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Difference between revisions of "2020 AMC 12A Problems"

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Creating 2020 AMC 12A Problems
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{{AMC12 Problems|year=2020|ab=A}}
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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.
 
  
 +
==Problem 1==
  
−
Problem 1
+
Carlos took <math>70\%</math> of a whole pie. Maria took one third of the remainder. What portion of the whole pie was left?
  
−
1-1=?
+
<math>\textbf{(A)}\ 10\%\qquad\textbf{(B)}\ 15\%\qquad\textbf{(C)}\ 20\%\qquad\textbf{(D)}\ 30\%\qquad\textbf{(E)}\ 35\%</math>
  
−
<math>A)100</math> <math>B)10</math> <math>C)1</math> <math>D)0</math> <math>E)</math>NONE OF THE ABOVE
+
[[2020 AMC 12A Problems/Problem 1|Solution]]
 +
 
 +
==Problem 2==
 +
 
 +
The acronym AMC is shown in the rectangular grid below with grid lines spaced <math>1</math> unit apart. In units, what is the sum of the lengths of the line segments that form the acronym AMC<math>?</math>
 +
 
 +
<asy>
 +
import olympiad;
 +
unitsize(25);
 +
for (int i = 0; i < 3; ++i) {
 +
for (int j = 0; j < 9; ++j) {
 +
pair A = (j,i);
 +
 
 +
}
 +
}
 +
for (int i = 0; i < 3; ++i) {
 +
for (int j = 0; j < 9; ++j) {
 +
if (j != 8) {
 +
draw((j,i)--(j+1,i), dashed);
 +
}
 +
if (i != 2) {
 +
draw((j,i)--(j,i+1), dashed);
 +
}
 +
}
 +
}
 +
draw((0,0)--(2,2),linewidth(2));
 +
draw((2,0)--(2,2),linewidth(2));
 +
draw((1,1)--(2,1),linewidth(2));
 +
draw((3,0)--(3,2),linewidth(2));
 +
draw((5,0)--(5,2),linewidth(2));
 +
draw((4,1)--(3,2),linewidth(2));
 +
draw((4,1)--(5,2),linewidth(2));
 +
draw((6,0)--(8,0),linewidth(2));
 +
draw((6,2)--(8,2),linewidth(2));
 +
draw((6,0)--(6,2),linewidth(2));
 +
</asy>
 +
 
 +
<math>\textbf{(A) } 17 \qquad \textbf{(B) } 15 + 2\sqrt{2} \qquad \textbf{(C) } 13 + 4\sqrt{2} \qquad \textbf{(D) } 11 + 6\sqrt{2} \qquad \textbf{(E) } 21</math>
 +
 
 +
[[2020 AMC 12A Problems/Problem 2|Solution]]
 +
 
 +
==Problem 3==
 +
 
 +
A driver travels for <math>2</math> hours at <math>60</math> miles per hour, during which her car gets <math>30</math> miles per gallon of gasoline. She is paid <math>\$0.50</math> per mile, and her only expense is gasoline at <math>\$2.00</math> per gallon. What is her net rate of pay, in dollars per hour, after this expense?
 +
 
 +
<math>\textbf{(A) }20 \qquad\textbf{(B) }22 \qquad\textbf{(C) }24 \qquad\textbf{(D) } 25\qquad\textbf{(E) } 26</math>
 +
 
 +
[[2020 AMC 12A Problems/Problem 3|Solution]]
 +
 
 +
==Problem 4==
 +
 
 +
How many <math>4</math>-digit positive integers (that is, integers between <math>1000</math> and <math>9999</math>, inclusive) having only even digits are divisible by <math>5?</math>
 +
 
 +
<math>\textbf{(A) } 80 \qquad \textbf{(B) } 100 \qquad \textbf{(C) } 125 \qquad \textbf{(D) } 200 \qquad \textbf{(E) } 500</math>
 +
 
 +
[[2020 AMC 12A Problems/Problem 4|Solution]]
 +
 
 +
==Problem 5==
 +
 
 +
The <math>25</math> integers from <math>-10</math> to <math>14,</math> inclusive, can be arranged to form a <math>5</math>-by-<math>5</math> square in which the sum of the numbers in each row, the sum of the numbers in each column, and the sum of the numbers along each of the main diagonals are all the same. What is the value of this common sum?
 +
 
