Difference between revisions of "2020 AMC 10B Problems"
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==Problem 5== | ==Problem 5== | ||
− | How many distinguishable arrangements are there of 1 brown tile, 1 purple tile, 2 green tiles, and 3 yellow tiles in a row from left to right? (Tiles of the same color are indistinguishable.) | + | How many distinguishable arrangements are there of <math>1</math> brown tile, <math>1</math> purple tile, <math>2</math> green tiles, and <math>3</math> yellow tiles in a row from left to right? (Tiles of the same color are indistinguishable.) |
<math>\textbf{(A)}\ 210 \qquad\textbf{(B)}\ 420 \qquad\textbf{(C)}\ 630 \qquad\textbf{(D)}\ 840 \qquad\textbf{(E)}\ 1050</math> | <math>\textbf{(A)}\ 210 \qquad\textbf{(B)}\ 420 \qquad\textbf{(C)}\ 630 \qquad\textbf{(D)}\ 840 \qquad\textbf{(E)}\ 1050</math> | ||
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==Problem 9== | ==Problem 9== | ||
− | + | How many ordered pairs of integers <math>(x,y)</math> satisfy the equation <cmath>x^{2020} + y^2 = 2y?</cmath> | |
<math>\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{infinitely many}</math> | <math>\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{infinitely many}</math> | ||
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A three-quarter sector of a circle of radius <math>4</math> inches together with its interior can be rolled up to form the lateral surface area of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches? | A three-quarter sector of a circle of radius <math>4</math> inches together with its interior can be rolled up to form the lateral surface area of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches? | ||
+ | |||
<asy> | <asy> | ||
− | + | draw(Arc((0,0), 4, 0, 270)); | |
− | draw(Arc((0,0), | + | draw((0,-4)--(0,0)--(4,0)); |
− | draw((0,- | ||
− | |||
label("$4$", (2,0), S); | label("$4$", (2,0), S); | ||
+ | </asy> | ||
− | |||
− | |||
<math>\textbf{(A)}\ 3\pi \sqrt5 \qquad\textbf{(B)}\ 4\pi \sqrt3 \qquad\textbf{(C)}\ 3 \pi \sqrt7 \qquad\textbf{(D)}\ 6\pi \sqrt3 \qquad\textbf{(E)}\ 6\pi \sqrt7</math> | <math>\textbf{(A)}\ 3\pi \sqrt5 \qquad\textbf{(B)}\ 4\pi \sqrt3 \qquad\textbf{(C)}\ 3 \pi \sqrt7 \qquad\textbf{(D)}\ 6\pi \sqrt3 \qquad\textbf{(E)}\ 6\pi \sqrt7</math> | ||
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==Problem 11== | ==Problem 11== | ||
− | |||
− | <math>\textbf{(A)}\ \frac{1}{8} \qquad\textbf{(B)}\ \frac{5}{36} \qquad\textbf{(C)}\ | + | Ms. Carr asks her students to read any <math>5</math> of the <math>10</math> books on a reading list. Harold randomly selects <math>5</math> books from this list, and Betty does the same. What is the probability that there are exactly <math>2</math> books that they both select? |
+ | |||
+ | <math>\textbf{(A)}\ \frac{1}{8} \qquad\textbf{(B)}\ \frac{5}{36} \qquad\textbf{(C)}\ \frac{14}{45} \qquad\textbf{(D)}\ \frac{25}{63} \qquad\textbf{(E)}\ \frac{1}{2}</math> | ||
[[2020 AMC 10B Problems/Problem 11|Solution]] | [[2020 AMC 10B Problems/Problem 11|Solution]] | ||
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==Problem 13== | ==Problem 13== | ||
− | Andy the Ant lives on a coordinate plane and is currently at <math>(-20, 20)</math> facing east (that is, in the positive <math>x</math>-direction). Andy moves <math>1</math> unit and then turns <math>90^{\circ}</math> | + | Andy the Ant lives on a coordinate plane and is currently at <math>(-20, 20)</math> facing east (that is, in the positive <math>x</math>-direction). Andy moves <math>1</math> unit and then turns <math>90^{\circ}</math> left. From there, Andy moves <math>2</math> units (north) and then turns <math>90^{\circ}</math> left. He then moves <math>3</math> units (west) and again turns <math>90^{\circ}</math> left. Andy continues his progress, increasing his distance each time by <math>1</math> unit and always turning left. What is the location of the point which Andy makes the <math> 2020</math> left turn? |
