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Difference between revisions of "2021 Fall AMC 12A Problems"

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I just want to say that instead of  
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{{AMC12 Problems|year=2021 Fall|ab=A}}
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'''
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You have followed a link to a page that does not exist yet. To create the page, start typing in the box below (see the help page for more info). If you are here by mistake, click your browser's back button.
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==Problem 1==
−
''',  
+
What is the value of <math>\frac{(2112-2021)^2}{169}</math>?
−
AoPS SHOULD AT LEAST put something like an auto-generated apology here.  
+
 
−
'''
+
<math>\textbf{(A) } 7 \qquad\textbf{(B) } 21 \qquad\textbf{(C) } 49 \qquad\textbf{(D) } 64 \qquad\textbf{(E) } 91</math>
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Sorry.  
+
 
−
This page is not created yet.  
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[[2021 Fall AMC 10A Problems/Problem 1|Solution]]
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Would you like to create it?  
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'''
+
==Problem 2==
−
is MUCH MORE HELPFUL.  
+
Menkara has a <math>4 \times 6</math> index card. If she shortens the length of one side of this card by <math>1</math> inch, the card would have area <math>18</math> square inches. What would the area of the card be in square inches if instead she shortens the length of the other side by <math>1</math> inch?
−
(Yes, I know there is a small print saying that it does not exist. )
+
 
 +
<math>\textbf{(A) }16\qquad\textbf{(B) }17\qquad\textbf{(C) }18\qquad\textbf{(D) }19\qquad\textbf{(E) }20</math>
 +
 
 +
[[2021 Fall AMC 10A Problems/Problem 2|Solution]]
 +
 
 +
==Problem 3==
 +
Mr. Lopez has a choice of two routes to get to work. Route A is <math>6</math> miles long, and his average speed along this route is <math>30</math> miles per hour. Route B is <math>5</math> miles long, and his average speed along this route is <math>40</math> miles per hour, except for a <math>\frac{1}{2}</math>-mile stretch in a school zone where his average speed is <math>20</math> miles per hour. By how many minutes is Route B quicker than Route A?
 +
 
 +
<math>\textbf{(A)}\ 2 \frac{3}{4}  \qquad\textbf{(B)}\  3 \frac{3}{4} \qquad\textbf{(C)}\  4 \frac{1}{2} \qquad\textbf{(D)}\
 +
5 \frac{1}{2} \qquad\textbf{(E)}\ 6 \frac{3}{4}</math>
 +
 
 +
[[2021 Fall AMC 10A Problems/Problem 4|Solution]]
 +
 
 +
==Problem 4==
 +
The six-digit number <math>\underline{2}\,\underline{0}\,\underline{2}\,\underline{1}\,\underline{0}\,\underline{A}</math> is prime for only one digit <math>A.</math> What is <math>A?</math>
 +
 
 +
<math>(\textbf{A})\: 1\qquad(\textbf{B}) \: 3\qquad(\textbf{C}) \: 5 \qquad(\textbf{D}) \: 7\qquad(\textbf{E}) \: 9</math>
 +
 
 +
[[2021 Fall AMC 10A Problems/Problem 5|Solution]]
 +
 
 +
==Problem 5==
 +
 
 +
Elmer the emu takes <math>44</math> equal strides to walk between consecutive telephone poles on a rural road. Oscar the ostrich can cover the same distance in <math>12</math> equal leaps. The telephone poles are evenly spaced, and the <math>41</math>st pole along this road is exactly one mile (<math>5280</math> feet) from the first pole. How much longer, in feet, is Oscar's leap than Elmer's stride?
 +
 
 +
<math>\textbf{(A) }6\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }11\qquad\textbf{(E) }15</math>
 +
 
 +
[[2021 Fall AMC 10A Problems/Problem 6|Solution]]
 +
 
 +
==Problem 6==
 +
 
 +
As shown in the figure below, point <math>E</math> lies on the opposite half-plane determined by line <math>CD</math> from point <math>A</math> so that <math>\angle CDE = 110^\circ</math>. Point <math>F</math> lies on <math>\overline{AD}</math> so that <math>DE=DF</math>, and <math>ABCD</math> is a square. What is the degree measure of <math>\angle AFE</math>?
 +
 
