Difference between revisions of "2021 Fall AMC 12B Problems"
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} | } | ||
label("$0$", O, 2*SW); | label("$0$", O, 2*SW); | ||
− | draw(O--X+(0. | + | draw(O--X+(0.35,0), black+1.5, EndArrow(10)); |
− | draw(O--Y+(0,0. | + | draw(O--Y+(0,0.35), black+1.5, EndArrow(10)); |
draw((1,0)--(3,5)--(5,0)--(3,2)--(1,0), black+1.5); | draw((1,0)--(3,5)--(5,0)--(3,2)--(1,0), black+1.5); | ||
</asy> | </asy> | ||
− | <math> | + | <math>\textbf{(A)}\: 4\qquad\textbf{(B)} \: 6\qquad\textbf{(C)} \: 8\qquad\textbf{(D)} \: 10\qquad\textbf{(E)} \: 12</math> |
[[2021 Fall AMC 12B Problems/Problem 2|Solution]] | [[2021 Fall AMC 12B Problems/Problem 2|Solution]] | ||
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At noon on a certain day, Minneapolis is <math>N</math> degrees warmer than St. Louis. At <math>4{:}00</math> the temperature in Minneapolis has fallen by <math>5</math> degrees while the temperature in St. Louis has risen by <math>3</math> degrees, at which time the temperatures in the two cities differ by <math>2</math> degrees. What is the product of all possible values of <math>N?</math> | At noon on a certain day, Minneapolis is <math>N</math> degrees warmer than St. Louis. At <math>4{:}00</math> the temperature in Minneapolis has fallen by <math>5</math> degrees while the temperature in St. Louis has risen by <math>3</math> degrees, at which time the temperatures in the two cities differ by <math>2</math> degrees. What is the product of all possible values of <math>N?</math> | ||
− | <math> | + | <math>\textbf{(A)}\: 10\qquad\textbf{(B)} \: 30\qquad\textbf{(C)} \: 60\qquad\textbf{(D)} \: 100\qquad\textbf{(E)} \: 120</math> |
[[2021 Fall AMC 12B Problems/Problem 3|Solution]] | [[2021 Fall AMC 12B Problems/Problem 3|Solution]] | ||
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Let <math>n=8^{2022}</math>. Which of the following is equal to <math>\frac{n}{4}?</math> | Let <math>n=8^{2022}</math>. Which of the following is equal to <math>\frac{n}{4}?</math> | ||
− | <math> | + | <math>\textbf{(A)}\: 4^{1010}\qquad\textbf{(B)} \: 2^{2022}\qquad\textbf{(C)} \: 8^{2018}\qquad\textbf{(D)} \: 4^{3031}\qquad\textbf{(E)} \: 4^{3032}</math> |
[[2021 Fall AMC 12B Problems/Problem 4|Solution]] | [[2021 Fall AMC 12B Problems/Problem 4|Solution]] | ||
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==Problem 6== | ==Problem 6== | ||
− | The | + | The greatest prime number that is a divisor of <math>16{,}384</math> is <math>2</math> because <math>16{,}384 = 2^{14}</math>. What is the sum of the digits of the greatest prime number that is a divisor of <math>16{,}383</math>? |
<math>\textbf{(A)} \: 3\qquad\textbf{(B)} \: 7\qquad\textbf{(C)} \: 10\qquad\textbf{(D)} \: 16\qquad\textbf{(E)} \: 22</math> | <math>\textbf{(A)} \: 3\qquad\textbf{(B)} \: 7\qquad\textbf{(C)} \: 10\qquad\textbf{(D)} \: 16\qquad\textbf{(E)} \: 22</math> | ||
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==Problem 10== | ==Problem 10== | ||
− | What is the sum of all possible values of <math>t</math> between <math>0</math> and <math>360</math> such that the triangle in the coordinate plane whose vertices are < | + | What is the sum of all possible values of <math>t</math> between <math>0</math> and <math>360</math> such that the triangle in the coordinate plane whose vertices are <cmath>(\cos 40^\circ,\sin 40^\circ), (\cos 60^\circ,\sin 60^\circ), \text{ and } (\cos t^\circ,\sin t^\circ)</cmath> |
