Difference between revisions of "2018 AMC 10A Problems"
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+ | {{AMC10 Problems|year=2018|ab=A}} | ||
==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> | ||
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[[2018 AMC 10A Problems/Problem 2|Solution]] | [[2018 AMC 10A Problems/Problem 2|Solution]] | ||
− | ==Problem 3== | + | == 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? | 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? | ||
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==Problem 4== | ==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.) | + | How many ways can a student schedule <math>3</math> mathematics courses -- algebra, geometry, and number theory -- in a <math>6</math>-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other <math>3</math> 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> | <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== | ==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>? | + | 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 <math>6</math> miles away," Bob replied, "We are at most <math>5</math> miles away." Charlie then remarked, "Actually the nearest town is at most <math>4</math> 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> | <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== | ==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? | + | 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 <math>0</math>, and the score increases by <math>1</math> for each like vote and decreases by <math>1</math> for each dislike vote. At one point Sangho saw that his video had a score of <math>90</math>, 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> | <math>\textbf{(A) } 200 \qquad \textbf{(B) } 300 \qquad \textbf{(C) } 400 \qquad \textbf{(D) } 500 \qquad \textbf{(E) } 600 </math> | ||
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==Problem 8== | ==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? | + | Joe has a collection of <math>23</math> coins, consisting of <math>5</math>-cent coins, <math>10</math>-cent coins, and <math>25</math>-cent coins. He has <math>3</math> more <math>10</math>-cent coins than <math>5</math>-cent coins, and the total value of his collection is <math>320</math> cents. How many more <math>25</math>-cent coins does Joe have than <math>5</math>-cent coins? |
<math>\textbf{(A) } 0 \qquad \textbf{(B) } 1 \qquad \textbf{(C) } 2 \qquad \textbf{(D) } 3 \qquad \textbf{(E) } 4 </math> | <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== | ==Problem 9== | ||
− | All of the triangles in the diagram below are similar to isosceles 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>? | + | All of the triangles in the diagram below are similar to isosceles triangle <math>ABC</math>, in which <math>AB=AC</math>. Each of the <math>7</math> smallest triangles has area <math>1</math>, and <math>\triangle ABC</math> has area <math>40</math>. What is the area of trapezoid <math>DBCE</math>? |
<asy> | <asy> | ||
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==Problem 10== | ==Problem 10== | ||
− | Suppose that real number <math>x</math> satisfies <cmath>\sqrt{49-x^2}-\sqrt{25-x^2}=3</cmath> | + | 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> | <math> | ||
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==Problem 11== | ==Problem 11== | ||
− | When <math>7</math> fair standard <math>6</math>-sided | + | When <math>7</math> fair standard <math>6</math>-sided dice are thrown, the probability that the sum of the numbers on the top faces is <math>10</math> can be written as <cmath>\frac{n}{6^{7}},</cmath> where <math>n</math> is a positive integer. What is <math>n</math>? |
<math> | <math> | ||
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==Problem 12== | ==Problem 12== | ||
How many ordered pairs of real numbers <math>(x,y)</math> satisfy the following system of equations? | How many ordered pairs of real numbers <math>(x,y)</math> satisfy the following system of equations? | ||
− | <cmath>x+3y=3 | + | <cmath>\begin{align*} |
− | + | x+3y&=3 \\ | |
− | + | \big||x|-|y|\big|&=1 | |
− | <math> | + | \end{align*}</cmath> |
− | \textbf{(A) }1\qquad | + | <math>\textbf{(A) } 1 \qquad |
− | \textbf{(B) }2\qquad | + | \textbf{(B) } 2 \qquad |
− | \textbf{(C) }3\qquad | + | \textbf{(C) } 3 \qquad |
− | \textbf{(D) }4\qquad | + | \textbf{(D) } 4 \qquad |
− | \textbf{(E) }8 | + | \textbf{(E) } 8 </math> |
− | </math> | ||
[[2018 AMC 10A Problems/Problem 12|Solution]] | [[2018 AMC 10A Problems/Problem 12|Solution]] | ||
==Problem 13== | ==Problem 13== | ||
− | A paper triangle with sides of lengths 3, 4, and 5 inches, as shown, is folded so that point <math>A</math> falls on point <math>B</math>. What is the length in inches of the crease? | + | A paper triangle with sides of lengths <math>3,4,</math> and <math>5</math> inches, as shown, is folded so that point <math>A</math> falls on point <math>B</math>. What is the length in inches of the crease? |
<asy> | <asy> | ||
draw((0,0)--(4,0)--(4,3)--(0,0)); | draw((0,0)--(4,0)--(4,3)--(0,0)); | ||
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==Problem 15== | ==Problem 15== | ||
