Difference between revisions of "2017 AMC 12B Problems"
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{{AMC12 Problems|year=2017|ab=B}} | {{AMC12 Problems|year=2017|ab=B}} | ||
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<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 20\qquad\textbf{(E)}\ 25</math> | <math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 20\qquad\textbf{(E)}\ 25</math> | ||
+ | |||
+ | [[2017 AMC 12B Problems/Problem 1|Solution]] | ||
==Problem 2== | ==Problem 2== | ||
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==Problem 5== | ==Problem 5== | ||
+ | |||
+ | The data set <math>[6,19,33,33,39,41,41,43,51,57]</math> has median <math>Q_2 = 40</math>, first quartile <math>Q_1 = 33</math>, and third quartile <math>Q_3=43</math>. An outlier in a data set is a value that is more than <math>1.5</math> times the interquartile range below the first quartile <math>(Q_1)</math> or more than <math>1.5</math> times the interquartile range above the third quartile <math>(Q_3)</math>, where the interquartile range is defined as <math>Q_3 - Q_1</math>. How many outliers does this data set have? | ||
+ | |||
+ | <math>\textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 4</math> | ||
+ | |||
+ | [[2017 AMC 12B Problems/Problem 5|Solution]] | ||
==Problem 6== | ==Problem 6== | ||
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The functions <math>\sin(x)</math> and <math>\cos(x)</math> are periodic with least period <math>2\pi</math>. What is the least period of the function <math>\cos(\sin(x))</math>? | The functions <math>\sin(x)</math> and <math>\cos(x)</math> are periodic with least period <math>2\pi</math>. What is the least period of the function <math>\cos(\sin(x))</math>? | ||
− | <math>\textbf{(A)}\ \frac{\ | + | <math>\textbf{(A)}\ \frac{\pi}{2}\qquad\textbf{(B)}\ \pi\qquad\textbf{(C)}\ 2\pi \qquad\textbf{(D)}\ 4\pi \qquad\textbf{(E)}</math> It's not periodic. |
[[2017 AMC 12B Problems/Problem 7|Solution]] | [[2017 AMC 12B Problems/Problem 7|Solution]] | ||
==Problem 8== | ==Problem 8== | ||
+ | The ratio of the short side of a certain rectangle to the long side is equal to the ratio of the long side to the diagonal. What is the square of the ratio of the short side to the long side of this rectangle? | ||
+ | |||
+ | <math>\textbf{(A)}\ \frac{\sqrt{3}-1}{2}\qquad\textbf{(B)}\ \frac{1}{2}\qquad\textbf{(C)}\ \frac{\sqrt{5}-1}{2} \qquad\textbf{(D)}\ \frac{\sqrt{2}}{2} \qquad\textbf{(E)}\ \frac{\sqrt{6}-1}{2}</math> | ||
+ | |||
+ | [[2017 AMC 12B Problems/Problem 8|Solution]] | ||
==Problem 9== | ==Problem 9== | ||
+ | |||
+ | A circle has center <math>(-10,-4)</math> and radius <math>13</math>. Another circle has center <math>(3,9)</math> and radius <math>\sqrt{65}</math>. The line passing through the two points of intersection of the two circles has equation <math>x + y = c</math>. What is <math>c</math>? | ||
+ | |||
+ | <math>\textbf{(A)}\ 3\qquad\textbf{(B)}\ 3\sqrt{3}\qquad\textbf{(C)}\ 4\sqrt{2}\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ \frac{13}{2}</math> | ||
+ | |||
+ | [[2017 AMC 12B Problems/Problem 9|Solution]] | ||
==Problem 10== | ==Problem 10== | ||
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What is the sum of the roots of <math>z^{12}=64</math> that have a positive real part? | What is the sum of the roots of <math>z^{12}=64</math> that have a positive real part? | ||
− | <math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ \sqrt{2}+2\sqrt{3} \qquad \textbf{(D)}\ 2\sqrt{2}+\sqrt{6} \qquad \textbf{(E)}\ (1+\sqrt{3})(1+\sqrt{3})i</math> | + | <math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ \sqrt{2}+2\sqrt{3} \qquad \textbf{(D)}\ 2\sqrt{2}+\sqrt{6} \qquad \textbf{(E)}\ (1+\sqrt{3}) + (1+\sqrt{3})i</math> |
[[2017 AMC 12B Problems/Problem 12|Solution]] | [[2017 AMC 12B Problems/Problem 12|Solution]] | ||
==Problem 13== | ==Problem 13== | ||
+ | |||
+ | 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) } 15</math> | ||
+ | |||
+ | [[2017 AMC 12B Problems/Problem 13|Solution]] | ||
==Problem 14== | ==Problem 14== | ||
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==Problem 15== | ==Problem 15== | ||
− | 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 | + | <math>\textbf{(A)}\ 9\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 36\qquad\textbf{(E)}\ 37</math> |
[[2017 AMC 12B Problems/Problem 15|Solution]] | [[2017 AMC 12B Problems/Problem 15|Solution]] | ||
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<math>\textbf{(E)}</math> The probability of winning Game A is <math>\frac{4}{81}</math> greater than the probability of winning Game B. | <math>\textbf{(E)}</math> The probability of winning Game A is <math>\frac{4}{81}</math> greater than the probability of winning Game B. | ||
+ | |||
[[2017 AMC 12B Problems/Problem 17|Solution]] | [[2017 AMC 12B Problems/Problem 17|Solution]] | ||
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==Problem 20== | ==Problem 20== | ||
− | Real numbers <math>x</math> and <math>y</math> are chosen independently and uniformly at random from the interval <math>(0,1)</math>. What is the probability that <math>\lfloor\log_2x\rfloor=\lfloor\log_2y\rfloor | + | Real numbers <math>x</math> and <math>y</math> are chosen independently and uniformly at random from the interval <math>(0,1)</math>. What is the probability that <math>\lfloor\log_2x\rfloor=\lfloor\log_2y\rfloor</math>? |