 +
<math>\textbf{(A) }2 \qquad\textbf{(B) } 5\qquad\textbf{(C) } 10\qquad\textbf{(D) } 25\qquad\textbf{(E) } 50</math>
 +
 
 +
[[2020 AMC 12A Problems/Problem 5|Solution]]
 +
 
 +
==Problem 6==
 +
 
 +
In the plane figure shown below, <math>3</math> of the unit squares have been shaded. What is the least number of additional unit squares that must be shaded so that the resulting figure has two lines of symmetry<math>?</math>
 +
 
 +
<asy>
 +
import olympiad;
 +
unitsize(25);
 +
filldraw((1,3)--(1,4)--(2,4)--(2,3)--cycle, gray(0.7));
 +
filldraw((2,1)--(2,2)--(3,2)--(3,1)--cycle, gray(0.7));
 +
filldraw((4,0)--(5,0)--(5,1)--(4,1)--cycle, gray(0.7));
 +
for (int i = 0; i < 5; ++i) {
 +
for (int j = 0; j < 6; ++j) {
 +
pair A = (j,i);
 +
 
 +
}
 +
}
 +
for (int i = 0; i < 5; ++i) {
 +
for (int j = 0; j < 6; ++j) {
 +
if (j != 5) {
 +
draw((j,i)--(j+1,i));
 +
}
 +
if (i != 4) {
 +
draw((j,i)--(j,i+1));
 +
}
 +
}
 +
}
 +
</asy>
 +
 
 +
<math>\textbf{(A) } 4 \qquad \textbf{(B) } 5 \qquad \textbf{(C) } 6 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 8</math>
 +
 
 +
[[2020 AMC 12A Problems/Problem 6|Solution]]
 +
 
 +
==Problem 7==
 +
 
 +
Seven cubes, whose volumes are <math>1</math>, <math>8</math>, <math>27</math>, <math>64</math>, <math>125</math>, <math>216</math>, and <math>343</math> cubic units, are stacked vertically to form a tower in which the volumes of the cubes decrease from bottom to top. Except for the bottom cube, the bottom face of each cube lies completely on top of the cube below it. What is the total surface area of the tower (including the bottom) in square units?
 +
 
 +
<math>\textbf{(A) } 644    \qquad \textbf{(B) } 658  \qquad \textbf{(C) } 664  \qquad \textbf{(D) } 720  \qquad \textbf{(E) } 749</math>
 +
 
 +
[[2020 AMC 12A Problems/Problem 7|Solution]]
 +
 
 +
==Problem 8==
 +
 
 +
What is the median of the following list of <math>4040</math> numbers<math>?</math>
 +
<cmath>1, 2, 3, \ldots, 2020, 1^2, 2^2, 3^2, \ldots, 2020^2</cmath>
 +
<math> \textbf{(A)}\ 1974.5\qquad\textbf{(B)}\ 1975.5\qquad\textbf{(C)}\ 1976.5\qquad\textbf{(D)}\ 1977.5\qquad\textbf{(E)}\ 1978.5 </math>
 +
 
 +
[[2020 AMC 12A Problems/Problem 8|Solution]]
 +
 
 +
==Problem 9==
 +
 
 +
How many solutions does the equation <math>\tan{(2x)} = \cos{(\tfrac{x}{2})}</math> have on the interval <math>[0, 2\pi]?</math>
 +
 
 +
<math>\textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 3 \qquad \textbf{(D) } 4 \qquad \textbf{(E) } 5</math>
 +
 
 +
[[2020 AMC 12A Problems/Problem 9|Solution]]
 +
 
 +
==Problem 10==
 +
 
 +
There is a unique positive integer <math>n</math> such that<cmath>\log_2{(\log_{16}{n})} = \log_4{(\log_4{n})}.</cmath>What is the sum of the digits of <math>n?</math>
 +
 
 +
<math>\textbf{(A) } 4 \qquad \textbf{(B) } 7 \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 11 \qquad \textbf{(E) } 13</math>
 +
 
