C) }\text{green}\qquad\textbf{(D) }\text{brown}\qquad\textbf{(E) }\text{purple}</math> | <math>\textbf{(A) }\text{red}\qquad\textbf{(B) }\text{white}\qquad\textbf{(C) }\text{green}\qquad\textbf{(D) }\text{brown}\qquad\textbf{(E) }\text{purple}</math> | ||
Line 261: | Line 332: | ||
== Problem 15 == | == Problem 15 == | ||
− | On a beach <math>50</math> people are wearing sunglasses and <math>35</math> 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 | + | On a beach <math>50</math> people are wearing sunglasses and <math>35</math> 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 <math>\frac{2}{5}</math>. If instead, someone wearing sunglasses is selected at random, what is the probability that this person is also wearing a cap? |
<math>\textbf{(A) }\frac{14}{85}\qquad\textbf{(B) }\frac{7}{25}\qquad\textbf{(C) }\frac{2}{5}\qquad\textbf{(D) }\frac{4}{7}\qquad\textbf{(E) }\frac{7}{10}</math> | <math>\textbf{(A) }\frac{14}{85}\qquad\textbf{(B) }\frac{7}{25}\qquad\textbf{(C) }\frac{2}{5}\qquad\textbf{(D) }\frac{4}{7}\qquad\textbf{(E) }\frac{7}{10}</math> | ||
Line 276: | Line 347: | ||
== Problem 17 == | == Problem 17 == | ||
What is the value of the product | What is the value of the product | ||
+ | |||
<cmath>\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)?</cmath> | <cmath>\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)?</cmath> | ||
Line 283: | Line 355: | ||
== Problem 18 == | == Problem 18 == | ||
− | The faces of each of two fair dice are numbered | + | The faces of 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? |
<math>\textbf{(A) }\frac{4}{9}\qquad\textbf{(B) }\frac{1}{2}\qquad\textbf{(C) }\frac{5}{9}\qquad\textbf{(D) }\frac{3}{5}\qquad\textbf{(E) }\frac{2}{3}</math> | <math>\textbf{(A) }\frac{4}{9}\qquad\textbf{(B) }\frac{1}{2}\qquad\textbf{(C) }\frac{5}{9}\qquad\textbf{(D) }\frac{3}{5}\qquad\textbf{(E) }\frac{2}{3}</math> | ||
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[[2019 AMC 8 Problems/Problem 19|Solution]] | [[2019 AMC 8 Problems/Problem 19|Solution]] | ||
− | == Problem 20 == | + | == Problem 20 == |
How many different real numbers <math>x</math> satisfy the equation <cmath>(x^{2}-5)^{2}=16?</cmath> | How many different real numbers <math>x</math> satisfy the equation <cmath>(x^{2}-5)^{2}=16?</cmath> | ||
Line 356: | Line 428: | ||
[[2019 AMC 8 Problems/Problem 24|Solution]] | [[2019 AMC 8 Problems/Problem 24|Solution]] | ||
− | == Problem 25 == | + | == Problem 25== |
− | Alice has | + | Alice has 24 apples. In how many ways can she share them with Becky and |
+ | Chris so that each of the three people has at least two apples? | ||
+ | |||
<math>\textbf{(A) }105\qquad\textbf{(B) }114\qquad\textbf{(C) }190\qquad\textbf{(D) }210\qquad\textbf{(E) }380</math> | <math>\textbf{(A) }105\qquad\textbf{(B) }114\qquad\textbf{(C) }190\qquad\textbf{(D) }210\qquad\textbf{(E) }380</math> | ||
− | [[2019 AMC 8 Problems/Problem 25 | + | [[2019 AMC 8 Problems/Problem 25|Solution]] |
− | |||
− | |||
− | + | ==See Also== | |
+ | {{AMC8 box|year=2019|before=[[2018 AMC 8 Problems|2018 AMC 8]]|after=[[2020 AMC 8 Problems|2020 AMC 8]]}} | ||
+ | * [[AMC 8]] | ||
+ | * [[AMC 8 Problems and Solutions]] | ||
+ | * [[Mathematics competition resources|Mathematics Competition Resources]] |
Latest revision as of 15:27, 4 May 2024
2019 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
Ike and Mike go into a sandwich shop with a total of to spend. Sandwiches cost
each and soft drinks cost
each. Ike and Mike plan to buy as many sandwiches as they can, and use any remaining money to buy soft drinks. Counting both sandwiches and soft drinks, how many items will they buy?
