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Difference between revisions of "2021 Fall AMC 12B Problems"
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==Problem 6==
 
==Problem 6==
−
The largest prime factor of <math>16384</math> is <math>2</math>, because <math>16384 = 2^{14}</math>. What is the sum of the digits of the largest prime factor of <math>16383</math>?
+
The largest prime factor of <math>16384</math> is <math>2</math> because <math>16384 = 2^{14}</math>. What is the sum of the digits of the greatest prime number that is a divisor of <math>16383</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>
  
 
[[2021 Fall AMC 12B Problems/Problem 7|Solution]]
 
[[2021 Fall AMC 12B Problems/Problem 7|Solution]]
Line 81: Line 81:
  
 
==Problem 8==
 
==Problem 8==
−
Let <math>M</math> be the least common multiple of all the integers <math>10</math> through <math>30,</math> inclusive. Let <math>N</math> be the least common multiple of <math>M,32,33,34,35,36,37,38,39,</math> and <math>40.</math> What is the value of <math>\frac{N}{M}?</math>
+
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?
  
−
<math>\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 37 \qquad\textbf{(D)}\ 74 \qquad\textbf{(E)}\ 2886</math>
+
<math>\textbf{(A)} \: 105 \qquad\textbf{(B)} \: 120 \qquad\textbf{(C)} \: 135 \qquad\textbf{(D)} \: 150 \qquad\textbf{(E)} \: 165</math>
  
 
[[2021 Fall AMC 12B Problems/Problem 8|Solution]]
 
[[2021 Fall AMC 12B Problems/Problem 8|Solution]]
Line 106: Line 106:
 
==Problem 12==
 
==Problem 12==
  
−
What is the number of terms with rational coefficients among the <math>1001</math> terms in the expansion of <math>\left(x\sqrt[3]{2}+y\sqrt{3}\right)^{1000}?</math>
 
−
 
−
<math>\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 166 \qquad\textbf{(C)}\ 167 \qquad\textbf{(D)}\ 500 \qquad\textbf{(E)}\ 501</math>
 
  
 
[[2021 Fall AMC 12B Problems/Problem 12|Solution]]
 
[[2021 Fall AMC 12B Problems/Problem 12|Solution]]
  
 
==Problem 13==
 
==Problem 13==
−
The angle bisector of the acute angle formed at the origin by the graphs of the lines <math>y = x</math> and <math>y=3x</math> has equation <math>y=kx.</math> What is <math>k?</math>
 
  
−
<math>\textbf{(A)} \ \frac{1+\sqrt{5}}{2} \qquad \textbf{(B)} \ \frac{1+\sqrt{7}}{2} \qquad \textbf{(C)} \ \frac{2+\sqrt{3}}{2} \qquad \textbf{(D)} \ 2\qquad \textbf{(E)} \ \frac{2+\sqrt{5}}{2}</math>
 
  
 
[[2021 Fall AMC 12B Problems/Problem 13|Solution]]
 
[[2021 Fall AMC 12B Problems/Problem 13|Solution]]
  
 
==Problem 14==
 
==Problem 14==
−
In the figure, equilateral hexagon <math>ABCDEF</math> has three nonadjacent acute interior angles that each measure <math>30^\circ</math>. The enclosed area of the hexagon is <math>6\sqrt{3}</math>. What is the perimeter of the hexagon?
+
 
−
<asy>
 
−
size(10cm);
 
−
pen p=black+linewidth(1),q=black+linewidth(5);
 
−
pair C=(0,0),D=(cos(pi/12),sin(pi/12)),E=rotate(150,D)*C,F=rotate(-30,E)*D,A=rotate(150,F)*E,B=rotate(-30,A)*F;
 
−
draw(C--D--E--F--A--B--cycle,p);
 
−
dot(A,q);
 
−
dot(B,q);
 
−
dot(C,q);
 
−
dot(D,q);
 
−
dot(E,q);
 
−
dot(F,q);
 
−
label("$C$",C,2*S);
 
−
label("$D$",D,2*S);
 
−
label("$E$",E,2*S);
 
−
label("$F$",F,2*dir(0));
 
−
label("$A$",A,2*N);
 
−
label("$B$",B,2*W);
 
−
</asy>
 
−
<math>\textbf{(A)} \: 4 \qquad \textbf{(B)} \: 4\sqrt3 \qquad \textbf{(C)} \: 12 \qquad \textbf{(D)} \: 18 \qquad \textbf{(E)} \: 12\sqrt3</math>
 
  
 