 +
[[2020 AMC 12A Problems/Problem 10|Solution]]
 +
 
 +
==Problem 11==
 +
 
 +
A frog sitting at the point <math>(1, 2)</math> begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length <math>1</math>, and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a side of the square with vertices <math>(0,0), (0,4), (4,4),</math> and <math>(4,0)</math>. What is the probability that the sequence of jumps ends on a vertical side of the square<math>?</math>
 +
 
 +
<math>\textbf{(A) } \frac{1}{2} \qquad \textbf{(B) } \frac{5}{8} \qquad \textbf{(C) } \frac{2}{3} \qquad \textbf{(D) } \frac{3}{4} \qquad \textbf{(E) } \frac{7}{8}</math>
 +
 
 +
[[2020 AMC 12A Problems/Problem 11|Solution]]
 +
 
 +
==Problem 12==
 +
 
 +
Line <math>\ell</math> in the coordinate plane has the equation <math>3x - 5y + 40 = 0</math>. This line is rotated <math>45^{\circ}</math> counterclockwise about the point <math>(20, 20)</math> to obtain line <math>k</math>. What is the <math>x</math>-coordinate of the <math>x</math>-intercept of line <math>k?</math>
 +
 
 +
<math>\textbf{(A) } 10 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 20 \qquad \textbf{(D) } 25 \qquad \textbf{(E) } 30</math>
 +
 
 +
[[2020 AMC 12A Problems/Problem 12|Solution]]
 +
 
 +
==Problem 13==
 +
 
 +
There are integers <math>a</math>, <math>b</math>, and <math>c</math>, each greater than 1, such that<cmath>\sqrt[a]{N \sqrt[b]{N \sqrt[c]{N}}} = \sqrt[36]{N^{25}}</cmath>for all <math>N > 1</math>. What is <math>b</math>?
 +
 
 +
<math>\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6</math>
 +
 
 +
[[2020 AMC 12A Problems/Problem 13|Solution]]
 +
 
 +
==Problem 14==
 +
 
 +
Regular octagon <math>ABCDEFGH</math> has area <math>n</math>. Let <math>m</math> be the area of quadrilateral <math>ACEG</math>. What is <math>\tfrac{m}{n}?</math>
 +
 
 +
<math>\textbf{(A) } \frac{\sqrt{2}}{4} \qquad \textbf{(B) } \frac{\sqrt{2}}{2} \qquad \textbf{(C) } \frac{3}{4} \qquad \textbf{(D) } \frac{3\sqrt{2}}{5} \qquad \textbf{(E) } \frac{2\sqrt{2}}{3}</math>
 +
 
 +
[[2020 AMC 12A Problems/Problem 14|Solution]]
 +
 
 +
==Problem 15==
 +
 
 +
In the complex plane, let <math>A</math> be the set of solutions to <math>z^3 - 8 = 0</math> and let <math>B</math> be the set of solutions to <math>z^3 - 8z^2 - 8z + 64 = 0</math>. What is the greatest distance between a point of <math>A</math> and a point of <math>B?</math>
 +
 
 +
<math>\textbf{(A) } 2\sqrt{3} \qquad \textbf{(B) } 6 \qquad \textbf{(C) } 9 \qquad \textbf{(D) } 2\sqrt{21} \qquad \textbf{(E) } 9 + \sqrt{3}</math>
 +
 
 +
[[2020 AMC 12A Problems/Problem 15|Solution]]
 +
 
 +
==Problem 16==
 +
 
 +
A point is chosen at random within the square in the coordinate plane whose vertices are <math>(0, 0), (2020, 0), (2020, 2020),</math> and <math>(0, 2020)</math>. The probability that the point is within <math>d</math> units of a lattice point is <math>\tfrac{1}{2}</math>. (A point <math>(x, y)</math> is a lattice point if <math>x</math> and <math>y</math> are both integers.) What is <math>d</math> to the nearest tenth<math>?</math>
 +
 
 +
<math>\textbf{(A) } 0.3 \qquad \textbf{(B) } 0.4 \qquad \textbf{(C) } 0.5 \qquad \textbf{(D) } 0.6 \qquad \textbf{(E) } 0.7</math>
 +
 
