Difference between revisions of "2017 AMC 10B Problems"
m (→Problem 3) |
m (→Problem 1) |
||
(51 intermediate revisions by 21 users not shown) | |||
Line 10: | Line 10: | ||
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]] | ||
Line 16: | Line 17: | ||
==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? | ||
Line 25: | Line 26: | ||
==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> | ||
[[2017 AMC 10B Problems/Problem 4|Solution]] | [[2017 AMC 10B Problems/Problem 4|Solution]] | ||
==Problem 5== | ==Problem 5== | ||
− | + | 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> | ||
[[2017 AMC 10B Problems/Problem 5|Solution]] | [[2017 AMC 10B Problems/Problem 5|Solution]] | ||
==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? | |
+ | |||
+ | <math>\textbf{(A)}\ 3\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 7</math> | ||
[[2017 AMC 10B Problems/Problem 6|Solution]] | [[2017 AMC 10B Problems/Problem 6|Solution]] | ||
==Problem 7== | ==Problem 7== | ||
− | + | 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 10B Problems/Problem 7|Solution]] | [[2017 AMC 10B Problems/Problem 7|Solution]] | ||
==Problem 8== | ==Problem 8== | ||
− | + | Points <math>A(11, 9)</math> and <math>B(2, -3)</math> are vertices of <math>\triangle ABC</math> with <math>AB=AC</math>. The altitude from <math>A</math> meets the opposite side at <math>D(-1, 3)</math>. What are the coordinates of point <math>C</math>? | |
+ | |||
+ | <math>\textbf{(A)}\ (-8, 9)\qquad\textbf{(B)}\ (-4, 8)\qquad\textbf{(C)}\ (-4, 9)\qquad\textbf{(D)}\ (-2, 3)\qquad\textbf{(E)}\ (-1, 0)</math> | ||
[[2017 AMC 10B Problems/Problem 8|Solution]] | [[2017 AMC 10B Problems/Problem 8|Solution]] | ||
==Problem 9== | ==Problem 9== | ||
− | + | A radio program has a quiz consisting of <math>3</math> multiple-choice questions, each with <math>3</math> choices. A contestant wins if he or she gets <math>2</math> or more of the questions right. The contestant answers randomly to each question. What is the probability of winning? | |
+ | |||
+ | <math>\textbf{(A)}\ \frac{1}{27}\qquad\textbf{(B)}\ \frac{1}{9}\qquad\textbf{(C)}\ \frac{2}{9}\qquad\textbf{(D)}\ \frac{7}{27}\qquad\textbf{(E)}\ \frac{1}{2}</math> | ||
[[2017 AMC 10B Problems/Problem 9|Solution]] | [[2017 AMC 10B Problems/Problem 9|Solution]] | ||
==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]] | ||
==Problem 11== | ==Problem 11== | ||
− | + | 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 10B Problems/Problem 11|Solution]] | [[2017 AMC 10B Problems/Problem 11|Solution]] | ||
==Problem 12== | ==Problem 12== | ||
− | + | 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> | ||
[[2017 AMC 10B Problems/Problem 12|Solution]] | [[2017 AMC 10B Problems/Problem 12|Solution]] | ||
==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 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> | ||
[[2017 AMC 10B Problems/Problem 15|Solution]] | [[2017 AMC 10B Problems/Problem 15|Solution]] | ||
==Problem 16== | ==Problem 16== | ||
− | + | How many of the base-ten numerals for the positive integers less than or equal to <math>2017</math> contain the digit <math>0</math>? | |
+ | |||
+ | <math>\textbf{(A)}\ 469\qquad\textbf{(B)}\ 471\qquad\textbf{(C)}\ 475\qquad\textbf{(D)}\ 478\qquad\textbf{(E)}\ 481</math> | ||
[[2017 AMC 10B Problems/Problem 16|Solution]] | [[2017 AMC 10B Problems/Problem 16|Solution]] | ||
==Problem 17== | ==Problem 17== | ||
− | + | 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> | ||
[[2017 AMC 10B Problems/Problem 17|Solution]] | [[2017 AMC 10B Problems/Problem 17|Solution]] | ||
==Problem 18== | ==Problem 18== | ||
− | + | 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 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'=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> | ||
[[2017 AMC 10B Problems/Problem 19|Solution]] | [[2017 AMC 10B Problems/Problem 19|Solution]] | ||
==Problem 20== | ==Problem 20== | ||
− | + | 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> | ||
[[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 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> | ||
[[2017 AMC 10B Problems/Problem 21|Solution]] | [[2017 AMC 10B Problems/Problem 21|Solution]] |
Latest revision as of 20:23, 9 September 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.