is isosceles? | is isosceles? | ||
Line 117: | Line 117: | ||
==Problem 12== | ==Problem 12== | ||
− | For <math>n</math> a positive integer, let <math>f(n)</math> be the quotient obtained when the sum of all positive divisors | + | For <math>n</math> a positive integer, let <math>f(n)</math> be the quotient obtained when the sum of all positive divisors of <math>n</math> is divided by <math>n.</math> For example, <cmath>f(14)=(1+2+7+14)\div 14=\frac{12}{7}</cmath> |
− | of n is divided by n. For example, | ||
− | <cmath>f(14)=(1+2+7+14)\div 14=\frac{12}{7}</cmath> | ||
What is <math>f(768)-f(384)?</math> | What is <math>f(768)-f(384)?</math> | ||
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<cmath>\frac{\sin 3c \cdot \sin 6c \cdot \sin 9c \cdot \sin 12c \cdot \sin 15c}{\sin c \cdot \sin 2c \cdot \sin 3c \cdot \sin 4c \cdot \sin 5c}?</cmath> | <cmath>\frac{\sin 3c \cdot \sin 6c \cdot \sin 9c \cdot \sin 12c \cdot \sin 15c}{\sin c \cdot \sin 2c \cdot \sin 3c \cdot \sin 4c \cdot \sin 5c}?</cmath> | ||
− | <math>\textbf{(A)}\ -1 \qquad\textbf{(B)}\ \frac{\sqrt{ | + | <math>\textbf{(A)}\ {-}1 \qquad\textbf{(B)}\ {-}\frac{\sqrt{11}}{5} \qquad\textbf{(C)}\ \frac{\sqrt{11}}{5} \qquad\textbf{(D)}\ |
\frac{10}{11} \qquad\textbf{(E)}\ 1</math> | \frac{10}{11} \qquad\textbf{(E)}\ 1</math> | ||
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==Problem 15== | ==Problem 15== | ||
Three identical square sheets of paper each with side length <math>6</math> are stacked on top of each other. The middle sheet is rotated clockwise <math>30^\circ</math> about its center and the top sheet is rotated clockwise <math>60^\circ</math> about its center, resulting in the <math>24</math>-sided polygon shown in the figure below. The area of this polygon can be expressed in the form <math>a-b\sqrt{c}</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are positive integers, and <math>c</math> is not divisible by the square of any prime. What is <math>a+b+c</math>? | Three identical square sheets of paper each with side length <math>6</math> are stacked on top of each other. The middle sheet is rotated clockwise <math>30^\circ</math> about its center and the top sheet is rotated clockwise <math>60^\circ</math> about its center, resulting in the <math>24</math>-sided polygon shown in the figure below. The area of this polygon can be expressed in the form <math>a-b\sqrt{c}</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are positive integers, and <math>c</math> is not divisible by the square of any prime. What is <math>a+b+c</math>? | ||
− | + | <center><asy> | |
− | + | defaultpen(fontsize(8)+0.8); size(150); | |
− | + | pair O,A1,B1,C1,A2,B2,C2,A3,B3,C3,A4,B4,C4; | |
+ | real x=45, y=90, z=60; O=origin; | ||
+ | A1=dir(x); A2=dir(x+y); A3=dir(x+2y); A4=dir(x+3y); | ||
+ | B1=dir(x-z); B2=dir(x+y-z); B3=dir(x+2y-z); B4=dir(x+3y-z); | ||
+ | C1=dir(x-2z); C2=dir(x+y-2z); C3=dir(x+2y-2z); C4=dir(x+3y-2z); | ||
+ | draw(A1--A2--A3--A4--A1, gray+0.25+dashed); | ||
+ | filldraw(B1--B2--B3--B4--cycle, white, gray+dashed+linewidth(0.25)); | ||
+ | filldraw(C1--C2--C3--C4--cycle, white, gray+dashed+linewidth(0.25)); | ||