 +
[[2020 AMC 12A Problems/Problem 16|Solution]]
 +
 
 +
==Problem 17==
 +
 
 +
The vertices of a quadrilateral lie on the graph of <math>y = \ln x</math>, and the <math>x</math>-coordinates of these vertices are consecutive positive integers. The area of the quadrilateral is <math>\ln \frac{91}{90}</math>. What is the <math>x</math>-coordinate of the leftmost vertex?
 +
 
 +
<math>\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 13</math>
 +
 
 +
[[2020 AMC 12A Problems/Problem 17|Solution]]
 +
 
 +
==Problem 18==
 +
 
 +
Quadrilateral <math>ABCD</math> satisfies <math>\angle ABC = \angle ACD = 90^{\circ}, AC = 20</math>, and <math>CD = 30</math>. Diagonals <math>\overline{AC}</math> and <math>\overline{BD}</math> intersect at point <math>E</math>, and <math>AE = 5</math>. What is the area of quadrilateral <math>ABCD</math>?
 +
 
 +
<math>\textbf{(A) } 330 \qquad\textbf{(B) } 340 \qquad\textbf{(C) } 350 \qquad\textbf{(D) } 360 \qquad\textbf{(E) } 370</math>
 +
 
 +
[[2020 AMC 12A Problems/Problem 18|Solution]]
 +
 
 +
==Problem 19==
 +
 
 +
There exists a unique strictly increasing sequence of nonnegative integers <math>a_1 < a_2 < … < a_k</math> such that<cmath>\frac{2^{289}+1}{2^{17}+1} = 2^{a_1} + 2^{a_2} + … + 2^{a_k}.</cmath>What is <math>k?</math>
 +
 
 +
<math>\textbf{(A) } 117 \qquad \textbf{(B) } 136 \qquad \textbf{(C) } 137 \qquad \textbf{(D) } 273 \qquad \textbf{(E) } 306</math>
 +
 
 +
[[2020 AMC 12A Problems/Problem 19|Solution]]
 +
 
 +
==Problem 20==
 +
 
 +
Let <math>T</math> be the triangle in the coordinate plane with vertices <math>\left(0,0\right)</math>, <math>\left(4,0\right)</math>, and <math>\left(0,3\right)</math>. Consider the following five isometries (rigid transformations) of the plane: rotations of <math>90^{\circ}</math>, <math>180^{\circ}</math>, and <math>270^{\circ}</math> counterclockwise around the origin, reflection across the <math>x</math>-axis, and reflection across the <math>y</math>-axis. How many of the <math>125</math> sequences of three of these transformations (not necessarily distinct) will return <math>T</math> to its original position? (For example, a <math>180^{\circ}</math> rotation, followed by a reflection across the <math>x</math>-axis, followed by a reflection across the <math>y</math>-axis will return <math>T</math> to its original position, but a <math>90^{\circ}</math> rotation, followed by a reflection across the <math>x</math>-axis, followed by another reflection across the <math>x</math>-axis will not return <math>T</math> to its original position.)
 +
 
 +
<math>\textbf{(A) } 12\qquad\textbf{(B) } 15\qquad\textbf{(C) }17 \qquad\textbf{(D) }20 \qquad\textbf{(E) }25</math>
 +
 
 +
[[2020 AMC 12A Problems/Problem 20|Solution]]
 +
 
 +
==Problem 21==
 +
 
 +
How many positive integers <math>n</math> are there such that <math>n</math> is a multiple of <math>5</math>, and the least common multiple of <math>5!</math> and <math>n</math> equals <math>5</math> times the greatest common divisor of <math>10!</math> and <math>n?</math>
 +
 
 +
<math>\textbf{(A) } 12 \qquad \textbf{(B) } 24 \qquad \textbf{(C) } 36 \qquad \textbf{(D) } 48 \qquad \textbf{(E) } 72</math>
 +
 
