Difference between revisions of "2017 AMC 12B Problems"
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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== | ||
Line 28: | Line 28: | ||
==Problem 4== | ==Problem 4== | ||
+ | Samia set off on her bicycle to visit her friend, traveling at an average speed of <math>17</math> 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 <math>5</math> kilometers per hour. In all it took her <math>44</math> minutes to reach her friend's house. In kilometers rounded to the nearest tenth, how far did Samia walk? | ||
+ | |||
+ | <math>\textbf{(A)}\ 2.0\qquad\textbf{(B)}\ 2.2\qquad\textbf{(C)}\ 2.8\qquad\textbf{(D)}\ 3.4\qquad\textbf{(E)}\ 4.4</math> | ||
+ | |||
+ | [[2017 AMC 12B Problems/Problem 4|Solution]] | ||
==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== | ||
+ | The circle having <math>(0,0)</math> and <math>(8,6)</math> as the endpoints of a diameter intersects the <math>x</math>-axis at a second point. What is the <math>x</math>-coordinate of this point? | ||
+ | |||
+ | <math>\textbf{(A)}\ 4\sqrt{2} \qquad \textbf{(B)}\ 6\qquad \textbf{(C)}\ 5\sqrt{2}\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 6\sqrt{2}</math> | ||
+ | |||
+ | [[2017 AMC 12B Problems/Problem 6|Solution]] | ||
==Problem 7== | ==Problem 7== | ||
+ | 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{\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]] | ||
==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== | ||
+ | At Typico High School, <math>60\%</math> of the students like dancing, and the rest dislike it. Of those who like dancing, <math>80\%</math> say that they like it, and the rest say that they dislike it. Of those who dislike dancing, <math>90\%</math> 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? | ||
+ | |||
+ | <math>\textbf{(A)}\ 10\%\qquad\textbf{(B)}\ 12\%\qquad\textbf{(C)}\ 20\%\qquad\textbf{(D)}\ 25\%\qquad\textbf{(E)}\ 33\frac{1}{3}\%</math> | ||
+ | |||
+ | [[2017 AMC 12B Problems/Problem 10|Solution]] | ||
==Problem 11== | ==Problem 11== | ||
+ | Call a positive integer <math>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> | ||
+ | |||
+ | [[2017 AMC 10B Problems/Problem 17|Solution]] | ||
==Problem 12== | ==Problem 12== | ||
+ | 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> | ||
+ | |||
+ | [[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== | ||
+ | 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? | ||
+ | |||
+ | <math>\textbf{(A)}\ 8\pi \qquad \textbf{(B)}\ \frac{28\pi}{3} \qquad \textbf{(C)}\ 12\pi \qquad \textbf{(D)}\ 14\pi \qquad \textbf{(E)}\ \frac{44\pi}{3}</math> | ||
+ | |||
+ | [[2017 AMC 12B Problems/Problem 14|Solution]] | ||
==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'=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\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 36\qquad\textbf{(E)}\ 37</math> | ||
+ | |||
+ | [[2017 AMC 12B Problems/Problem 15|Solution]] | ||
==Problem 16== | ==Problem 16== | ||
Line 59: | Line 139: | ||
==Problem 17== | ==Problem 17== | ||
+ | A coin is biased in such a way that on each toss the probability of heads is <math>\frac{2}{3}</math> and the probability of tails is <math>\frac{1}{3}</math>. 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? | ||
+ | |||
+ | <math>\textbf{(A)}</math> The probability of winning Game A is <math>\frac{4}{81}</math> less than the probability of winning Game B. | ||
+ | |||
+ | <math>\textbf{(B)}</math> The probability of winning Game A is <math>\frac{2}{81}</math> less than the probability of winning Game B. | ||
+ | |||
+ | <math>\textbf{(C)}</math> The probabilities are the same. | ||
+ | |||
+ | <math>\textbf{(D)}</math> The probability of winning Game A is <math>\frac{2}{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]] | ||
==Problem 18== | ==Problem 18== | ||
+ | The diameter <math>AB</math> of a circle of radius <math>2</math> is extended to a point <math>D</math> outside the circle so that <math>BD=3</math>. Point <math>E</math> is chosen so that <math>ED=5</math> and line <math>ED</math> is perpendicular to line <math>AD</math>. Segment <math>AE</math> intersects the circle at a point <math>C</math> between <math>A</math> and <math>E</math>. What is the area of <math>\triangle | ||
+ | ABC</math>? | ||
+ | |||
+ | <math>\textbf{(A)}\ \frac{120}{37}\qquad\textbf{(B)}\ \frac{140}{39}\qquad\textbf{(C)}\ \frac{145}{39}\qquad\textbf{(D)}\ \frac{140}{37}\qquad\textbf{(E)}\ \frac{120}{31}</math> | ||
+ | |||
+ | [[2017 AMC 12B Problems/Problem 18|Solution]] | ||
==Problem 19== | ==Problem 19== | ||
+ | Let <math>N=123456789101112\dots4344</math> be the <math>79</math>-digit number that is formed by writing the integers from <math>1</math> to <math>44</math> in order, one after the other. What is the remainder when <math>N</math> is divided by <math>45</math>? | ||
+ | |||
+ | <math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 44</math> | ||
+ | |||
+ | [[2017 AMC 12B Problems/Problem 19|Solution]] | ||
==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</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> | ||
+ | |||
+ | [[2017 AMC 12B Problems/Problem 20|Solution]] | ||
==Problem 21== | ==Problem 21== | ||
+ | Last year Isabella took <math>7</math> math tests and received <math>7</math> different scores, each an integer between <math>91</math> and <math>100</math>, inclusive. After each test she noticed that the average of her test scores was an integer. Her score on the seventh test was <math>95</math>. What was her score on the sixth test? | ||
+ | |||
+ | <math>\textbf{(A)}\ 92\qquad\textbf{(B)}\ 94\qquad\textbf{(C)}\ 96\qquad\textbf{(D)}\ 98\qquad\textbf{(E)}\ 100</math> | ||
+ | |||
+ | [[2017 AMC 12B Problems/Problem 21|Solution]] | ||
==Problem 22== | ==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? | ||
+ | |||
+ | <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 <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 18:26, 25 December 2020
2017 AMC 12B (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
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.