[[2021 Fall AMC 12B Problems/Problem 14|Solution]]
 
[[2021 Fall AMC 12B Problems/Problem 14|Solution]]
Line 153: Line 129:
  
 
==Problem 16==
 
==Problem 16==
−
An organization has <math>30</math> employees, <math>20</math> of whom have a brand A computer while the other <math>10</math> have a brand B computer. For security, the computers can only be connected to each other and only by cables. The cables can only connect a brand A computer to a brand B computer. Employees can communicate with each other if their computers are directly connected by a cable or by relaying messages through a series of connected computers. Initially, no computer is connected to any other. A technician arbitrarily selects one computer of each brand and installs a cable between them, provided there is not already a cable between that pair. The technician stops once every employee can communicate with each other. What is the maximum possible number of cables used?
 
  
−
<math>\textbf{(A)}\ 190  \qquad\textbf{(B)}\  191 \qquad\textbf{(C)}\  192 \qquad\textbf{(D)}\
 
−
195 \qquad\textbf{(E)}\ 196</math>
 
  
 
[[2021 Fall AMC 12B Problems/Problem 16|Solution]]
 
[[2021 Fall AMC 12B Problems/Problem 16|Solution]]
  
 
==Problem 17==
 
==Problem 17==
−
For how many ordered pairs <math>(b,c)</math> of positive integers does neither <math>x^2+bx+c=0</math> nor <math>x^2+cx+b=0</math> have two distinct real solutions?
 
  
−
<math>\textbf{(A) } 4 \qquad \textbf{(B) } 6 \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 12 \qquad \textbf{(E) } 16 \qquad</math>
 
  
 
[[2021 Fall AMC 12B Problems/Problem 17|Solution]]
 
[[2021 Fall AMC 12B Problems/Problem 17|Solution]]
  
 
==Problem 18==
 
==Problem 18==
−
Each of <math>20</math> balls is tossed independently and at random into one of <math>5</math> bins. Let <math>p</math> be the probability that some bin ends up with <math>3</math> balls, another with <math>5</math> balls, and the other three with <math>4</math> balls each. Let <math>q</math> be the probability that every bin ends up with <math>4</math> balls. What is <math>\frac{p}{q}</math>?
 
  
−
<math>\textbf{(A)}\ 1 \qquad\textbf{(B)}\  4 \qquad\textbf{(C)}\  8 \qquad\textbf{(D)}\  12 \qquad\textbf{(E)}\ 16</math>
 
  
 
[[2021 Fall AMC 12B Problems/Problem 18|Solution]]
 
[[2021 Fall AMC 12B Problems/Problem 18|Solution]]
Line 194: Line 163:
  
 
==Problem 22==
 
==Problem 22==
−
Azar and Carl play a game of tic-tac-toe. Azar places an in <math>X</math> one of the boxes in a 3-by-3 array of boxes, then Carl places an <math>O</math> in one of the remaining boxes. After that, Azar places an <math>X</math> in one of the remaining boxes, and so on until all boxes are filled or one of the players has of their symbols in a row—horizontal, vertical, or diagonal—whichever comes first, in which case that player wins the game. Suppose the players make their moves at random, rather than trying to follow a rational strategy, and that Carl wins the game when he places his third <math>O</math>. How many ways can the board look after the game is over?
 
  
−
<math>\textbf{(A) } 36 \qquad\textbf{(B) } 112 \qquad\textbf{(C) } 120 \qquad\textbf{(D) } 148 \qquad\textbf{(E) } 160</math>
 
  
 
[[2021 Fall AMC 12B Problems/Problem 22|Solution]]
 
[[2021 Fall AMC 12B Problems/Problem 22|Solution]]
Line 202: Line 169:
 
==Problem 23==
 
==Problem 23==
  
−
A quadratic polynomial with real coefficients and leading coefficient <math>1</math> is called <math>\emph{disrespectful}</math> if the equation <math>p(p(x))=0</math> is satisfied by exactly three real numbers. Among all the disrespectful quadratic polynomials, there is a unique such polynomial <math>\tilde{p}(x)</math> for which the sum of the roots is maximized. What is <math>\tilde{p}(1)</math>?
 