 +
[[2020 AMC 12A Problems/Problem 21|Solution]]
 +
 
 +
==Problem 22==
 +
 
 +
Let <math>(a_n)</math> and <math>(b_n)</math> be the sequences of real numbers such that
 +
<cmath>\[
 +
(2 + i)^n = a_n + b_ni
 +
\]</cmath>for all integers <math>n\geq 0</math>, where <math>i = \sqrt{-1}</math>. What is<cmath>\sum_{n=0}^\infty\frac{a_nb_n}{7^n}\,?</cmath>
 +
<math>\textbf{(A) }\frac 38\qquad\textbf{(B) }\frac7{16}\qquad\textbf{(C) }\frac12\qquad\textbf{(D) }\frac9{16}\qquad\textbf{(E) }\frac47</math>
 +
 
 +
[[2020 AMC 12A Problems/Problem 22|Solution]]
 +
 
 +
==Problem 23==
 +
 
 +
Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly <math>7</math>. Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice?
 +
 
 +
<math>\textbf{(A) } \frac{7}{36} \qquad\textbf{(B) } \frac{5}{24} \qquad\textbf{(C) } \frac{2}{9} \qquad\textbf{(D) } \frac{17}{72} \qquad\textbf{(E) } \frac{1}{4}</math>
 +
 
 +
[[2020 AMC 12A Problems/Problem 23|Solution]]
 +
 
 +
==Problem 24==
 +
 
 +
Suppose that <math>\triangle ABC</math> is an equilateral triangle of side length <math>s</math>, with the property that there is a unique point <math>P</math> inside the triangle such that <math>AP = 1</math>, <math>BP = \sqrt{3}</math>, and <math>CP = 2</math>. What is <math>s?</math>
 +
 
 +
<math>\textbf{(A) } 1 + \sqrt{2} \qquad \textbf{(B) } \sqrt{7} \qquad \textbf{(C) } \frac{8}{3} \qquad \textbf{(D) } \sqrt{5 + \sqrt{5}} \qquad \textbf{(E) } 2\sqrt{2}</math>
 +
 
 +
[[2020 AMC 12A Problems/Problem 24|Solution]]
 +
 
 +
==Problem 25==
 +
 
 +
The number <math>a = \tfrac{p}{q}</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers, has the property that the sum of all real numbers <math>x</math> satisfying<cmath>\lfloor x \rfloor \cdot \{x\} = a \cdot x^2</cmath>is <math>420</math>, where <math>\lfloor x \rfloor</math> denotes the greatest integer less than or equal to <math>x</math> and <math>\{x\} = x - \lfloor x \rfloor</math> denotes the fractional part of <math>x</math>. What is <math>p + q?</math>
 +
 
 +
<math>\textbf{(A) } 245 \qquad \textbf{(B) } 593 \qquad \textbf{(C) } 929 \qquad \textbf{(D) } 1331 \qquad \textbf{(E) } 1332</math>
 +
 
 +
[[2020 AMC 12A Problems/Problem 25|Solution]]
 +
 
 +
==See also==
 +
{{AMC12 box|year=2020|ab=A|before=[[2019 AMC 12B Problems]]|after=[[2020 AMC 12B Problems]]}}
 +
{{MAA Notice}}

Latest revision as of 14:15, 21 October 2021

2020 AMC 12A (Answer Key)
Printable versions: Wiki • AoPS Resources • 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

Carlos took $70\%$ of a whole pie. Maria took one third of the remainder. What portion of the whole pie was left?

$\textbf{(A)}\ 10\%\qquad\textbf{(B)}\ 15\%\qquad\textbf{(C)}\ 20\%\qquad\textbf{(D)}\ 30\%\qquad\textbf{(E)}\ 35\%$

Solution

Problem 2

The acronym AMC is shown in the rectangular grid below with grid lines spaced $1$ unit apart. In units, what is the sum of the lengths of the line segments that form the acronym AMC$?$

[asy] import olympiad; unitsize(25); for (int i = 0; i < 3; ++i) { for (int j = 0; j < 9; ++j) { pair A = (j,i); } } for (int i = 0; i < 3; ++i) { for (int j = 0; j < 9; ++j) { if (j != 8) { draw((j,i)--(j+1,i), dashed); } if (i != 2) { draw((j,i)--(j,i+1), dashed); } } } draw((0,0)--(2,2),linewidth(2)); draw((2,0)--(2,2),linewidth(2)); draw((1,1)--(2,1),linewidth(2)); draw((3,0)--(3,2),linewidth(2)); draw((5,0)--(5,2),linewidth(2)); draw((4,1)--(3,2),linewidth(2)); draw((4,1)--(5,2),linewidth(2)); draw((6,0)--(8,0),linewidth(2)); draw((6,2)--(8,2),linewidth(2)); draw((6,0)--(6,2),linewidth(2)); [/asy]