 +
<asy>
 +
usepackage("mathptmx");
 +
size(6cm);
 +
pair A = (0,10);
 +
label("$A$", A, N);
 +
pair B = (0,0);
 +
label("$B$", B, S);
 +
pair C = (10,0);
 +
label("$C$", C, S);
 +
pair D = (10,10);
 +
label("$D$", D, SW);
 +
pair EE = (15,11.8);
 +
label("$E$", EE, N);
 +
pair F = (3,10);
 +
label("$F$", F, N);
 +
filldraw(D--arc(D,2.5,270,380)--cycle,lightgray);
 +
dot(A^^B^^C^^D^^EE^^F);
 +
draw(A--B--C--D--cycle);
 +
draw(D--EE--F--cycle);
 +
label("$110^\circ$", (15,9), SW);
 +
</asy>
 +
 
 +
<math>\textbf{(A) }160\qquad\textbf{(B) }164\qquad\textbf{(C) }166\qquad\textbf{(D) }170\qquad\textbf{(E) }174</math>
 +
 
 +
[[2021 Fall AMC 10A Problems/Problem 7|Solution]]
 +
 
 +
==Problem 7==
 +
 
 +
A school has <math>100</math> students and <math>5</math> teachers. In the first period, each student is taking one class, and each teacher is teaching one class. The enrollments in the classes are <math>50, 20, 20, 5, </math> and <math>5</math>. Let <math>t</math> be the average value obtained if a teacher is picked at random and the number of students in their class is noted. Let <math>s</math> be the average value obtained if a student was picked at random and the number of students in their class, including the student, is noted. What is <math>t-s</math>?
 +
 
 +
<math>\textbf{(A)}\ {-}18.5  \qquad\textbf{(B)}\  {-}13.5 \qquad\textbf{(C)}\  0 \qquad\textbf{(D)}\
 +
13.5 \qquad\textbf{(E)}\ 18.5</math>
 +
 
 +
[[2021 Fall AMC 10A Problems/Problem 10|Solution]]
 +
 
 +
==Problem 8==
 +
<math>\textbf{(A) } \qquad\textbf{(B) } \qquad\textbf{(C) } \qquad\textbf{(D) } \qquad\textbf{(E) }</math>
 +
 
 +
[[2021 Fall AMC 10A Problems/Problem 8|Solution]]
 +
 
 +
==Problem 9==
 +
<math>\textbf{(A) } \qquad\textbf{(B) } \qquad\textbf{(C) } \qquad\textbf{(D) } \qquad\textbf{(E) }</math>
 +
 
 +
[[2021 Fall AMC 10A Problems/Problem 9|Solution]]
 +
 
 +
==Problem 10==
 +
The base-nine representation of the number <math>N</math> is <math>27{,}006{,}000{,}052_{\text{nine}}.</math> What is the remainder when <math>N</math> is divided by <math>5?</math>
 +
 
 +
<math>\textbf{(A) } 0\qquad\textbf{(B) } 1\qquad\textbf{(C) } 2\qquad\textbf{(D) } 3\qquad\textbf{(E) }4</math>
 +
 
 +
[[2021 Fall AMC 10A Problems/Problem 10|Solution]]
 +
 
 +
==Problem 11==
 +
Consider two concentric circles of radius <math>17</math> and <math>19.</math> The larger circle has a chord, half of which lies inside the smaller circle. What is the length of the chord in the larger circle?
 +
 
 +
<math>\textbf{(A)}\ 12\sqrt{2} \qquad\textbf{(B)}\ 10\sqrt{3} \qquad\textbf{(C)}\ \sqrt{17 \cdot 19} \qquad\textbf{(D)}\ 18 \qquad\textbf{(E)}\ 8\sqrt{6}</math>
 +
 
