Difference between revisions of "2017 AMC 10B Problems"
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Sofia ran <math>5</math> laps around the <math>400</math>-meter track at her school. For each lap, she ran the first <math>100</math> meters at an average speed of <math>4</math> meters per second and the remaining <math>300</math> meters at an average speed of <math>5</math> meters per second. How much time did Sofia take running the <math>5</math> laps? | Sofia ran <math>5</math> laps around the <math>400</math>-meter track at her school. For each lap, she ran the first <math>100</math> meters at an average speed of <math>4</math> meters per second and the remaining <math>300</math> meters at an average speed of <math>5</math> meters per second. How much time did Sofia take running the <math>5</math> laps? | ||
− | <math>\textbf{(A)}\ 5 | + | <math>\textbf{(A)}\ \text{5 minutes and 35 seconds}\qquad\textbf{(B)}\ \text{6 minutes and 40 seconds}\qquad\textbf{(C)}\ \text{7 minutes and 5 seconds}\qquad</math> |
+ | <math>\textbf{(D)}\ \text{7 minutes and 25 seconds}\ \qquad\textbf{(E)}\ \text{8 minutes and 10 seconds}</math> | ||
[[2017 AMC 10B Problems/Problem 2|Solution]] | [[2017 AMC 10B Problems/Problem 2|Solution]] | ||
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==Problem 3== | ==Problem 3== | ||
− | Real numbers <math>x</math>, <math>y</math>, and <math>z</math> | + | Real numbers <math>x</math>, <math>y</math>, and <math>z</math> satisfy the inequalities |
<math>0<x<1</math>, <math>-1<y<0</math>, and <math>1<z<2</math>. | <math>0<x<1</math>, <math>-1<y<0</math>, and <math>1<z<2</math>. | ||
Which of the following numbers is necessarily positive? | Which of the following numbers is necessarily positive? | ||
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==Problem 4== | ==Problem 4== | ||
− | + | Suppose that <math>x</math> and <math>y</math> are nonzero real numbers such that <math>\frac{3x+y}{x-3y}=-2</math>. What is the value of <math>\frac{x+3y}{3x-y}</math>? | |
<math>\textbf{(A)}\ -3\qquad\textbf{(B)}\ -1\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ 2\qquad\textbf{(E)}\ 3</math> | <math>\textbf{(A)}\ -3\qquad\textbf{(B)}\ -1\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ 2\qquad\textbf{(E)}\ 3</math> | ||
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==Problem 5== | ==Problem 5== | ||
− | Camilla had twice as many blueberry jelly beans as cherry jelly beans. After eating 10 pieces of each kind, she now has three times as many blueberry jelly beans as cherry jelly beans. How many blueberry jelly beans did she originally have? | + | Camilla had twice as many blueberry jelly beans as cherry jelly beans. After eating <math>10</math> pieces of each kind, she now has three times as many blueberry jelly beans as cherry jelly beans. How many blueberry jelly beans did she originally have? |
<math>\textbf{(A)}\ 10\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 30\qquad\textbf{(D)}\ 40\qquad\textbf{(E)}\ 50</math> | <math>\textbf{(A)}\ 10\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 30\qquad\textbf{(D)}\ 40\qquad\textbf{(E)}\ 50</math> | ||
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==Problem 6== | ==Problem 6== | ||
− | What is the largest number of solid <math>2\text{ in}</math> by <math>2\text{ in}</math> by <math>1\text{ in}</math> blocks that can fit in a <math>3\text{ in}</math> by <math>2\text{ in}</math> by <math>3\text{ in}</math> box? | + | What is the largest number of solid <math>2\text{ in.}</math> by <math>2\text{ in.}</math> by <math>1\text{ in.}</math> blocks that can fit in a <math>3\text{ in.}</math> by <math>2\text{ in.}</math> by <math>3\text{ in.}</math> box? |
<math>\textbf{(A)}\ 3\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 7</math> | <math>\textbf{(A)}\ 3\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 7</math> | ||
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==Problem 10== | ==Problem 10== | ||
− | + | The lines with equations <math>ax-2y=c</math> and <math>2x+by=-c</math> are perpendicular and intersect at <math>(1, -5)</math>. What is <math>c</math>? | |
+ | |||
+ | <math>\textbf{(A)}\ -13\qquad\textbf{(B)}\ -8\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 13</math> | ||
[[2017 AMC 10B Problems/Problem 10|Solution]] | [[2017 AMC 10B Problems/Problem 10|Solution]] | ||