<math>\textbf{(A)}\ (-1030, -994)\qquad\textbf{(B)}\ (-1030, -990)\qquad\textbf{(C)}\ (-1026, -994)\qquad\textbf{(D)}\ (-1026, -990)\qquad\textbf{(E)}\ (-1022, -994)</math> | <math>\textbf{(A)}\ (-1030, -994)\qquad\textbf{(B)}\ (-1030, -990)\qquad\textbf{(C)}\ (-1026, -994)\qquad\textbf{(D)}\ (-1026, -990)\qquad\textbf{(E)}\ (-1022, -994)</math> | ||
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<math> \textbf {(A) } 6\sqrt{3}-3\pi \qquad \textbf {(B) } \frac{9\sqrt{3}}{2} - 2\pi\ \qquad \textbf {(C) } \frac{3\sqrt{3}}{2} - \frac{\pi}{3} \qquad \textbf {(D) } 3\sqrt{3} - \pi \qquad \textbf {(E) } \frac{9\sqrt{3}}{2} - \pi </math> | <math> \textbf {(A) } 6\sqrt{3}-3\pi \qquad \textbf {(B) } \frac{9\sqrt{3}}{2} - 2\pi\ \qquad \textbf {(C) } \frac{3\sqrt{3}}{2} - \frac{\pi}{3} \qquad \textbf {(D) } 3\sqrt{3} - \pi \qquad \textbf {(E) } \frac{9\sqrt{3}}{2} - \pi </math> | ||
− | |||
[[2020 AMC 10B Problems/Problem 14|Solution]] | [[2020 AMC 10B Problems/Problem 14|Solution]] | ||
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Steve wrote the digits <math>1</math>, <math>2</math>, <math>3</math>, <math>4</math>, and <math>5</math> in order repeatedly from left to right, forming a list of <math>10,000</math> digits, beginning <math>123451234512\ldots.</math> He then erased every third digit from his list (that is, the <math>3</math>rd, <math>6</math>th, <math>9</math>th, <math>\ldots</math> digits from the left), then erased every fourth digit from the resulting list (that is, the <math>4</math>th, <math>8</math>th, <math>12</math>th, <math>\ldots</math> digits from the left in what remained), and then erased every fifth digit from what remained at that point. What is the sum of the three digits that were then in the positions <math>2019, 2020, 2021</math>? | Steve wrote the digits <math>1</math>, <math>2</math>, <math>3</math>, <math>4</math>, and <math>5</math> in order repeatedly from left to right, forming a list of <math>10,000</math> digits, beginning <math>123451234512\ldots.</math> He then erased every third digit from his list (that is, the <math>3</math>rd, <math>6</math>th, <math>9</math>th, <math>\ldots</math> digits from the left), then erased every fourth digit from the resulting list (that is, the <math>4</math>th, <math>8</math>th, <math>12</math>th, <math>\ldots</math> digits from the left in what remained), and then erased every fifth digit from what remained at that point. What is the sum of the three digits that were then in the positions <math>2019, 2020, 2021</math>? | ||
− | <math>\textbf{(A)} \ | + | <math>\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 9 \qquad\textbf{(C)}\ 10 \qquad\textbf{(D)}\ 11 \qquad\textbf{(E)}\ 12</math> |
[[2020 AMC 10B Problems/Problem 15|Solution]] | [[2020 AMC 10B Problems/Problem 15|Solution]] | ||
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==Problem 18== | ==Problem 18== | ||
− | An urn contains one red ball and one blue ball. A box of extra red and blue balls | + | An urn contains one red ball and one blue ball. A box of extra red and blue balls lies nearby. George performs the following operation four times: he draws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to the urn. After the four iterations the urn contains six balls. What is the probability that the urn contains three balls of each color? |
<math>\textbf{(A) } \frac16 \qquad \textbf{(B) }\frac15 \qquad \textbf{(C) } \frac14 \qquad \textbf{(D) } \frac13 \qquad \textbf{(E) } \frac12</math> | <math>\textbf{(A) } \frac16 \qquad \textbf{(B) }\frac15 \qquad \textbf{(C) } \frac14 \qquad \textbf{(D) } \frac13 \qquad \textbf{(E) } \frac12</math> | ||
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==Problem 20== | ==Problem 20== | ||