 +
[[2021 Fall AMC 10A Problems/Problem 11|Solution]]
 +
 
 +
==Problem 12==
 +
<math>\textbf{(A) } \qquad\textbf{(B) } \qquad\textbf{(C) } \qquad\textbf{(D) } \qquad\textbf{(E) }</math>
 +
 
 +
[[2021 Fall AMC 10A Problems/Problem 12|Solution]]
 +
 
 +
==Problem 13==
 +
Each of <math>6</math> balls is randomly and independently painted either black or white with equal probability. What is the probability that every ball is different in color from more than half of the other <math>5</math> balls?
 +
 
 +
<math>\textbf{(A) } \frac{1}{64}\qquad\textbf{(B) } \frac{1}{6}\qquad\textbf{(C) } \frac{1}{4}\qquad\textbf{(D) } \frac{5}{16}\qquad\textbf{(E) }\frac{1}{2}</math>
 +
 
 +
[[2021 Fall AMC 10A Problems/Problem 13|Solution]]
 +
 
 +
==Problem 14==
 +
<math>\textbf{(A) } \qquad\textbf{(B) } \qquad\textbf{(C) } \qquad\textbf{(D) } \qquad\textbf{(E) }</math>
 +
 
 +
[[2021 Fall AMC 10A Problems/Problem 14|Solution]]
 +
 
 +
==Problem 15==
 +
Isosceles triangle <math>ABC</math> has <math>AB = AC = 3\sqrt6</math>, and a circle with radius <math>5\sqrt2</math> is tangent to line <math>AB</math> at <math>B</math> and to line <math>AC</math> at <math>C</math>. What is the area of the circle that passes through vertices <math>A</math>, <math>B</math>, and <math>C?</math>
 +
 
 +
<math>\textbf{(A) }24\pi\qquad\textbf{(B) }25\pi\qquad\textbf{(C) }26\pi\qquad\textbf{(D) }27\pi\qquad\textbf{(E) }28\pi</math>
 +
 
 +
[[2021 Fall AMC 10A Problems/Problem 15|Solution]]
 +
 
 +
==Problem 16==
 +
 
 +
The graph of
 +
<cmath>f(x)=|\lfloor x \rfloor|-|\lfloor 1-x\rfloor |</cmath>
 +
is symmetric about which of the following? (Here <math>\lfloor x\rfloor</math> is the greatest integer not exeeding <math>x</math>.
 +
 
 +
<math>\textbf{(A) }</math>the <math>y</math> -axis<math> \qquad\textbf{(B) } </math>the line <math>x=1 \qquad\textbf{(C) }</math>the origin<math> \qquad\textbf{(D) }</math>the point <math>\left(\frac{1}{2},0\right) \qquad\textbf{(E) }</math>the point <math>(1,0)</math>
 +
 
 +
[[2021 Fall AMC 10A Problems/Problem 16|Solution]]
 +
 
 +
==Problem 17==
 +
An architect is building a structure that will place vertical pillars at the vertices of regular hexagon <math>ABCDEF</math>, which is lying horizontally on the ground. The six pillars will hold up a flat solar panel that will not be parallel to the ground. The heights of pillars at <math>A</math>, <math>B</math>, and <math>C</math> are <math>12</math>, <math>9</math>, and <math>10</math> meters, respectively. What is the height, in meters, of the pillar at <math>E</math>?
 +
 
 +
<math>\textbf{(A) }9 \qquad\textbf{(B) } 6\sqrt{3} \qquad\textbf{(C) } 8\sqrt{3} \qquad\textbf{(D) } 17 \qquad\textbf{(E) }12\sqrt{3}</math>
 +
 