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==Problem 12== | ==Problem 12== | ||
− | Elmer's new car gives <math>50\%</math> | + | Elmer's new car gives <math>50\%</math> better fuel efficiency. However, the new car uses diesel fuel, which is <math>20\%</math> more expensive per liter than the gasoline the old car used. By what percent will Elmer save money if he uses his new car instead of his old car for a long trip? |
− | |||
<math>\textbf{(A) } 20\% \qquad \textbf{(B) } 26\tfrac23\% \qquad \textbf{(C) } 27\tfrac79\% \qquad \textbf{(D) } 33\tfrac13\% \qquad \textbf{(E) } 66\tfrac23\%</math> | <math>\textbf{(A) } 20\% \qquad \textbf{(B) } 26\tfrac23\% \qquad \textbf{(C) } 27\tfrac79\% \qquad \textbf{(D) } 33\tfrac13\% \qquad \textbf{(E) } 66\tfrac23\%</math> | ||
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==Problem 13== | ==Problem 13== | ||
− | + | There are <math>20</math> students participating in an after-school program offering classes in yoga, bridge, and painting. Each student must take at least one of these three classes, but may take two or all three. There are <math>10</math> students taking yoga, <math>13</math> taking bridge, and <math>9</math> taking painting. There are <math>9</math> students taking at least two classes. How many students are taking all three classes? | |
+ | |||
+ | <math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math> | ||
[[2017 AMC 10B Problems/Problem 13|Solution]] | [[2017 AMC 10B Problems/Problem 13|Solution]] | ||
==Problem 14== | ==Problem 14== | ||
− | + | An integer <math>N</math> is selected at random in the range <math>1\leq N \leq 2020</math>. What is the probability that the remainder when <math>N^{16}</math> is divided by <math>5</math> is <math>1</math>? | |
+ | |||
+ | <math>\textbf{(A)}\ \frac{1}{5}\qquad\textbf{(B)}\ \frac{2}{5}\qquad\textbf{(C)}\ \frac{3}{5}\qquad\textbf{(D)}\ \frac{4}{5}\qquad\textbf{(E)}\ 1</math> | ||
[[2017 AMC 10B Problems/Problem 14|Solution]] | [[2017 AMC 10B Problems/Problem 14|Solution]] | ||
==Problem 15== | ==Problem 15== | ||
− | Rectangle <math>ABCD</math> has <math>AB=3</math> and <math>BC=4</math>. Point <math>E</math> is the foot of the perpendicular from <math>B</math> to diagonal <math>\overline{AC}</math>. What is the area of <math>\triangle | + | Rectangle <math>ABCD</math> has <math>AB=3</math> and <math>BC=4</math>. Point <math>E</math> is the foot of the perpendicular from <math>B</math> to diagonal <math>\overline{AC}</math>. What is the area of <math>\triangle AED</math>? |
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ \frac{42}{25}\qquad\textbf{(C)}\ \frac{28}{15}\qquad\textbf{(D)}\ 2\qquad\textbf{(E)}\ \frac{54}{25}</math> | <math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ \frac{42}{25}\qquad\textbf{(C)}\ \frac{28}{15}\qquad\textbf{(D)}\ 2\qquad\textbf{(E)}\ \frac{54}{25}</math> | ||
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==Problem 17== | ==Problem 17== | ||
− | Call a positive integer monotonous if it is a one-digit number or its digits, when read from left to right, form either a strictly increasing or a strictly decreasing sequence. For example, <math>3</math>, <math>23578</math>, and <math>987620</math> are monotonous, but <math>88</math>, <math>7434</math>, and <math>23557</math> are not. How many monotonous positive integers are there? | + | Call a positive integer <math>\textbf{monotonous}</math> if it is a one-digit number or its digits, when read from left to right, form either a strictly increasing or a strictly decreasing sequence. For example, <math>3</math>, <math>23578</math>, and <math>987620</math> are monotonous, but <math>88</math>, <math>7434</math>, and <math>23557</math> are not. How many monotonous positive integers are there? |
<math>\textbf{(A)}\ 1024\qquad\textbf{(B)}\ 1524\qquad\textbf{(C)}\ 1533\qquad\textbf{(D)}\ 1536\qquad\textbf{(E)}\ 2048</math> | <math>\textbf{(A)}\ 1024\qquad\textbf{(B)}\ 1524\qquad\textbf{(C)}\ 1533\qquad\textbf{(D)}\ 1536\qquad\textbf{(E)}\ 2048</math> | ||
+ | |||
[[2017 AMC 10B Problems/Problem 17|Solution]] | [[2017 AMC 10B Problems/Problem 17|Solution]] | ||
==Problem 18== | ==Problem 18== | ||