− | Let <math>B</math> be a right rectangular prism (box) with edges lengths <math>1,</math> <math>3,</math> and <math>4</math>, together with its interior. For real <math>r\geq0</math>, let <math>S(r)</math> be the set of points in <math>3</math>-dimensional space that lie within a distance <math>r</math> of some point <math>B</math>. The volume of <math>S(r)</math> can be expressed as <math>ar^{3} + br^{2} + cr +d</math>, where <math>a,</math> <math>b,</math> <math>c,</math> and <math>d</math> are positive real numbers. What is <math>\frac{bc}{ad}?</math> | + | Let <math>B</math> be a right rectangular prism (box) with edges lengths <math>1,</math> <math>3,</math> and <math>4</math>, together with its interior. For real <math>r\geq0</math>, let <math>S(r)</math> be the set of points in <math>3</math>-dimensional space that lie within a distance <math>r</math> of some point in <math>B</math>. The volume of <math>S(r)</math> can be expressed as <math>ar^{3} + br^{2} + cr +d</math>, where <math>a,</math> <math>b,</math> <math>c,</math> and <math>d</math> are positive real numbers. What is <math>\frac{bc}{ad}?</math> |
<math>\textbf{(A) } 6 \qquad\textbf{(B) } 19 \qquad\textbf{(C) } 24 \qquad\textbf{(D) } 26 \qquad\textbf{(E) } 38</math> | <math>\textbf{(A) } 6 \qquad\textbf{(B) } 19 \qquad\textbf{(C) } 24 \qquad\textbf{(D) } 26 \qquad\textbf{(E) } 38</math> | ||
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Square <math>ABCD</math> in the coordinate plane has vertices at the points <math>A(1,1), B(-1,1), C(-1,-1),</math> and <math>D(1,-1).</math> Consider the following four transformations: | Square <math>ABCD</math> in the coordinate plane has vertices at the points <math>A(1,1), B(-1,1), C(-1,-1),</math> and <math>D(1,-1).</math> Consider the following four transformations: | ||
+ | <math>\quad\bullet\qquad</math> <math>L,</math> a rotation of <math>90^{\circ}</math> counterclockwise around the origin; | ||
− | + | <math>\quad\bullet\qquad</math> <math>R,</math> a rotation of <math>90^{\circ}</math> clockwise around the origin; | |
− | |||
− | |||
− | |||
− | |||
− | + | <math>\quad\bullet\qquad</math> <math>H,</math> a reflection across the <math>x</math>-axis; and | |
+ | <math>\quad\bullet\qquad</math> <math>V,</math> a reflection across the <math>y</math>-axis. | ||
Each of these transformations maps the squares onto itself, but the positions of the labeled vertices will change. For example, applying <math>R</math> and then <math>V</math> would send the vertex <math>A</math> at <math>(1,1)</math> to <math>(-1,-1)</math> and would send the vertex <math>B</math> at <math>(-1,1)</math> to itself. How many sequences of <math>20</math> transformations chosen from <math>\{L, R, H, V\}</math> will send all of the labeled vertices back to their original positions? (For example, <math>R, R, V, H</math> is one sequence of <math>4</math> transformations that will send the vertices back to their original positions.) | Each of these transformations maps the squares onto itself, but the positions of the labeled vertices will change. For example, applying <math>R</math> and then <math>V</math> would send the vertex <math>A</math> at <math>(1,1)</math> to <math>(-1,-1)</math> and would send the vertex <math>B</math> at <math>(-1,1)</math> to itself. How many sequences of <math>20</math> transformations chosen from <math>\{L, R, H, V\}</math> will send all of the labeled vertices back to their original positions? (For example, <math>R, R, V, H</math> is one sequence of <math>4</math> transformations that will send the vertices back to their original positions.) |
Latest revision as of 15:43, 30 June 2024
2020 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
What is the value of
Problem 2
Carl has cubes each having side length
, and Kate has
cubes each having side length
. What is the total volume of these
cubes?
Problem 3
The ratio of to
is
, the ratio of
to
is
, and the ratio of
to
is
. What is the ratio of
to
Problem 4
The acute angles of a right triangle are and
, where
and both
and
are prime numbers. What is the least possible value of
?
Problem 5
How many distinguishable arrangements are there of brown tile,
purple tile,
green tiles, and
yellow tiles in a row from left to right? (Tiles of the same color are indistinguishable.)