+ | dot(O); | ||
+ | pair P1,P2,P3,P4,Q1,Q2,Q3,Q4,R1,R2,R3,R4; | ||
+ | P1=extension(A1,A2,B1,B2); Q1=extension(A1,A2,C3,C4); | ||
+ | P2=extension(A2,A3,B2,B3); Q2=extension(A2,A3,C4,C1); | ||
+ | P3=extension(A3,A4,B3,B4); Q3=extension(A3,A4,C1,C2); | ||
+ | P4=extension(A4,A1,B4,B1); Q4=extension(A4,A1,C2,C3); | ||
+ | R1=extension(C2,C3,B2,B3); R2=extension(C3,C4,B3,B4); | ||
+ | R3=extension(C4,C1,B4,B1); R4=extension(C1,C2,B1,B2); | ||
+ | draw(A1--P1--B2--R1--C3--Q1--A2); | ||
+ | draw(A2--P2--B3--R2--C4--Q2--A3); | ||
+ | draw(A3--P3--B4--R3--C1--Q3--A4); | ||
+ | draw(A4--P4--B1--R4--C2--Q4--A1); | ||
+ | </asy></center> | ||
<math>(\textbf{A})\: 75\qquad(\textbf{B}) \: 93\qquad(\textbf{C}) \: 96\qquad(\textbf{D}) \: 129\qquad(\textbf{E}) \: 147</math> | <math>(\textbf{A})\: 75\qquad(\textbf{B}) \: 93\qquad(\textbf{C}) \: 96\qquad(\textbf{D}) \: 129\qquad(\textbf{E}) \: 147</math> | ||
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Suppose <math>a</math>, <math>b</math>, <math>c</math> are positive integers such that <cmath>a+b+c=23</cmath> and <cmath>\gcd(a,b)+\gcd(b,c)+\gcd(c,a)=9.</cmath> What is the sum of all possible distinct values of <math>a^2+b^2+c^2</math>? | Suppose <math>a</math>, <math>b</math>, <math>c</math> are positive integers such that <cmath>a+b+c=23</cmath> and <cmath>\gcd(a,b)+\gcd(b,c)+\gcd(c,a)=9.</cmath> What is the sum of all possible distinct values of <math>a^2+b^2+c^2</math>? | ||
− | <math> | + | <math>\textbf{(A)}\: 259\qquad\textbf{(B)} \: 438\qquad\textbf{(C)} \: 516\qquad\textbf{(D)} \: 625\qquad\textbf{(E)} \: 687</math> |
[[2021 Fall AMC 12B Problems/Problem 16|Solution]] | [[2021 Fall AMC 12B Problems/Problem 16|Solution]] | ||
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This sequence tends to a limit; call it <math>L</math>. What is the least value of <math>k</math> such that <cmath>|u_k-L| \le \frac{1}{2^{1000}}?</cmath> | This sequence tends to a limit; call it <math>L</math>. What is the least value of <math>k</math> such that <cmath>|u_k-L| \le \frac{1}{2^{1000}}?</cmath> | ||
− | <math> | + | <math>\textbf{(A)}\: 10\qquad\textbf{(B)}\: 87\qquad\textbf{(C)}\: 123\qquad\textbf{(D)}\: 329\qquad\textbf{(E)}\: 401</math> |
[[2021 Fall AMC 12B Problems/Problem 18|Solution]] | [[2021 Fall AMC 12B Problems/Problem 18|Solution]] | ||
==Problem 19== | ==Problem 19== | ||
− | Regular polygons with <math>5 | + | Regular polygons with <math>5,6,7,</math> and <math>8</math> sides are inscribed in the same circle. No two of the polygons share a vertex, and no three of their sides intersect at a common point. At how many points inside the circle do two of their sides intersect? |
<math>(\textbf{A})\: 52\qquad(\textbf{B}) \: 56\qquad(\textbf{C}) \: 60\qquad(\textbf{D}) \: 64\qquad(\textbf{E}) \: 68</math> | <math>(\textbf{A})\: 52\qquad(\textbf{B}) \: 56\qquad(\textbf{C}) \: 60\qquad(\textbf{D}) \: 64\qquad(\textbf{E}) \: 68</math> | ||
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==Problem 22== | ==Problem 22== | ||
− | Right triangle <math>ABC</math> has side lengths <math>BC=6</math>, <math>AC=8</math>, and <math>AB=10</math>. | + | Right triangle <math>ABC</math> has side lengths <math>BC=6</math>, <math>AC=8</math>, and <math>AB=10</math>. A circle centered at <math>O</math> is tangent to line <math>BC</math> at <math>B</math> and passes through <math>A</math>. A circle centered at <math>P</math> is tangent to line <math>AC</math> at <math>A</math> and passes through <math>B</math>. What is <math>OP</math>? |
