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>
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 +
==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>
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 +
==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>
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 +
==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==
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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
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\textbf{(B) }4 \qquad
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\textbf{(C) }6 \qquad
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\textbf{(D) }8 \qquad
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\textbf{(E) }9 \qquad
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</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>
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 +
==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>?
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<asy>
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unitsize(5);
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dot((0,0));
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dot((60,0));
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dot((50,10));
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dot((10,10));
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dot((30,30));
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draw((0,0)--(60,0)--(50,10)--(30,30)--(10,10)--(0,0));
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draw((10,10)--(50,10));
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label("$B$",(0,0),SW);
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label("$C$",(60,0),SE);
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label("$E$",(50,10),E);
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label("$D$",(10,10),W);
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label("$A$",(30,30),N);
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draw((10,10)--(15,15)--(20,10)--(25,15)--(30,10)--(35,15)--(40,10)--(45,15)--(50,10));
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draw((15,15)--(45,15));
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</asy>
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<math>\textbf{(A) }  16  \qquad        \textbf{(B) }  18  \qquad    \textbf{(C) }  20  \qquad  \textbf{(D) }  22 \qquad  \textbf{(E) }  24 </math>
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 +
==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
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\textbf{(B) }\sqrt{33}+8\qquad
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\textbf{(C) }9 \qquad
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\textbf{(D) }2\sqrt{10}+4 \qquad
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\textbf{(E) }12 \qquad
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</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

Problem 1

What is the value of \[\left(\left((2+1)^{-1}+1\right)^{-1}+1\right)^{-1}+1?\]$\textbf{(A) } \frac58 \qquad \textbf{(B) }\frac{11}7 \qquad \textbf{(C) } \frac85 \qquad \textbf{(D) } \frac{18}{11} \qquad \textbf{(E) } \frac{15}8$

Problem 2

Liliane has $50\%$ more soda than Jacqueline, and Alice has $25\%$ more soda than Jacqueline. What is the relationship between the amounts of soda that Liliane and Alica have?

$\textbf{(A)}$ Liliane has $20\%$ more soda than Alice. $\textbf{(B)}$ Liliane has $25\%$ more soda than Alice. $\textbf{(C)}$ Liliane has $45\%$ more soda than Alice. $\textbf{(D)}$ Liliane has $75\%$ more soda than Alice. $\textbf{(E)}$ Liliane has $100\%$ more soda than Alice.

Problem 3

A unit of blood expires after $10!=10\cdot 9 \cdot 8 \cdots 1$ seconds. Yasin donates a unit of blood at noon of January 1. On what day does his unit of blood expire?

$\textbf{(A) }\text{January 2}\qquad\textbf{(B) }\text{January 12}\qquad\textbf{(C) }\text{January 22}\qquad\textbf{(D) }\text{Febuary 11}\qquad\textbf{(E) }\text{Febuary 12}$

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.)

$\textbf{(A) }3\qquad\textbf{(B) }6\qquad\textbf{(C) }12\qquad\textbf{(D) }18\qquad\textbf{(E) }24$

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 $d$ be the distance in miles to the nearest town. Which of the following intervals is the set of all possible values of $d$? $\textbf{(A) }   (0,4)   \qquad        \textbf{(B) }   (4,5)   \qquad    \textbf{(C) }   (4,6)   \qquad   \textbf{(D) }  (5,6)  \qquad  \textbf{(E) }   (5,\infty)$

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 $65\%$ of the votes cast on his video were like votes. How many votes had been cast on Sangho's video at that point?

$\textbf{(A) }   200   \qquad        \textbf{(B) }   300   \qquad    \textbf{(C) }   400   \qquad   \textbf{(D) }  500  \qquad  \textbf{(E) }   600$

Problem 7

For how many (not necessarily positive) integer values of $n$ is the value of $4000\cdot \left(\tfrac{2}{5}\right)^n$ an integer?

$\textbf{(A) }3 \qquad \textbf{(B) }4 \qquad \textbf{(C) }6 \qquad \textbf{(D) }8 \qquad \textbf{(E) }9 \qquad$

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?

$\textbf{(A) }   0   \qquad        \textbf{(B) }   1   \qquad    \textbf{(C) }   2   \qquad   \textbf{(D) }  3  \qquad  \textbf{(E) }   4$

Problem 9

All of the triangles in the diagram below are similar to iscoceles triangle $ABC$, in which $AB=AC$. Each of the 7 smallest triangles has area 1, and $\triangle ABC$ has area 40. What is the area of trapezoid $DBCE$?

[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]

$\textbf{(A) }   16   \qquad        \textbf{(B) }   18   \qquad    \textbf{(C) }   20   \qquad   \textbf{(D) }  22 \qquad  \textbf{(E) }   24$

Problem 10

Suppose that real number $x$ satisfies \[\sqrt{49-x^2}-\sqrt{25-x^2}=3\]. What is the value of $\sqrt{49-x^2}+\sqrt{25-x^2}$?

$\textbf{(A) }8 \qquad \textbf{(B) }\sqrt{33}+8\qquad \textbf{(C) }9 \qquad \textbf{(D) }2\sqrt{10}+4 \qquad \textbf{(E) }12 \qquad$

See also

2018 AMC 10A (ProblemsAnswer KeyResources)
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

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