 +
[[2021 Fall AMC 10A Problems/Problem 17|Solution]]
 +
 
 +
==Problem 18==
 +
A farmer's rectangular field is partitioned into <math>2</math> by <math>2</math> grid of <math>4</math> rectangular sections as shown in the figure. In each section the farmer will plant one crop: corn, wheat, soybeans, or potatoes. The farmer does not want to grow corn and wheat in any two sections that share a border, and the farmer does not want to grow soybeans and potatoes in any two sections that share a border. Given these restrictions, in how many ways can the farmer choose crops to plant in each of the four sections of the field?
 +
<asy>
 +
draw((0,0)--(100,0)--(100,50)--(0,50)--cycle);
 +
draw((50,0)--(50,50));
 +
draw((0,25)--(100,25));
 +
</asy>
 +
<math>\textbf{(A)}\ 12 \qquad \textbf{(B)}\ 64 \qquad \textbf{(C)}\ 84 \qquad \textbf{(D)}\ 90 \qquad \textbf{(E)}\ 144</math>
 +
 
 +
[[2021 Fall AMC 10A Problems/Problem 18|Solution]]
 +
 
 +
==Problem 19==
 +
A disk of radius <math>1</math> rolls all the way around the inside of a square of side length <math>s>4</math> and sweeps out a region of area <math>A</math>. A second disk of radius <math>1</math> rolls all the way around the outside of the same square and sweeps out a region of area <math>2A</math>. The value of <math>s</math> can be written as <math>a+\frac{b\pi}{c}</math>, where <math>a,b</math>, and <math>c</math> are positive integers and <math>b</math> and <math>c</math> are relatively prime. What is <math>a+b+c</math>?
 +
 
 +
<math>\textbf{(A)} ~10\qquad\textbf{(B)} ~11\qquad\textbf{(C)} ~12\qquad\textbf{(D)} ~13\qquad\textbf{(E)} ~14</math>
 +
 
 +
[[2021 Fall AMC 10A Problems/Problem 19|Solution]]
 +
 
 +
==Problem 20==
 +
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 10A Problems/Problem 20|Solution]]
 +
 
 +
==Problem 21==
 +
Each of the <math>20</math> balls is tossed independently and at random into one of the <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 10A Problems/Problem 21|Solution]]
 +
 
 +
==Problem 22==
 +
Inside a right circular cone with base radius <math>5</math> and height <math>12</math> are three congruent spheres with radius <math>r</math>. Each sphere is tangent to the other two spheres and also tangent to the base and side of the cone. What is <math>r</math>?
 +
 
 +
<math>\textbf{(A)}\ \frac{3}{2} \qquad\textbf{(B)}\ \frac{90-40\sqrt{3}}{11} \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ \frac{144-25\sqrt{3}}{44} \qquad\textbf{(E)}\ \frac{5}{2}</math>
 +
 
 +
[[2021 Fall AMC 10A Problems/Problem 22|Solution]]
 +
 
 +
==Problem 23==
 +
 
 +
For each positive integer <math>n</math>, let <math>f_1(n)</math> be twice the number of positive integer divisors of <math>n</math>, and for <math>j \ge 2</math>, let <math>f_j(n) = f_1(f_{j-1}(n))</math>. For how many values of <math>n \le 50</math> is <math>f_{50}(n) = 12?</math>
 +
 
 +
<math>\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }9\qquad\textbf{(D) }10\qquad\textbf{(E) }11</math>
 +
 
 +
[[2021 Fall AMC 10A Problems/Problem 23|Solution]]
 +
 
 +
==Problem 24==
 +
Each of the <math>12</math> edges of a cube is labeled <math>0</math> or <math>1</math>. Two labelings are considered different even if one can be obtained from the other by a sequence of one or more rotations and/or reflections. For how many such labelings is the sum of the labels on the edges of each of the <math>6</math> faces of the cube equal to <math>2</math>?
 +
 
 +
<math>\textbf{(A) } 8 \qquad\textbf{(B) } 10 \qquad\textbf{(C) } 12 \qquad\textbf{(D) } 16 \qquad\textbf{(E) } 20</math>
 +
 
 +
[[2021 Fall AMC 10A Problems/Problem 24|Solution]]
 +
 
 +
==Problem 25==
 +
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 10A Problems/Problem 25|Solution]]
 +
 
 +
==See also==
 +
{{AMC12 box|year=2021 Fall|ab=A|before=[[2021 AMC 12B Problems]]|after=[[2021 Fall AMC 12B Problems]]}}
 +
{{MAA Notice}}