<math>\textbf{(A)}\ \frac{1}{8}\qquad\textbf{(B)}\ \frac{1}{6}\qquad\textbf{(C)}\ \frac{1}{4}\qquad\textbf{(D)}\ \frac{1}{3}\qquad\textbf{(E)}\ \frac{1}{2}</math> | <math>\textbf{(A)}\ \frac{1}{8}\qquad\textbf{(B)}\ \frac{1}{6}\qquad\textbf{(C)}\ \frac{1}{4}\qquad\textbf{(D)}\ \frac{1}{3}\qquad\textbf{(E)}\ \frac{1}{2}</math> | ||
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Abby, Bernardo, Carl, and Debra play a game in which each of them starts with four coins. The game consists of four rounds. In each round, four balls are placed in an urn---one green, one red, and two white. The players each draw a ball at random without replacement. Whoever gets the green ball gives one coin to whoever gets the red ball. What is the probability that, at the end of the fourth round, each of the players has four coins? | Abby, Bernardo, Carl, and Debra play a game in which each of them starts with four coins. The game consists of four rounds. In each round, four balls are placed in an urn---one green, one red, and two white. The players each draw a ball at random without replacement. Whoever gets the green ball gives one coin to whoever gets the red ball. What is the probability that, at the end of the fourth round, each of the players has four coins? | ||
− | <math>\textbf{(A)}\ | + | <math>\textbf{(A)}\ \frac{7}{576} \qquad \textbf{(B)}\ \frac{5}{192} \qquad \textbf{(C)}\ \frac{1}{36} \qquad \textbf{(D)}\ \frac{5}{144} \qquad\textbf{(E)}\ \frac{7}{48}</math> |
+ | |||
+ | [[2017 AMC 12B Problems/Problem 22|Solution]] | ||
==Problem 23== | ==Problem 23== | ||
− | The graph of <math>y=f(x)</math>, where <math>f(x)</math> is a polynomial of degree 3, contains points <math>A(2,4)</math>, <math>B(3,9)</math>, and <math>C(4,16)</math>. Lines <math>AB</math>, <math>AC</math>, and <math>BC</math> intersect the graph again at points <math>D</math>, <math>E</math>, and <math>F</math>, respectively, and the sum of the <math>x</math>-coordinates of <math>D</math>, <math>E</math>, and <math>F</math> is 24. What is <math>f(0)</math>? | + | The graph of <math>y=f(x)</math>, where <math>f(x)</math> is a polynomial of degree <math>3</math>, contains points <math>A(2,4)</math>, <math>B(3,9)</math>, and <math>C(4,16)</math>. Lines <math>AB</math>, <math>AC</math>, and <math>BC</math> intersect the graph again at points <math>D</math>, <math>E</math>, and <math>F</math>, respectively, and the sum of the <math>x</math>-coordinates of <math>D</math>, <math>E</math>, and <math>F</math> is <math>24</math>. What is <math>f(0)</math>? |
+ | <math>\textbf{(A)}\ -2 \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ \frac{24}{5} \qquad\textbf{(E)}\ 8</math> | ||
+ | |||
+ | [[2017 AMC 12B Problems/Problem 23|Solution]] | ||
− | |||
==Problem 24== | ==Problem 24== | ||
+ | |||
+ | Quadrilateral <math>ABCD</math> has right angles at <math>B</math> and <math>C</math>, <math>\triangle ABC \sim \triangle BCD</math>, and <math>AB > BC</math>. There is a point <math>E</math> in the interior of <math>ABCD</math> such that <math>\triangle ABC \sim \triangle CEB</math> and the area of <math>\triangle AED</math> is <math>17</math> times the area of <math>\triangle CEB</math>. What is <math>\frac{AB}{BC}</math>? | ||
+ | |||
+ | <math>\textbf{(A)}\ 1 + \sqrt{2} \qquad \textbf{(B)}\ 2 + \sqrt{2} \qquad \textbf{(C)}\ \sqrt{17} \qquad \textbf{(D)}\ 2 + \sqrt{5} \qquad\textbf{(E)}\ 1 + 2\sqrt{3}</math> | ||
+ | |||
+ | [[2017 AMC 12B Problems/Problem 24|Solution]] | ||
==Problem 25== | ==Problem 25== | ||
+ | A set of <math>n</math> people participate in an online video basketball tournament. Each person may be a member of any number of <math>5</math>-player teams, but no teams may have exactly the same <math>5</math> members. The site statistics show a curious fact: The average, over all subsets of size <math>9</math> of the set of <math>n</math> participants, of the number of complete teams whose members are among those 9 people is equal to the reciprocal of the average, over all subsets of size <math>8</math> of the set of <math>n</math> participants, of the number of complete teams whose members are among those <math>8</math> people. How many values <math>n</math>, <math>9 \leq n \leq 2017</math>, can be the number of participants? | ||
+ | |||
+ | <math>\textbf{(A)}\ 477 \qquad \textbf{(B)}\ 482 \qquad \textbf{(C)}\ 487 \qquad \textbf{(D)}\ 557 \qquad\textbf{(E)}\ 562</math> | ||
+ | |||
+ | [[2017 AMC 12B Problems/Problem 25|Solution]] | ||
==See also== | ==See also== | ||
{{AMC12 box|year=2017|ab=B|before=[[2017 AMC 12A Problems]]|after=[[2018 AMC 12A Problems]]}} | {{AMC12 box|year=2017|ab=B|before=[[2017 AMC 12A Problems]]|after=[[2018 AMC 12A Problems]]}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 17:26, 25 December 2020
2017 AMC 12B (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
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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