− | |||
− | A circle centered at <math>O</math> is tangent to line <math>BC</math> at <math>B</math> and passes through <math>A</math>. A circle centered at <math>P</math> is tangent to line <math>AC</math> at <math>A</math> and passes through <math>B</math>. What is <math>OP</math>? | ||
<math>\textbf{(A)}\ \frac{23}{8} \qquad\textbf{(B)}\ \frac{29}{10} \qquad\textbf{(C)}\ \frac{35}{12} \qquad\textbf{(D)}\ | <math>\textbf{(A)}\ \frac{23}{8} \qquad\textbf{(B)}\ \frac{29}{10} \qquad\textbf{(C)}\ \frac{35}{12} \qquad\textbf{(D)}\ | ||
Line 240: | Line 256: | ||
==See also== | ==See also== | ||
− | {{AMC12 box|year=2021 Fall|ab= | + | {{AMC12 box|year=2021 Fall|ab=B|before=[[2021 Fall AMC 12A Problems]]|after=[[2022 AMC 12A Problems]]}} |
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 19:57, 12 November 2023
2021 Fall AMC 12B (Answer Key) Printable versions: • Fall AoPS Resources • Fall 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
What is the area of the shaded figure shown below?
Problem 3
At noon on a certain day, Minneapolis is degrees warmer than St. Louis. At
the temperature in Minneapolis has fallen by
degrees while the temperature in St. Louis has risen by
degrees, at which time the temperatures in the two cities differ by
degrees. What is the product of all possible values of
Problem 4
Let . Which of the following is equal to
Problem 5
Call a fraction , not necessarily in the simplest form, special if
and
are positive integers whose sum is
. How many distinct integers can be written as the sum of two, not necessarily different, special fractions?
Problem 6
The greatest prime number that is a divisor of is
because
. What is the sum of the digits of the greatest prime number that is a divisor of
?
Problem 7
Which of the following conditions is sufficient to guarantee that integers ,
, and
satisfy the equation
and
and
and
and
Problem 8
The product of the lengths of the two congruent sides of an obtuse isosceles triangle is equal to the product of the base and twice the triangle's height to the base. What is the measure, in degrees, of the vertex angle of this triangle?
Problem 9
Triangle is equilateral with side length
. Suppose that
is the center of the inscribed
circle of this triangle. What is the area of the circle passing through
,
, and
?
Problem 10
What is the sum of all possible values of between
and
such that the triangle in the coordinate plane whose vertices are
is isosceles?
Problem 11
Una rolls standard
-sided dice simultaneously and calculates the product of the
numbers obtained. What is the probability that the product is divisible by
Problem 12
For a positive integer, let
be the quotient obtained when the sum of all positive divisors of
is divided by
For example,
What is
Problem 13
Let What is the value of
Problem 14
Suppose that , and
are polynomials with real coefficients, having degrees
,
, and
, respectively, and constant terms
,
, and
, respectively. Let
be the number of distinct complex numbers
that satisfy the equation
. What is the minimum possible value of
?