− | Two circles of radius 5 are externally tangent to each other and are internally tangent to a circle of radius 13 at points <math>A</math> and <math>B</math>, as shown in the diagram. The distance <math>AB</math> can be written in the form <math>\tfrac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. What is <math>m+n</math>? | + | Two circles of radius <math>5</math> are externally tangent to each other and are internally tangent to a circle of radius <math>13</math> at points <math>A</math> and <math>B</math>, as shown in the diagram. The distance <math>AB</math> can be written in the form <math>\tfrac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. What is <math>m+n</math>? |
<asy> | <asy> | ||
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==Problem 17== | ==Problem 17== | ||
− | Let <math>S</math> be a set of 6 integers taken from <math>\{1,2,\dots,12\}</math> with the property that if <math>a</math> and <math>b</math> are elements of <math>S</math> with <math>a<b</math>, then <math>b</math> is not a multiple of <math>a</math>. What is the least possible value of an element in <math>S | + | Let <math>S</math> be a set of <math>6</math> integers taken from <math>\{1,2,\dots,12\}</math> with the property that if <math>a</math> and <math>b</math> are elements of <math>S</math> with <math>a<b</math>, then <math>b</math> is not a multiple of <math>a</math>. What is the least possible value of an element in <math>S</math>? |
<math>\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 7</math> | <math>\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 7</math> | ||
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\textbf{(C) } 1094 \qquad | \textbf{(C) } 1094 \qquad | ||
\textbf{(D) } 3281 \qquad | \textbf{(D) } 3281 \qquad | ||
− | \textbf{(E) } 59,048 </math> | + | \textbf{(E) } 59,048 |
+ | </math> | ||
[[2018 AMC 10A Problems/Problem 18|Solution]] | [[2018 AMC 10A Problems/Problem 18|Solution]] | ||
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==Problem 20== | ==Problem 20== | ||
− | A scanning code consists of a <math>7 \times 7</math> grid of squares, with some of its squares colored black and the rest colored white. There must be at least one square of each color in this grid of <math>49</math> squares. A scanning code is called <math>\ | + | A scanning code consists of a <math>7 \times 7</math> grid of squares, with some of its squares colored black and the rest colored white. There must be at least one square of each color in this grid of <math>49</math> squares. A scanning code is called <math>\textit{symmetric}</math> if its look does not change when the entire square is rotated by a multiple of <math>90 ^{\circ}</math> counterclockwise around its center, nor when it is reflected across a line joining opposite corners or a line joining midpoints of opposite sides. What is the total number of possible symmetric scanning codes? |
<math>\textbf{(A)} \text{ 510} \qquad \textbf{(B)} \text{ 1022} \qquad \textbf{(C)} \text{ 8190} \qquad \textbf{(D)} \text{ 8192} \qquad \textbf{(E)} \text{ 65,534}</math> | <math>\textbf{(A)} \text{ 510} \qquad \textbf{(B)} \text{ 1022} \qquad \textbf{(C)} \text{ 8190} \qquad \textbf{(D)} \text{ 8192} \qquad \textbf{(E)} \text{ 65,534}</math> | ||
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==Problem 23== | ==Problem 23== | ||
− | Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths 3 and 4 units. In the corner where those sides meet at a right angle, he leaves a small unplanted square <math>S</math> so that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest distance from <math>S</math> to the hypotenuse is 2 units. What fraction of the field is planted? | + | Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths <math>3</math> and <math>4</math> units. In the corner where those sides meet at a right angle, he leaves a small unplanted square <math>S</math> so that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest distance from <math>S</math> to the hypotenuse is <math>2</math> units. What fraction of the field is planted? |
<asy> | <asy> | ||
− | + | /* Edited by MRENTHUSIASM */ | |
− | + | size(160); | |
− | + | pair A, B, C, D, F; | |
− | label("$4$", ( | + | A = origin; |
− | label("$3$", ( | + | B = (4,0); |
− | label("$2$", ( | + | C = (0,3); |
− | label("$S$", ( | + | D = (2/7,2/7); |
− | draw( | + | F = foot(D,B,C); |
+ | fill(A--(2/7,0)--D--(0,2/7)--cycle, lightgray); | ||
+ | draw(A--B--C--cycle); | ||
+ | draw((2/7,0)--D--(0,2/7)); | ||
+ | label("$4$", midpoint(A--B), N); | ||
+ | label("$3$", midpoint(A--C), E); | ||
+ | label("$2$", midpoint(D--F), SE); | ||
+ | label("$S$", midpoint(A--D)); | ||
+ | draw(D--F, dashed); | ||
</asy> | </asy> | ||