Problem 6
Driving along a highway, Megan noticed that her odometer showed (miles). This number is a palindrome-it reads the same forward and backward. Then
hours later, the odometer displayed the next higher palindrome. What was her average speed, in miles per hour, during this
-hour period?
Problem 7
How many positive even multiples of less than
are perfect squares?
Problem 8
Points and
lie in a plane with
. How many locations for point
in this plane are there such that the triangle with vertices
,
, and
is a right triangle with area
square units?
Problem 9
How many ordered pairs of integers satisfy the equation
Problem 10
A three-quarter sector of a circle of radius inches together with its interior can be rolled up to form the lateral surface area of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches?
Problem 11
Ms. Carr asks her students to read any of the
books on a reading list. Harold randomly selects
books from this list, and Betty does the same. What is the probability that there are exactly
books that they both select?
Problem 12
The decimal representation of
consists of a string of zeros after the decimal point, followed by a
and then several more digits. How many zeros are in that initial string of zeros after the decimal point?
Problem 13
Andy the Ant lives on a coordinate plane and is currently at facing east (that is, in the positive
-direction). Andy moves
unit and then turns
left. From there, Andy moves
units (north) and then turns
left. He then moves
units (west) and again turns
left. Andy continues his progress, increasing his distance each time by
unit and always turning left. What is the location of the point which Andy makes the
left turn?
Problem 14
As shown in the figure below, six semicircles lie in the interior of a regular hexagon with side length 2 so that the diameters of the semicircles coincide with the sides of the hexagon. What is the area of the shaded region — inside the hexagon but outside all of the semicircles?
Problem 15
Steve wrote the digits ,
,
,
, and
in order repeatedly from left to right, forming a list of
digits, beginning
He then erased every third digit from his list (that is, the
rd,
th,
th,
digits from the left), then erased every fourth digit from the resulting list (that is, the
th,
th,
th,
digits from the left in what remained), and then erased every fifth digit from what remained at that point. What is the sum of the three digits that were then in the positions
?
Problem 16
Bela and Jenn play the following game on the closed interval of the real number line, where
is a fixed integer greater than
. They take turns playing, with Bela going first. At his first turn, Bela chooses any real number in the interval
. Thereafter, the player whose turn it is chooses a real number that is more than one unit away from all numbers previously chosen by either player. A player unable to choose such a number loses. Using optimal strategy, which player will win the game?
Problem 17
There are people standing equally spaced around a circle. Each person knows exactly
of the other
people: the
people standing next to her or him, as well as the person directly across the circle. How many ways are there for the
people to split up into
pairs so that the members of each pair know each other?
Problem 18
An urn contains one red ball and one blue ball. A box of extra red and blue balls lies nearby. George performs the following operation four times: he draws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to the urn. After the four iterations the urn contains six balls. What is the probability that the urn contains three balls of each color?
Problem 19
In a certain card game, a player is dealt a hand of cards from a deck of
distinct cards. The number of distinct (unordered) hands that can be dealt to the player can be written as
. What is the digit
?
Problem 20
Let be a right rectangular prism (box) with edges lengths
and
, together with its interior. For real
, let
be the set of points in
-dimensional space that lie within a distance
of some point in
. The volume of
can be expressed as
, where
and
are positive real numbers. What is
Problem 21
In square , points
and
lie on
and
, respectively, so that
Points
and
lie on
and
, respectively, and points
and
lie on
so that
and
. See the figure below. Triangle
, quadrilateral
, quadrilateral
, and pentagon
each has area
What is
?
Problem 22
What is the remainder when is divided by
?
Problem 23
Square in the coordinate plane has vertices at the points
and
Consider the following four transformations:
a rotation of
counterclockwise around the origin;
a rotation of
clockwise around the origin;
a reflection across the
-axis; and
a reflection across the
-axis.
Each of these transformations maps the squares onto itself, but the positions of the labeled vertices will change. For example, applying and then
would send the vertex
at
to
and would send the vertex
at
to itself. How many sequences of
transformations chosen from
will send all of the labeled vertices back to their original positions? (For example,
is one sequence of
transformations that will send the vertices back to their original positions.)
Problem 24
How many positive integers satisfy
(Recall that
is the greatest integer not exceeding
.)
Problem 25
Let denote the number of ways of writing the positive integer
as a product
where , the
are integers strictly greater than
, and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are counted as distinct). For example, the number
can be written as
,
, and
, so
. What is
?
See also
2020 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by 2020 AMC 10A Problems |
Followed by 2021 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.