− | In the figure below, 3 of the 6 disks are to be painted blue, 2 are to be painted red, and 1 is to be painted green. Two paintings that can be obtained from one another by a rotation or a reflection of the entire figure are considered the same. How many different paintings are possible? | + | In the figure below, <math>3</math> of the <math>6</math> disks are to be painted blue, <math>2</math> are to be painted red, and <math>1</math> is to be painted green. Two paintings that can be obtained from one another by a rotation or a reflection of the entire figure are considered the same. How many different paintings are possible? |
+ | |||
+ | <asy> | ||
+ | size(100); | ||
+ | pair A, B, C, D, E, F; | ||
+ | A = (0,0); | ||
+ | B = (1,0); | ||
+ | C = (2,0); | ||
+ | D = rotate(60, A)*B; | ||
+ | E = B + D; | ||
+ | F = rotate(60, A)*C; | ||
+ | draw(Circle(A, 0.5)); | ||
+ | draw(Circle(B, 0.5)); | ||
+ | draw(Circle(C, 0.5)); | ||
+ | draw(Circle(D, 0.5)); | ||
+ | draw(Circle(E, 0.5)); | ||
+ | draw(Circle(F, 0.5)); | ||
+ | </asy> | ||
− | <math>\textbf{(A)}\ 6\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ | + | <math>\textbf{(A)}\ 6\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 15</math> |
[[2017 AMC 10B Problems/Problem 18|Solution]] | [[2017 AMC 10B Problems/Problem 18|Solution]] | ||
==Problem 19== | ==Problem 19== | ||
− | Let <math>ABC</math> be an equilateral triangle. Extend side <math>\overline{AB}</math> beyond <math>B</math> to a point <math>B'</math> so that <math>BB'= | + | Let <math>ABC</math> be an equilateral triangle. Extend side <math>\overline{AB}</math> beyond <math>B</math> to a point <math>B'</math> so that <math>BB'=3 \cdot AB</math>. Similarly, extend side <math>\overline{BC}</math> beyond <math>C</math> to a point <math>C'</math> so that <math>CC'=3 \cdot BC</math>, and extend side <math>\overline{CA}</math> beyond <math>A</math> to a point <math>A'</math> so that <math>AA'=3 \cdot CA</math>. What is the ratio of the area of <math>\triangle A'B'C'</math> to the area of <math>\triangle ABC</math>? |
<math>\textbf{(A)}\ 9:1\qquad\textbf{(B)}\ 16:1\qquad\textbf{(C)}\ 25:1\qquad\textbf{(D)}\ 36:1\qquad\textbf{(E)}\ 37:1</math> | <math>\textbf{(A)}\ 9:1\qquad\textbf{(B)}\ 16:1\qquad\textbf{(C)}\ 25:1\qquad\textbf{(D)}\ 36:1\qquad\textbf{(E)}\ 37:1</math> | ||
− | |||
[[2017 AMC 10B Problems/Problem 19|Solution]] | [[2017 AMC 10B Problems/Problem 19|Solution]] | ||
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The number <math>21!=51,090,942,171,709,440,000</math> has over <math>60,000</math> positive integer divisors. One of them is chosen at random. What is the probability that it is odd? | The number <math>21!=51,090,942,171,709,440,000</math> has over <math>60,000</math> positive integer divisors. One of them is chosen at random. What is the probability that it is odd? | ||
− | <math>\textbf{(A)} \frac{1}{21} \qquad \textbf{(B)} \frac{1}{19} \qquad \textbf{(C)} \frac{1}{18} \qquad \textbf{(D)} \frac{1}{2} \qquad \textbf{(E)} \frac{11}{21}</math> | + | <math>\textbf{(A)}\ \frac{1}{21} \qquad \textbf{(B)}\ \frac{1}{19} \qquad \textbf{(C)}\ \frac{1}{18} \qquad \textbf{(D)}\ \frac{1}{2} \qquad \textbf{(E)}\ \frac{11}{21}</math> |
− | |||
[[2017 AMC 10B Problems/Problem 20|Solution]] | [[2017 AMC 10B Problems/Problem 20|Solution]] | ||
==Problem 21== | ==Problem 21== | ||
− | In <math>\triangle ABC</math>, <math>AB=6</math>, <math>AC=8</math>, <math>BC=10</math>, and <math>D</math> is the midpoint of <math>\overline{BC}</math>. What is the sum of the radii of the circles | + | In <math>\triangle ABC</math>, <math>AB=6</math>, <math>AC=8</math>, <math>BC=10</math>, and <math>D</math> is the midpoint of <math>\overline{BC}</math>. What is the sum of the radii of the circles inscribed in <math>\triangle ADB</math> and <math>\triangle ADC</math>? |
<math>\textbf{(A)}\ \sqrt{5}\qquad\textbf{(B)}\ \frac{11}{4}\qquad\textbf{(C)}\ 2\sqrt{2}\qquad\textbf{(D)}\ \frac{17}{6}\qquad\textbf{(E)}\ 3</math> | <math>\textbf{(A)}\ \sqrt{5}\qquad\textbf{(B)}\ \frac{11}{4}\qquad\textbf{(C)}\ 2\sqrt{2}\qquad\textbf{(D)}\ \frac{17}{6}\qquad\textbf{(E)}\ 3</math> |
Latest revision as of 18:27, 1 March 2022
2017 AMC 10B (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
Mary thought of a positive two-digit number. She multiplied it by and added . Then she switched the digits of the result, obtaining a number between and , inclusive. What was Mary's number?