Kymbrea's comic book collection currently has comic books in it, and she is adding to her collection at the rate of comic books per month. LaShawn's collection currently has comic books in it, and he is adding to his collection at the rate of comic books per month. After how many months will LaShawn's collection have twice as many comic books as Kymbrea's?
Problem 2
Real numbers , , and satify the inequalities , , and . Which of the following numbers is necessarily positive?
Problem 3
Supposed that and are nonzero real numbers such that . What is the value of ?
Problem 4
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 5
The data set has median , first quartile , and third quartile . An outlier in a data set is a value that is more than times the interquartile range below the first quartile or more than times the interquartile range above the third quartile , where the interquartile range is defined as . How many outliers does this data set have?
Problem 6
The circle having and as the endpoints of a diameter intersects the -axis at a second point. What is the -coordinate of this point?
Problem 7
The functions and are periodic with least period . What is the least period of the function ?
It's not periodic.
Problem 8
The ratio of the short side of a certain rectangle to the long side is equal to the ratio of the long side to the diagonal. What is the square of the ratio of the short side to the long side of this rectangle?
Problem 9
A circle has center and radius . Another circle has center and radius . The line passing through the two points of intersection of the two circles has equation . What is ?
Problem 10
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 11
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 12
What is the sum of the roots of that have a positive real part?
Problem 13
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 14
An ice-cream novelty item consists of a cup in the shape of a 4-inch-tall frustum of a right circular cone, with a 2-inch-diameter base at the bottom and a 4-inch-diameter base at the top, packed solid with ice cream, together with a solid cone of ice cream of height 4 inches, whose base, at the bottom, is the top base of the frustum. What is the total volume of the ice cream, in cubic inches?
Problem 15
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 16
The number has over positive integer divisors. One of them is chosen at random. What is the probability that it is odd?
Problem 17
A coin is biased in such a way that on each toss the probability of heads is and the probability of tails is . The outcomes of the tosses are independent. A player has the choice of playing Game A or Game B. In Game A she tosses the coin three times and wins if all three outcomes are the same. In Game B she tosses the coin four times and wins if both the outcomes of the first and second tosses are the same and the outcomes of the third and fourth tosses are the same. How do the chances of winning Game A compare to the chances of winning Game B?
The probability of winning Game A is less than the probability of winning Game B.
The probability of winning Game A is less than the probability of winning Game B.
The probabilities are the same.
The probability of winning Game A is greater than the probability of winning Game B.
The probability of winning Game A is greater than the probability of winning Game B.
Problem 18
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 19
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 20
Real numbers and are chosen independently and uniformly at random from the interval . What is the probability that ?
Problem 21
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?
Problem 22
Abby, Bernardo, Carl, and Debra play a game in which each of them starts with four coins. The game consists of four rounds. In each round, four balls are placed in an urn---one green, one red, and two white. The players each draw a ball at random without replacement. Whoever gets the green ball gives one coin to whoever gets the red ball. What is the probability that, at the end of the fourth round, each of the players has four coins?
Problem 23
The graph of , where is a polynomial of degree , contains points , , and . Lines , , and intersect the graph again at points , , and , respectively, and the sum of the -coordinates of , , and is . What is ?
Problem 24
Quadrilateral has right angles at and , , and . There is a point in the interior of such that and the area of is times the area of . What is ?
Problem 25
A set of people participate in an online video basketball tournament. Each person may be a member of any number of -player teams, but no teams may have exactly the same members. The site statistics show a curious fact: The average, over all subsets of size of the set of participants, of the number of complete teams whose members are among those 9 people is equal to the reciprocal of the average, over all subsets of size of the set of participants, of the number of complete teams whose members are among those people. How many values , , can be the number of participants?
See also
2017 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by 2017 AMC 12A Problems |
Followed by 2018 AMC 12A 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 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.