Revision as of 18:09, 23 November 2021

2021 Fall AMC 12A (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 $\frac{(2112-2021)^2}{169}$?

$\textbf{(A) } 7 \qquad\textbf{(B) } 21 \qquad\textbf{(C) } 49 \qquad\textbf{(D) } 64 \qquad\textbf{(E) } 91$

Solution

Problem 2

Menkara has a $4 \times 6$ index card. If she shortens the length of one side of this card by $1$ inch, the card would have area $18$ square inches. What would the area of the card be in square inches if instead she shortens the length of the other side by $1$ inch?

$\textbf{(A) }16\qquad\textbf{(B) }17\qquad\textbf{(C) }18\qquad\textbf{(D) }19\qquad\textbf{(E) }20$

Solution

Problem 3

Mr. Lopez has a choice of two routes to get to work. Route A is $6$ miles long, and his average speed along this route is $30$ miles per hour. Route B is $5$ miles long, and his average speed along this route is $40$ miles per hour, except for a $\frac{1}{2}$-mile stretch in a school zone where his average speed is $20$ miles per hour. By how many minutes is Route B quicker than Route A?

$\textbf{(A)}\ 2 \frac{3}{4} \qquad\textbf{(B)}\ 3 \frac{3}{4} \qquad\textbf{(C)}\ 4 \frac{1}{2} \qquad\textbf{(D)}\ 5 \frac{1}{2} \qquad\textbf{(E)}\ 6 \frac{3}{4}$

Solution

Problem 4

The six-digit number $\underline{2}\,\underline{0}\,\underline{2}\,\underline{1}\,\underline{0}\,\underline{A}$ is prime for only one digit $A.$ What is $A?$

$(\textbf{A})\: 1\qquad(\textbf{B}) \: 3\qquad(\textbf{C}) \: 5 \qquad(\textbf{D}) \: 7\qquad(\textbf{E}) \: 9$

Solution

Problem 5

Elmer the emu takes $44$ equal strides to walk between consecutive telephone poles on a rural road. Oscar the ostrich can cover the same distance in $12$ equal leaps. The telephone poles are evenly spaced, and the $41$st pole along this road is exactly one mile ($5280$ feet) from the first pole. How much longer, in feet, is Oscar's leap than Elmer's stride?

$\textbf{(A) }6\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }11\qquad\textbf{(E) }15$

Solution

Problem 6

As shown in the figure below, point $E$ lies on the opposite half-plane determined by line $CD$ from point $A$ so that $\angle CDE = 110^\circ$. Point $F$ lies on $\overline{AD}$ so that $DE=DF$, and $ABCD$ is a square. What is the degree measure of $\angle AFE$?

[asy] usepackage("mathptmx"); size(6cm); pair A = (0,10); label("$A$", A, N); pair B = (0,0); label("$B$", B, S); pair C = (10,0); label("$C$", C, S); pair D = (10,10); label("$D$", D, SW); pair EE = (15,11.8); label("$E$", EE, N); pair F = (3,10); label("$F$", F, N); filldraw(D--arc(D,2.5,270,380)--cycle,lightgray); dot(A^^B^^C^^D^^EE^^F); draw(A--B--C--D--cycle); draw(D--EE--F--cycle); label("$110^\circ$", (15,9), SW); [/asy]

$\textbf{(A) }160\qquad\textbf{(B) }164\qquad\textbf{(C) }166\qquad\textbf{(D) }170\qquad\textbf{(E) }174$

Solution

Problem 7

A school has $100$ students and $5$ teachers. In the first period, each student is taking one class, and each teacher is teaching one class. The enrollments in the classes are $50, 20, 20, 5,$ and $5$. Let $t$ be the average value obtained if a teacher is picked at random and the number of students in their class is noted. Let $s$ be the average value obtained if a student was picked at random and the number of students in their class, including the student, is noted. What is $t-s$?