$\textbf{(A) } 17 \qquad \textbf{(B) } 15 + 2\sqrt{2} \qquad \textbf{(C) } 13 + 4\sqrt{2} \qquad \textbf{(D) } 11 + 6\sqrt{2} \qquad \textbf{(E) } 21$

Solution

Problem 3

A driver travels for $2$ hours at $60$ miles per hour, during which her car gets $30$ miles per gallon of gasoline. She is paid $$0.50$ per mile, and her only expense is gasoline at $$2.00$ per gallon. What is her net rate of pay, in dollars per hour, after this expense?

$\textbf{(A) }20 \qquad\textbf{(B) }22 \qquad\textbf{(C) }24 \qquad\textbf{(D) } 25\qquad\textbf{(E) } 26$

Solution

Problem 4

How many $4$-digit positive integers (that is, integers between $1000$ and $9999$, inclusive) having only even digits are divisible by $5?$

$\textbf{(A) } 80 \qquad \textbf{(B) } 100 \qquad \textbf{(C) } 125 \qquad \textbf{(D) } 200 \qquad \textbf{(E) } 500$

Solution

Problem 5

The $25$ integers from $-10$ to $14,$ inclusive, can be arranged to form a $5$-by-$5$ square in which the sum of the numbers in each row, the sum of the numbers in each column, and the sum of the numbers along each of the main diagonals are all the same. What is the value of this common sum?

$\textbf{(A) }2 \qquad\textbf{(B) } 5\qquad\textbf{(C) } 10\qquad\textbf{(D) } 25\qquad\textbf{(E) } 50$

Solution

Problem 6

In the plane figure shown below, $3$ of the unit squares have been shaded. What is the least number of additional unit squares that must be shaded so that the resulting figure has two lines of symmetry$?$

[asy] import olympiad; unitsize(25); filldraw((1,3)--(1,4)--(2,4)--(2,3)--cycle, gray(0.7)); filldraw((2,1)--(2,2)--(3,2)--(3,1)--cycle, gray(0.7)); filldraw((4,0)--(5,0)--(5,1)--(4,1)--cycle, gray(0.7)); for (int i = 0; i < 5; ++i) { for (int j = 0; j < 6; ++j) { pair A = (j,i); } } for (int i = 0; i < 5; ++i) { for (int j = 0; j < 6; ++j) { if (j != 5) { draw((j,i)--(j+1,i)); } if (i != 4) { draw((j,i)--(j,i+1)); } } } [/asy]

$\textbf{(A) } 4 \qquad \textbf{(B) } 5 \qquad \textbf{(C) } 6 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 8$

Solution

Problem 7

Seven cubes, whose volumes are $1$, $8$, $27$, $64$, $125$, $216$, and $343$ cubic units, are stacked vertically to form a tower in which the volumes of the cubes decrease from bottom to top. Except for the bottom cube, the bottom face of each cube lies completely on top of the cube below it. What is the total surface area of the tower (including the bottom) in square units?

$\textbf{(A) } 644 \qquad \textbf{(B) } 658 \qquad \textbf{(C) } 664 \qquad \textbf{(D) } 720 \qquad \textbf{(E) } 749$

Solution

Problem 8

What is the median of the following list of $4040$ numbers$?$ \[1, 2, 3, \ldots, 2020, 1^2, 2^2, 3^2, \ldots, 2020^2\] $\textbf{(A)}\ 1974.5\qquad\textbf{(B)}\ 1975.5\qquad\textbf{(C)}\ 1976.5\qquad\textbf{(D)}\ 1977.5\qquad\textbf{(E)}\ 1978.5$

Solution

Problem 9

How many solutions does the equation $\tan{(2x)} = \cos{(\tfrac{x}{2})}$ have on the interval $[0, 2\pi]?$

