Difference between revisions of "2018 AMC 10A Problems"
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==Problem 1== | ==Problem 1== | ||
What is the value of <cmath>\left(\left((2+1)^{-1}+1\right)^{-1}+1\right)^{-1}+1?</cmath><math>\textbf{(A) } \frac58 \qquad \textbf{(B) }\frac{11}7 \qquad \textbf{(C) } \frac85 \qquad \textbf{(D) } \frac{18}{11} \qquad \textbf{(E) } \frac{15}8</math> | What is the value of <cmath>\left(\left((2+1)^{-1}+1\right)^{-1}+1\right)^{-1}+1?</cmath><math>\textbf{(A) } \frac58 \qquad \textbf{(B) }\frac{11}7 \qquad \textbf{(C) } \frac85 \qquad \textbf{(D) } \frac{18}{11} \qquad \textbf{(E) } \frac{15}8</math> | ||
+ | |||
+ | ==Problem 2== | ||
+ | Liliane has <math>50\%</math> more soda than Jacqueline, and Alice has <math>25\%</math> more soda than Jacqueline. What is the relationship between the amounts of soda that Liliane and Alica have? | ||
+ | |||
+ | <math>\textbf{(A)}</math> Liliane has <math>20\%</math> more soda than Alice. <math>\textbf{(B)}</math> Liliane has <math>25\%</math> more soda than Alice. <math>\textbf{(C)}</math> Liliane has <math>45\%</math> more soda than Alice. <math>\textbf{(D)}</math> Liliane has <math>75\%</math> more soda than Alice. <math>\textbf{(E)}</math> Liliane has <math>100\%</math> more soda than Alice. | ||
+ | |||
+ | ==Problem 3== | ||
+ | A unit of blood expires after <math>10!=10\cdot 9 \cdot 8 \cdots 1</math> seconds. Yasin donates a unit of blood at noon of January 1. On what day does his unit of blood expire? | ||
+ | |||
+ | <math>\textbf{(A) }\text{January 2}\qquad\textbf{(B) }\text{January 12}\qquad\textbf{(C) }\text{January 22}\qquad\textbf{(D) }\text{Febuary 11}\qquad\textbf{(E) }\text{Febuary 12}</math> | ||
+ | |||
+ | ==Problem 4== | ||
+ | How many ways can a student schedule 3 mathematics courses -- algebra, geometry, and number theory -- in a 6-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other 3 periods is of no concern here.) | ||
+ | |||
+ | <math>\textbf{(A) }3\qquad\textbf{(B) }6\qquad\textbf{(C) }12\qquad\textbf{(D) }18\qquad\textbf{(E) }24</math> | ||
+ | |||
+ | ==Problem 5== | ||
+ | Alice, Bob, and Charlie were on a hike and were wondering how far away the nearest town was. When Alice said, "We are at least 6 miles away," Bob replied, "We are at most 5 miles away." Charlie then remarked, "Actually the nearest town is at most 4 miles away." It turned out that none of the three statements were true. Let <math>d</math> be the distance in miles to the nearest town. Which of the following intervals is the set of all possible values of <math>d</math>? | ||
+ | <math>\textbf{(A) } (0,4) \qquad \textbf{(B) } (4,5) \qquad \textbf{(C) } (4,6) \qquad \textbf{(D) } (5,6) \qquad \textbf{(E) } (5,\infty) </math> | ||
+ | |||
+ | ==Problem 6== | ||
+ | Sangho uploaded a video to a website where viewers can vote that they like or dislike a video. Each video begins with a score of 0, and the score increases by 1 for each like vote and decreases by 1 for each dislike vote. At one point Sangho saw that his video had a score of 90, and that <math>65\%</math> of the votes cast on his video were like votes. How many votes had been cast on Sangho's video at that point? | ||
+ | |||
+ | <math>\textbf{(A) } 200 \qquad \textbf{(B) } 300 \qquad \textbf{(C) } 400 \qquad \textbf{(D) } 500 \qquad \textbf{(E) } 600 </math> | ||
+ | |||
+ | ==Problem 7== | ||
+ | For how many (not necessarily positive) integer values of <math>n</math> is the value of <math>4000\cdot \left(\tfrac{2}{5}\right)^n</math> an integer? | ||
+ | |||
+ | <math> | ||
+ | \textbf{(A) }3 \qquad | ||
+ | \textbf{(B) }4 \qquad | ||
+ | \textbf{(C) }6 \qquad | ||
+ | \textbf{(D) }8 \qquad | ||
+ | \textbf{(E) }9 \qquad | ||
+ | </math> | ||
+ | |||
+ | ==Problem 8== | ||
+ | Joe has a collection of 23 coins, consisting of 5-cent coins, 10-cent coins, and 25-cent coins. He has 3 more 10-cent coins than 5-cent coins, and the total value of his collection is 320 cents. How many more 25-cent coins does Joe have than 5-cent coins? | ||
+ | |||
+ | <math>\textbf{(A) } 0 \qquad \textbf{(B) } 1 \qquad \textbf{(C) } 2 \qquad \textbf{(D) } 3 \qquad \textbf{(E) } 4 </math> | ||