$\textbf{(A)}\ {-}18.5 \qquad\textbf{(B)}\ {-}13.5 \qquad\textbf{(C)}\ 0 \qquad\textbf{(D)}\ 13.5 \qquad\textbf{(E)}\ 18.5$

Solution

Problem 8

$\textbf{(A) } \qquad\textbf{(B) } \qquad\textbf{(C) } \qquad\textbf{(D) } \qquad\textbf{(E) }$

Solution

Problem 9

$\textbf{(A) } \qquad\textbf{(B) } \qquad\textbf{(C) } \qquad\textbf{(D) } \qquad\textbf{(E) }$

Solution

Problem 10

The base-nine representation of the number $N$ is $27{,}006{,}000{,}052_{\text{nine}}.$ What is the remainder when $N$ is divided by $5?$

$\textbf{(A) } 0\qquad\textbf{(B) } 1\qquad\textbf{(C) } 2\qquad\textbf{(D) } 3\qquad\textbf{(E) }4$

Solution

Problem 11

Consider two concentric circles of radius $17$ and $19.$ The larger circle has a chord, half of which lies inside the smaller circle. What is the length of the chord in the larger circle?

$\textbf{(A)}\ 12\sqrt{2} \qquad\textbf{(B)}\ 10\sqrt{3} \qquad\textbf{(C)}\ \sqrt{17 \cdot 19} \qquad\textbf{(D)}\ 18 \qquad\textbf{(E)}\ 8\sqrt{6}$

Solution

Problem 12

$\textbf{(A) } \qquad\textbf{(B) } \qquad\textbf{(C) } \qquad\textbf{(D) } \qquad\textbf{(E) }$

Solution

Problem 13

Each of $6$ balls is randomly and independently painted either black or white with equal probability. What is the probability that every ball is different in color from more than half of the other $5$ balls?

$\textbf{(A) } \frac{1}{64}\qquad\textbf{(B) } \frac{1}{6}\qquad\textbf{(C) } \frac{1}{4}\qquad\textbf{(D) } \frac{5}{16}\qquad\textbf{(E) }\frac{1}{2}$

Solution

Problem 14

$\textbf{(A) } \qquad\textbf{(B) } \qquad\textbf{(C) } \qquad\textbf{(D) } \qquad\textbf{(E) }$

Solution

Problem 15

Isosceles triangle $ABC$ has $AB = AC = 3\sqrt6$, and a circle with radius $5\sqrt2$ is tangent to line $AB$ at $B$ and to line $AC$ at $C$. What is the area of the circle that passes through vertices $A$, $B$, and $C?$

$\textbf{(A) }24\pi\qquad\textbf{(B) }25\pi\qquad\textbf{(C) }26\pi\qquad\textbf{(D) }27\pi\qquad\textbf{(E) }28\pi$

Solution

Problem 16

The graph of \[f(x)=|\lfloor x \rfloor|-|\lfloor 1-x\rfloor |\] is symmetric about which of the following? (Here $\lfloor x\rfloor$ is the greatest integer not exeeding $x$.

$\textbf{(A) }$the $y$ -axis$\qquad\textbf{(B) }$the line $x=1 \qquad\textbf{(C) }$the origin$\qquad\textbf{(D) }$the point $\left(\frac{1}{2},0\right) \qquad\textbf{(E) }$the point $(1,0)$

Solution

Problem 17

An architect is building a structure that will place vertical pillars at the vertices of regular hexagon $ABCDEF$, which is lying horizontally on the ground. The six pillars will hold up a flat solar panel that will not be parallel to the ground. The heights of pillars at $A$, $B$, and $C$ are $12$, $9$, and $10$ meters, respectively. What is the height, in meters, of the pillar at $E$?