−
 
−
<math>\textbf{(A) } \frac{5}{16} \qquad\textbf{(B) } \frac{1}{2} \qquad\textbf{(C) } \frac{5}{8} \qquad\textbf{(D) } 1 \qquad\textbf{(E) } \frac{9}{8}</math>
 
  
 
[[2021 Fall AMC 12B Problems/Problem 23|Solution]]
 
[[2021 Fall AMC 12B Problems/Problem 23|Solution]]
  
 
==Problem 24==
 
==Problem 24==
−
Convex quadrilateral <math>ABCD</math> has <math>AB = 18, \angle{A} = 60 \textdegree</math>, and <math>\overline{AB} \parallel \overline{CD}</math>. In some order, the lengths of the four sides form an arithmetic progression, and side <math>\overline{AB}</math> is a side of maximum length. The length of another side is <math>a</math>. What is the sum of all possible values of <math>a</math>?
 
  
−
<math>\textbf{(A) } 24 \qquad \textbf{(B) } 42 \qquad \textbf{(C) } 60 \qquad \textbf{(D) } 66 \qquad \textbf{(E) } 84</math>
 
  
 
[[2021 Fall AMC 12B Problems/Problem 24|Solution]]
 
[[2021 Fall AMC 12B Problems/Problem 24|Solution]]
  
 
==Problem 25==
 
==Problem 25==
−
Let <math>m\ge 5</math> be an odd integer, and let <math>D(m)</math> denote the number of quadruples <math>\big(a_1, a_2, a_3, a_4\big)</math> of distinct integers with <math>1\le a_i \le m</math> for all <math>i</math> such that <math>m</math> divides <math>a_1+a_2+a_3+a_4</math>. There is a polynomial
 
−
<cmath>q(x) = c_3x^3+c_2x^2+c_1x+c_0</cmath>such that <math>D(m) = q(m)</math> for all odd integers <math>m\ge 5</math>. What is <math>c_1?</math>
 
  
−
<math>(\textbf{A})\: {-}6\qquad(\textbf{B}) \: {-}1\qquad(\textbf{C}) \: 4\qquad(\textbf{D}) \: 6\qquad(\textbf{E}) \: 11</math>
 
  
 
[[2021 Fall AMC 12B Problems/Problem 25|Solution]]
 
[[2021 Fall AMC 12B Problems/Problem 25|Solution]]

Revision as of 20:51, 23 November 2021

2021 Fall AMC 12B (Answer Key)
Printable versions: Wiki • Fall AoPS Resources • Fall PDF

Instructions

  1. This is a 25-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct.
  2. You will receive 6 points for each correct answer, 2.5 points for each problem left unanswered if the year is before 2006, 1.5 points for each problem left unanswered if the year is after 2006, and 0 points for each incorrect answer.
  3. No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers (and calculators that are accepted for use on the test if before 2006. No problems on the test will require the use of a calculator).
  4. Figures are not necessarily drawn to scale.
  5. You will have 75 minutes working time to complete the test.
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

Problem 1

What is the value of $1234+2341+3412+4123?$

$(\textbf{A})\: 10{,}000\qquad(\textbf{B}) \: 10{,}010\qquad(\textbf{C}) \: 10{,}110\qquad(\textbf{D}) \: 11{,}000\qquad(\textbf{E}) \: 11{,}110$

Solution

Problem 2

What is the area of the shaded figure shown below? [asy] size(200); defaultpen(linewidth(0.4)+fontsize(12)); pen s = linewidth(0.8)+fontsize(8); pair O,X,Y; O = origin; X = (6,0); Y = (0,5); fill((1,0)--(3,5)--(5,0)--(3,2)--cycle, palegray+opacity(0.2)); for (int i=1; i<7; ++i) { draw((i,0)--(i,5), gray+dashed); label("${"+string(i)+"}$", (i,0), 2*S); if (i<6) { draw((0,i)--(6,i), gray+dashed); label("${"+string(i)+"}$", (0,i), 2*W); } } label("$0$", O, 2*SW); draw(O--X+(0.15,0), EndArrow); draw(O--Y+(0,0.15), EndArrow); draw((1,0)--(3,5)--(5,0)--(3,2)--(1,0), black+1.5); [/asy]

$(\textbf{A})\: 4\qquad(\textbf{B}) \: 6\qquad(\textbf{C}) \: 8\qquad(\textbf{D}) \: 10\qquad(\textbf{E}) \: 12$

Solution

Problem 3

At noon on a certain day, Minneapolis is $N$ degrees warmer than St. Louis. At $4{:}00$ the temperature in Minneapolis has fallen by $5$ degrees while the temperature in St. Louis has risen by $3$ degrees, at which time the temperatures in the two cities differ by $2$ degrees. What is the product of all possible values of $N?$