Problem 15
Three identical square sheets of paper each with side length are stacked on top of each other. The middle sheet is rotated clockwise
about its center and the top sheet is rotated clockwise
about its center, resulting in the
-sided polygon shown in the figure below. The area of this polygon can be expressed in the form
, where
,
, and
are positive integers, and
is not divisible by the square of any prime. What is
?
![[asy] defaultpen(fontsize(8)+0.8); size(150); pair O,A1,B1,C1,A2,B2,C2,A3,B3,C3,A4,B4,C4; real x=45, y=90, z=60; O=origin; A1=dir(x); A2=dir(x+y); A3=dir(x+2y); A4=dir(x+3y); B1=dir(x-z); B2=dir(x+y-z); B3=dir(x+2y-z); B4=dir(x+3y-z); C1=dir(x-2z); C2=dir(x+y-2z); C3=dir(x+2y-2z); C4=dir(x+3y-2z); draw(A1--A2--A3--A4--A1, gray+0.25+dashed); filldraw(B1--B2--B3--B4--cycle, white, gray+dashed+linewidth(0.25)); filldraw(C1--C2--C3--C4--cycle, white, gray+dashed+linewidth(0.25)); dot(O); pair P1,P2,P3,P4,Q1,Q2,Q3,Q4,R1,R2,R3,R4; P1=extension(A1,A2,B1,B2); Q1=extension(A1,A2,C3,C4); P2=extension(A2,A3,B2,B3); Q2=extension(A2,A3,C4,C1); P3=extension(A3,A4,B3,B4); Q3=extension(A3,A4,C1,C2); P4=extension(A4,A1,B4,B1); Q4=extension(A4,A1,C2,C3); R1=extension(C2,C3,B2,B3); R2=extension(C3,C4,B3,B4); R3=extension(C4,C1,B4,B1); R4=extension(C1,C2,B1,B2); draw(A1--P1--B2--R1--C3--Q1--A2); draw(A2--P2--B3--R2--C4--Q2--A3); draw(A3--P3--B4--R3--C1--Q3--A4); draw(A4--P4--B1--R4--C2--Q4--A1); [/asy]](http://latex.artofproblemsolving.com/7/e/b/7eb8841b95d6e1c451addf20403c4e540f176116.png)
Problem 16
Suppose ,
,
are positive integers such that
and
What is the sum of all possible distinct values of
?
Problem 17
A bug starts at a vertex of a grid made of equilateral triangles of side length . At each step the bug moves in one of the
possible directions along the grid lines randomly and independently with equal probability. What is the probability that after
moves the bug never will have been more than
unit away from the starting position?
Problem 18
Set , and for
let
be determined by the recurrence
This sequence tends to a limit; call it . What is the least value of
such that
Problem 19
Regular polygons with and
sides are inscribed in the same circle. No two of the polygons share a vertex, and no three of their sides intersect at a common point. At how many points inside the circle do two of their sides intersect?
Problem 20
A cube is constructed from white unit cubes and
blue unit cubes. How many different ways are there to construct the
cube using these smaller cubes? (Two constructions are considered the same if one can be rotated to match the other.)
Problem 21
For real numbers , let
where
. For how many values of
with
does
Problem 22
Right triangle has side lengths
,
, and
. A circle centered at
is tangent to line
at
and passes through
. A circle centered at
is tangent to line
at
and passes through
. What is
?
Problem 23
What is the average number of pairs of consecutive integers in a randomly selected subset of distinct integers chosen from the set
? (For example the set
has
pairs of consecutive integers.)
Problem 24
Triangle has side lengths
, and
. The bisector of
intersects
in point
, and intersects the circumcircle of
in point
. The circumcircle of
intersects the line
in points
and
. What is
?
Problem 25
For a positive integer, let
be the sum of the remainders when
is divided by
,
,
,
,
,
,
,
, and
. For example,
. How many two-digit positive integers
satisfy
See also
2021 Fall AMC 12B (Problems • Answer Key • Resources) | |
Preceded by 2021 Fall AMC 12A Problems |
Followed by 2022 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.