+ | |||
+ | ==Problem 9== | ||
+ | All of the triangles in the diagram below are similar to iscoceles triangle <math>ABC</math>, in which <math>AB=AC</math>. Each of the 7 smallest triangles has area 1, and <math>\triangle ABC</math> has area 40. What is the area of trapezoid <math>DBCE</math>? | ||
+ | |||
+ | <asy> | ||
+ | unitsize(5); | ||
+ | dot((0,0)); | ||
+ | dot((60,0)); | ||
+ | dot((50,10)); | ||
+ | dot((10,10)); | ||
+ | dot((30,30)); | ||
+ | draw((0,0)--(60,0)--(50,10)--(30,30)--(10,10)--(0,0)); | ||
+ | draw((10,10)--(50,10)); | ||
+ | label("$B$",(0,0),SW); | ||
+ | label("$C$",(60,0),SE); | ||
+ | label("$E$",(50,10),E); | ||
+ | label("$D$",(10,10),W); | ||
+ | label("$A$",(30,30),N); | ||
+ | draw((10,10)--(15,15)--(20,10)--(25,15)--(30,10)--(35,15)--(40,10)--(45,15)--(50,10)); | ||
+ | draw((15,15)--(45,15)); | ||
+ | </asy> | ||
+ | |||
+ | <math>\textbf{(A) } 16 \qquad \textbf{(B) } 18 \qquad \textbf{(C) } 20 \qquad \textbf{(D) } 22 \qquad \textbf{(E) } 24 </math> | ||
+ | |||
+ | ==Problem 10== | ||
+ | Suppose that real number <math>x</math> satisfies <cmath>\sqrt{49-x^2}-\sqrt{25-x^2}=3</cmath>. What is the value of <math>\sqrt{49-x^2}+\sqrt{25-x^2}</math>? | ||
+ | |||
+ | <math> | ||
+ | \textbf{(A) }8 \qquad | ||
+ | \textbf{(B) }\sqrt{33}+8\qquad | ||
+ | \textbf{(C) }9 \qquad | ||
+ | \textbf{(D) }2\sqrt{10}+4 \qquad | ||
+ | \textbf{(E) }12 \qquad | ||
+ | </math> | ||
+ | |||
==See also== | ==See also== | ||
{{AMC10 box|year=2018|ab=A|before=[[2017 AMC 10B]]|after=[[2018 AMC 10B]]}} | {{AMC10 box|year=2018|ab=A|before=[[2017 AMC 10B]]|after=[[2018 AMC 10B]]}} |
Revision as of 16:52, 8 February 2018
Contents
Problem 1
What is the value of
Problem 2
Liliane has more soda than Jacqueline, and Alice has more soda than Jacqueline. What is the relationship between the amounts of soda that Liliane and Alica have?
Liliane has more soda than Alice. Liliane has more soda than Alice. Liliane has more soda than Alice. Liliane has more soda than Alice. Liliane has more soda than Alice.
Problem 3
A unit of blood expires after seconds. Yasin donates a unit of blood at noon of January 1. On what day does his unit of blood expire?
Problem 4
How many ways can a student schedule 3 mathematics courses -- algebra, geometry, and number theory -- in a 6-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other 3 periods is of no concern here.)
Problem 5
Alice, Bob, and Charlie were on a hike and were wondering how far away the nearest town was. When Alice said, "We are at least 6 miles away," Bob replied, "We are at most 5 miles away." Charlie then remarked, "Actually the nearest town is at most 4 miles away." It turned out that none of the three statements were true. Let be the distance in miles to the nearest town. Which of the following intervals is the set of all possible values of ?
Problem 6
Sangho uploaded a video to a website where viewers can vote that they like or dislike a video. Each video begins with a score of 0, and the score increases by 1 for each like vote and decreases by 1 for each dislike vote. At one point Sangho saw that his video had a score of 90, and that of the votes cast on his video were like votes. How many votes had been cast on Sangho's video at that point?
Problem 7
For how many (not necessarily positive) integer values of is the value of an integer?
Problem 8
Joe has a collection of 23 coins, consisting of 5-cent coins, 10-cent coins, and 25-cent coins. He has 3 more 10-cent coins than 5-cent coins, and the total value of his collection is 320 cents. How many more 25-cent coins does Joe have than 5-cent coins?
Problem 9
All of the triangles in the diagram below are similar to iscoceles triangle , in which . Each of the 7 smallest triangles has area 1, and has area 40. What is the area of trapezoid ?
Problem 10
Suppose that real number satisfies . What is the value of ?
See also
2018 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by 2017 AMC 10B |
Followed by 2018 AMC 10B | |
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 AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.