Problem 2
Sofia ran laps around the -meter track at her school. For each lap, she ran the first meters at an average speed of meters per second and the remaining meters at an average speed of meters per second. How much time did Sofia take running the laps?
Problem 3
Real numbers , , and satisfy the inequalities , , and . Which of the following numbers is necessarily positive?
Problem 4
Suppose that and are nonzero real numbers such that . What is the value of ?
Problem 5
Camilla had twice as many blueberry jelly beans as cherry jelly beans. After eating pieces of each kind, she now has three times as many blueberry jelly beans as cherry jelly beans. How many blueberry jelly beans did she originally have?
Problem 6
What is the largest number of solid by by blocks that can fit in a by by box?
Problem 7
Samia set off on her bicycle to visit her friend, traveling at an average speed of kilometers per hour. When she had gone half the distance to her friend's house, a tire went flat, and she walked the rest of the way at kilometers per hour. In all it took her minutes to reach her friend's house. In kilometers rounded to the nearest tenth, how far did Samia walk?
Problem 8
Points and are vertices of with . The altitude from meets the opposite side at . What are the coordinates of point ?
Problem 9
A radio program has a quiz consisting of multiple-choice questions, each with choices. A contestant wins if he or she gets or more of the questions right. The contestant answers randomly to each question. What is the probability of winning?
Problem 10
The lines with equations and are perpendicular and intersect at . What is ?
Problem 11
At Typico High School, of the students like dancing, and the rest dislike it. Of those who like dancing, say that they like it, and the rest say that they dislike it. Of those who dislike dancing, say that they dislike it, and the rest say that they like it. What fraction of students who say they dislike dancing actually like it?
Problem 12
Elmer's new car gives better fuel efficiency. However, the new car uses diesel fuel, which is more expensive per liter than the gasoline the old car used. By what percent will Elmer save money if he uses his new car instead of his old car for a long trip?
Problem 13
There are students participating in an after-school program offering classes in yoga, bridge, and painting. Each student must take at least one of these three classes, but may take two or all three. There are students taking yoga, taking bridge, and taking painting. There are students taking at least two classes. How many students are taking all three classes?
Problem 14
An integer is selected at random in the range . What is the probability that the remainder when is divided by is ?
Problem 15
Rectangle has and . Point is the foot of the perpendicular from to diagonal . What is the area of ?
Problem 16
How many of the base-ten numerals for the positive integers less than or equal to contain the digit ?
Problem 17
Call a positive integer if it is a one-digit number or its digits, when read from left to right, form either a strictly increasing or a strictly decreasing sequence. For example, , , and are monotonous, but , , and are not. How many monotonous positive integers are there?
Problem 18
In the figure below, of the disks are to be painted blue, are to be painted red, and is to be painted green. Two paintings that can be obtained from one another by a rotation or a reflection of the entire figure are considered the same. How many different paintings are possible?
Problem 19
Let be an equilateral triangle. Extend side beyond to a point so that . Similarly, extend side beyond to a point so that , and extend side beyond to a point so that . What is the ratio of the area of to the area of ?
Problem 20
The number has over positive integer divisors. One of them is chosen at random. What is the probability that it is odd?
Problem 21
In , , , , and is the midpoint of . What is the sum of the radii of the circles inscribed in and ?
Problem 22
The diameter of a circle of radius is extended to a point outside the circle so that . Point is chosen so that and line is perpendicular to line . Segment intersects the circle at a point between and . What is the area of ?
Problem 23
Let be the -digit number that is formed by writing the integers from to in order, one after the other. What is the remainder when is divided by ?
Problem 24
The vertices of an equilateral triangle lie on the hyperbola , and a vertex of this hyperbola is the centroid of the triangle. What is the square of the area of the triangle?
Problem 25
Last year Isabella took math tests and received different scores, each an integer between and , inclusive. After each test she noticed that the average of her test scores was an integer. Her score on the seventh test was . What was her score on the sixth test?
See also
2017 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by 2017 AMC 10A Problems |
Followed by 2018 AMC 10A Problems | |
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.