Problem 2
Three identical rectangles are put together to form rectangle , as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles is 5 feet, what is the area in square feet of rectangle
?
Problem 3
Which of the following is the correct order of the fractions and
from least to greatest?
Problem 4
Quadrilateral is a rhombus with perimeter
meters. The length of diagonal
is
meters. What is the area in square meters of rhombus
?
Problem 5
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 traveled by the two animals over time
from start to finish?
Problem 6
There are grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point
is in the center of the square. Given that point
is randomly chosen among the other
points, what is the probability that the line
is a line of symmetry for the square?
Problem 7
Shauna takes five tests, each worth a maximum of points. Her scores on the first three tests are
,
, and
. In order to average
for all five tests, what is the lowest score she could earn on one of the other two tests?
Problem 8
Gilda has a bag of marbles. She gives of them to her friend Pedro. Then Gilda gives
of her original to another friend, Ebony. Finally, Gilda gives
of her original in the bag to her brother Jimmy. What percentage of her original bag of marbles does Gilda have left for herself?
Problem 9
Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are cm in diameter and
cm high. Felicia buys cat food in cylindrical cans that are
cm in diameter and
cm high. What is the ratio of the volume of one of Alex's cans to the volume one of Felicia's cans?
Problem 10
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 participants on Wednesday. Which of the following statements describes the change in the mean and median after the correction is made?
The mean increases by
and the median does not change.
The mean increases by
and the median increases by
.
The mean increases by
and the median increases by
.
The mean increases by
and the median increases by
.
The mean increases by
and the median increases by
.
Problem 11
The third-grade class at Lincoln Elementary School has students. Each student takes a math class or a foreign language class or both. There are
third graders taking a math class, and there are
third graders taking a foreign language class. How many third graders take only a math class and not a foreign language class?
Problem 12
The faces of a cube are painted in six different colors: red , white
, green
, brown
, aqua
, and purple
. Three views of the cube are shown below. What is the color of the face opposite the aqua face?
Problem 13
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 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
?
Problem 14
Isabella has 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
days until she has used them all. She knows that Pete's is closed on Sundays, but as she circles the
dates on her calendar, she realizes that no circled date falls on a Sunday. On what day of the week does Isabella redeem her first coupon?
Problem 15
On a beach people are wearing sunglasses and
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
. If instead, someone wearing sunglasses is selected at random, what is the probability that this person is also wearing a cap?
Problem 16
Qiang drives miles at an average speed of
miles per hour. How many additional miles will he have to drive at
miles per hour to average
miles per hour for the entire trip?
Problem 17
What is the value of the product
Problem 18
The faces of 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?
Problem 19
In a tournament there are six teams that play each other twice. A team earns points for a win,
point for a draw, and
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?
Problem 20
How many different real numbers satisfy the equation
Problem 21
What is the area of the triangle formed by the lines ,
, and
?
Problem 22
A store increased the original price of a shirt by a certain percent and then decreased the new price by the same amount. Given that the resulting price was of the original price, by what percent was the price increased and decreased?
Problem 23
After Euclid High School's last basketball game, it was determined that of the team's points were scored by Alexa and
were scored by Brittany. Chelsea scored
points. None of the other
team members scored more than
points. What was the total number of points scored by the other
team members?
Problem 24
In triangle , point
divides side
so that
. Let
be the midpoint of
and let
be the point of intersection of line
and line
. Given that the area of
is
, what is the area of
?
Problem 25
Alice has 24 apples. In how many ways can she share them with Becky and Chris so that each of the three people has at least two apples?
See Also
2019 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by 2018 AMC 8 |
Followed by 2020 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 |