$(\textbf{A})\: 10\qquad(\textbf{B}) \: 30\qquad(\textbf{C}) \: 60\qquad(\textbf{D}) \: 100\qquad(\textbf{E}) \: 120$

Solution

Problem 4

Let $n=8^{2022}$. Which of the following is equal to $\frac{n}{4}?$

$(\textbf{A})\: 4^{1010}\qquad(\textbf{B}) \: 2^{2022}\qquad(\textbf{C}) \: 8^{2018}\qquad(\textbf{D}) \: 4^{3031}\qquad(\textbf{E}) \: 4^{3032}$

Solution

Problem 5

Call a fraction $\frac{a}{b}$, not necessarily in the simplest form, special if $a$ and $b$ are positive integers whose sum is $15$. How many distinct integers can be written as the sum of two, not necessarily different, special fractions?

$\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 11 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 13$

Solution

Problem 6

The largest prime factor of $16384$ is $2$ because $16384 = 2^{14}$. What is the sum of the digits of the greatest prime number that is a divisor of $16383$?

$\textbf{(A)} \: 3\qquad\textbf{(B)} \: 7\qquad\textbf{(C)} \: 10\qquad\textbf{(D)} \: 16\qquad\textbf{(E)} \: 22$

Solution

Problem 7

Which of the following conditions is sufficient to guarantee that integers $x$, $y$, and $z$ satisfy the equation \[x(x-y)+y(y-z)+z(z-x) = 1?\]

$\textbf{(A)} \: x>y$ and $y=z$ $\textbf{(B)} \: x=y-1$ and $y=z-1$ $\textbf{(C)} \: x=z+1$ and $y=x+1$ $\textbf{(D)} \: x=z$ and $y-1=x$ $\textbf{(E)} \: x+y+z=1$

Solution

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?

$\textbf{(A)} \: 105 \qquad\textbf{(B)} \: 120 \qquad\textbf{(C)} \: 135 \qquad\textbf{(D)} \: 150 \qquad\textbf{(E)} \: 165$

Solution

Problem 9

Solution

Problem 10

Solution

Problem 11

Una rolls $6$ standard $6$-sided dice simultaneously and calculates the product of the $6{ }$ numbers obtained. What is the probability that the product is divisible by $4?$

$\textbf{(A)}\: \frac34\qquad\textbf{(B)} \: \frac{57}{64}\qquad\textbf{(C)} \: \frac{59}{64}\qquad\textbf{(D)} \: \frac{187}{192}\qquad\textbf{(E)} \: \frac{63}{64}$

Solution

Problem 12

Solution

Problem 13

Solution

Problem 14

Solution

Problem 15

Three identical square sheets of paper each with side length $6$ are stacked on top of each other. The middle sheet is rotated clockwise $30^\circ$ about its center and the top sheet is rotated clockwise $60^\circ$ about its center, resulting in the $24$-sided polygon shown in the figure below. The area of this polygon can be expressed in the form $a-b\sqrt{c}$, where $a$, $b$, and $c$ are positive integers, and $c$ is not divisible by the square of any prime. What is $a+b+c$?

IMAGE

$(\textbf{A})\: 75\qquad(\textbf{B}) \: 93\qquad(\textbf{C}) \: 96\qquad(\textbf{D}) \: 129\qquad(\textbf{E}) \: 147$

Solution

Problem 16

Solution

Problem 17

Solution

Problem 18

Solution

Problem 19

Regular polygons with $5$, $6$, $7$, and $8$ 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?

$(\textbf{A})\: 52\qquad(\textbf{B}) \: 56\qquad(\textbf{C}) \: 60\qquad(\textbf{D}) \: 64\qquad(\textbf{E}) \: 68$

Solution

Problem 20

A cube is constructed from $4$ white unit cubes and $4$ blue unit cubes. How many different ways are there to construct the $2 \times 2 \times 2$ cube using these smaller cubes? (Two constructions are considered the same if one can be rotated to match the other.)

$(\textbf{A})\: 7\qquad(\textbf{B}) \: 8\qquad(\textbf{C}) \: 9\qquad(\textbf{D}) \: 10\qquad(\textbf{E}) \: 11$

Solution

Problem 21

Solution

Problem 22

Solution

Problem 23

Solution

Problem 24

Solution

Problem 25

Solution

See also

2021 Fall AMC 12A (Problems • Answer Key • Resources)
Preceded by
2021 Fall AMC 12B Problems 		Followed by
[[2021 Fall AMC 12A Problems/Problem {{{num-a}}}|Problem {{{num-a}}}]]
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. AMC logo.png

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