$\textbf{(A) }9 \qquad\textbf{(B) } 6\sqrt{3} \qquad\textbf{(C) } 8\sqrt{3} \qquad\textbf{(D) } 17 \qquad\textbf{(E) }12\sqrt{3}$

Solution

Problem 18

A farmer's rectangular field is partitioned into $2$ by $2$ grid of $4$ rectangular sections as shown in the figure. In each section the farmer will plant one crop: corn, wheat, soybeans, or potatoes. The farmer does not want to grow corn and wheat in any two sections that share a border, and the farmer does not want to grow soybeans and potatoes in any two sections that share a border. Given these restrictions, in how many ways can the farmer choose crops to plant in each of the four sections of the field? [asy] draw((0,0)--(100,0)--(100,50)--(0,50)--cycle); draw((50,0)--(50,50)); draw((0,25)--(100,25)); [/asy] $\textbf{(A)}\ 12 \qquad \textbf{(B)}\ 64 \qquad \textbf{(C)}\ 84 \qquad \textbf{(D)}\ 90 \qquad \textbf{(E)}\ 144$

Solution

Problem 19

A disk of radius $1$ rolls all the way around the inside of a square of side length $s>4$ and sweeps out a region of area $A$. A second disk of radius $1$ rolls all the way around the outside of the same square and sweeps out a region of area $2A$. The value of $s$ can be written as $a+\frac{b\pi}{c}$, where $a,b$, and $c$ are positive integers and $b$ and $c$ are relatively prime. What is $a+b+c$?

$\textbf{(A)} ~10\qquad\textbf{(B)} ~11\qquad\textbf{(C)} ~12\qquad\textbf{(D)} ~13\qquad\textbf{(E)} ~14$

Solution

Problem 20

For how many ordered pairs $(b,c)$ of positive integers does neither $x^2+bx+c=0$ nor $x^2+cx+b=0$ have two distinct real solutions?

$\textbf{(A) } 4 \qquad \textbf{(B) } 6 \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 12 \qquad \textbf{(E) } 16 \qquad$

Solution

Problem 21

Each of the $20$ balls is tossed independently and at random into one of the $5$ bins. Let $p$ be the probability that some bin ends up with $3$ balls, another with $5$ balls, and the other three with $4$ balls each. Let $q$ be the probability that every bin ends up with $4$ balls. What is $\frac{p}{q}$?

$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 16$

Solution

Problem 22

Inside a right circular cone with base radius $5$ and height $12$ are three congruent spheres with radius $r$. Each sphere is tangent to the other two spheres and also tangent to the base and side of the cone. What is $r$?

$\textbf{(A)}\ \frac{3}{2} \qquad\textbf{(B)}\ \frac{90-40\sqrt{3}}{11} \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ \frac{144-25\sqrt{3}}{44} \qquad\textbf{(E)}\ \frac{5}{2}$

Solution

Problem 23

For each positive integer $n$, let $f_1(n)$ be twice the number of positive integer divisors of $n$, and for $j \ge 2$, let $f_j(n) = f_1(f_{j-1}(n))$. For how many values of $n \le 50$ is $f_{50}(n) = 12?$

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

Solution

Problem 24

Each of the $12$ edges of a cube is labeled $0$ or $1$. Two labelings are considered different even if one can be obtained from the other by a sequence of one or more rotations and/or reflections. For how many such labelings is the sum of the labels on the edges of each of the $6$ faces of the cube equal to $2$?

$\textbf{(A) } 8 \qquad\textbf{(B) } 10 \qquad\textbf{(C) } 12 \qquad\textbf{(D) } 16 \qquad\textbf{(E) } 20$

Solution

Problem 25

A quadratic polynomial with real coefficients and leading coefficient $1$ is called $\emph{disrespectful}$ if the equation $p(p(x))=0$ is satisfied by exactly three real numbers. Among all the disrespectful quadratic polynomials, there is a unique such polynomial $\tilde{p}(x)$ for which the sum of the roots is maximized. What is $\tilde{p}(1)$?

$\textbf{(A) } \frac{5}{16} \qquad\textbf{(B) } \frac{1}{2} \qquad\textbf{(C) } \frac{5}{8} \qquad\textbf{(D) } 1 \qquad\textbf{(E) } \frac{9}{8}$

Solution

See also

2021 Fall AMC 12A (Problems • Answer Key • Resources)
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All AMC 12 Problems and Solutions

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