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For a positive integer <math>n</math> and nonzero digits <math>a</math>, <math>b</math>, and <math>c</math>, let <math>A_n</math> be the <math>n</math>-digit integer each of whose digits is equal to <math>a</math>; let <math>B_n</math> be the <math>n</math>-digit integer each of whose digits is equal to <math>b</math>, and let <math>C_n</math> be the <math>2n</math>-digit (not <math>n</math>-digit) integer each of whose digits is equal to <math>c</math>. What is the greatest possible value of <math>a + b + c</math> for which there are at least two values of <math>n</math> such that <math>C_n - B_n = A_n^2</math>? | For a positive integer <math>n</math> and nonzero digits <math>a</math>, <math>b</math>, and <math>c</math>, let <math>A_n</math> be the <math>n</math>-digit integer each of whose digits is equal to <math>a</math>; let <math>B_n</math> be the <math>n</math>-digit integer each of whose digits is equal to <math>b</math>, and let <math>C_n</math> be the <math>2n</math>-digit (not <math>n</math>-digit) integer each of whose digits is equal to <math>c</math>. What is the greatest possible value of <math>a + b + c</math> for which there are at least two values of <math>n</math> such that <math>C_n - B_n = A_n^2</math>? | ||
− | <math>\textbf{(A)} | + | <math>\textbf{(A) } 12 \qquad \textbf{(B) } 14 \qquad \textbf{(C) } 16 \qquad \textbf{(D) } 18 \qquad \textbf{(E) } 20</math> |
[[2018 AMC 10A Problems/Problem 25|Solution]] | [[2018 AMC 10A Problems/Problem 25|Solution]] |
Latest revision as of 11:06, 20 September 2021
2018 AMC 10A (Answer Key) Printable version: | 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
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 Alice 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 mathematics courses -- algebra, geometry, and number theory -- in a -period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other 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 miles away," Bob replied, "We are at most miles away." Charlie then remarked, "Actually the nearest town is at most 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 , and the score increases by for each like vote and decreases by for each dislike vote. At one point Sangho saw that his video had a score of , 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 coins, consisting of -cent coins, -cent coins, and -cent coins. He has more -cent coins than -cent coins, and the total value of his collection is cents. How many more -cent coins does Joe have than -cent coins?
Problem 9
All of the triangles in the diagram below are similar to isosceles triangle , in which . Each of the smallest triangles has area , and has area . What is the area of trapezoid ?
Problem 10
Suppose that real number satisfies What is the value of ?
Problem 11
When fair standard -sided dice are thrown, the probability that the sum of the numbers on the top faces is can be written as where is a positive integer. What is ?
Problem 12
How many ordered pairs of real numbers satisfy the following system of equations?
Problem 13
A paper triangle with sides of lengths and inches, as shown, is folded so that point falls on point . What is the length in inches of the crease?
Problem 14
What is the greatest integer less than or equal to
Problem 15
Two circles of radius are externally tangent to each other and are internally tangent to a circle of radius at points and , as shown in the diagram. The distance can be written in the form , where and are relatively prime positive integers. What is ?
Problem 16
Right triangle has leg lengths and . Including and , how many line segments with integer length can be drawn from vertex to a point on hypotenuse ?
Problem 17
Let be a set of integers taken from with the property that if and are elements of with , then is not a multiple of . What is the least possible value of an element in ?
Problem 18
How many nonnegative integers can be written in the form where for ?
Problem 19
A number is randomly selected from the set , and a number is randomly selected from . What is the probability that has a units digit of ?
Problem 20
A scanning code consists of a grid of squares, with some of its squares colored black and the rest colored white. There must be at least one square of each color in this grid of squares. A scanning code is called if its look does not change when the entire square is rotated by a multiple of counterclockwise around its center, nor when it is reflected across a line joining opposite corners or a line joining midpoints of opposite sides. What is the total number of possible symmetric scanning codes?
Problem 21
Which of the following describes the set of values of for which the curves and in the real -plane intersect at exactly points?
Problem 22
Let and be positive integers such that , , , and . Which of the following must be a divisor of ?
Problem 23
Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths and units. In the corner where those sides meet at a right angle, he leaves a small unplanted square so that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest distance from to the hypotenuse is units. What fraction of the field is planted?
Problem 24
Triangle with and has area . Let be the midpoint of , and let be the midpoint of . The angle bisector of intersects and at and , respectively. What is the area of quadrilateral ?
Problem 25
For a positive integer and nonzero digits , , and , let be the -digit integer each of whose digits is equal to ; let be the -digit integer each of whose digits is equal to , and let be the -digit (not -digit) integer each of whose digits is equal to . What is the greatest possible value of for which there are at least two values of such that ?
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.