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

Solution

Problem 10

There is a unique positive integer $n$ such that\[\log_2{(\log_{16}{n})} = \log_4{(\log_4{n})}.\]What is the sum of the digits of $n?$

$\textbf{(A) } 4 \qquad \textbf{(B) } 7 \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 11 \qquad \textbf{(E) } 13$

Solution

Problem 11

A frog sitting at the point $(1, 2)$ begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length $1$, and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a side of the square with vertices $(0,0), (0,4), (4,4),$ and $(4,0)$. What is the probability that the sequence of jumps ends on a vertical side of the square$?$

$\textbf{(A) } \frac{1}{2} \qquad \textbf{(B) } \frac{5}{8} \qquad \textbf{(C) } \frac{2}{3} \qquad \textbf{(D) } \frac{3}{4} \qquad \textbf{(E) } \frac{7}{8}$

Solution

Problem 12

Line $\ell$ in the coordinate plane has the equation $3x - 5y + 40 = 0$. This line is rotated $45^{\circ}$ counterclockwise about the point $(20, 20)$ to obtain line $k$. What is the $x$-coordinate of the $x$-intercept of line $k?$

$\textbf{(A) } 10 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 20 \qquad \textbf{(D) } 25 \qquad \textbf{(E) } 30$

Solution

Problem 13

There are integers $a$, $b$, and $c$, each greater than 1, such that\[\sqrt[a]{N \sqrt[b]{N \sqrt[c]{N}}} = \sqrt[36]{N^{25}}\]for all $N > 1$. What is $b$?

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

Solution

Problem 14

Regular octagon $ABCDEFGH$ has area $n$. Let $m$ be the area of quadrilateral $ACEG$. What is $\tfrac{m}{n}?$

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

Solution

Problem 15

In the complex plane, let $A$ be the set of solutions to $z^3 - 8 = 0$ and let $B$ be the set of solutions to $z^3 - 8z^2 - 8z + 64 = 0$. What is the greatest distance between a point of $A$ and a point of $B?$

$\textbf{(A) } 2\sqrt{3} \qquad \textbf{(B) } 6 \qquad \textbf{(C) } 9 \qquad \textbf{(D) } 2\sqrt{21} \qquad \textbf{(E) } 9 + \sqrt{3}$

Solution

Problem 16

A point is chosen at random within the square in the coordinate plane whose vertices are $(0, 0), (2020, 0), (2020, 2020),$ and $(0, 2020)$. The probability that the point is within $d$ units of a lattice point is $\tfrac{1}{2}$. (A point $(x, y)$ is a lattice point if $x$ and $y$ are both integers.) What is $d$ to the nearest tenth$?$

$\textbf{(A) } 0.3 \qquad \textbf{(B) } 0.4 \qquad \textbf{(C) } 0.5 \qquad \textbf{(D) } 0.6 \qquad \textbf{(E) } 0.7$

Solution

Problem 17

The vertices of a quadrilateral lie on the graph of $y = \ln x$, and the $x$-coordinates of these vertices are consecutive positive integers. The area of the quadrilateral is $\ln \frac{91}{90}$. What is the $x$-coordinate of the leftmost vertex?

$\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 13$

Solution

Problem 18

Quadrilateral $ABCD$ satisfies $\angle ABC = \angle ACD = 90^{\circ}, AC = 20$, and $CD = 30$. Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at point $E$, and $AE = 5$. What is the area of quadrilateral $ABCD$?

$\textbf{(A) } 330 \qquad\textbf{(B) } 340 \qquad\textbf{(C) } 350 \qquad\textbf{(D) } 360 \qquad\textbf{(E) } 370$

Solution

Problem 19

There exists a unique strictly increasing sequence of nonnegative integers $a_1 < a_2 < … < a_k$ such that\[\frac{2^{289}+1}{2^{17}+1} = 2^{a_1} + 2^{a_2} + … + 2^{a_k}.\]What is $k?$

$\textbf{(A) } 117 \qquad \textbf{(B) } 136 \qquad \textbf{(C) } 137 \qquad \textbf{(D) } 273 \qquad \textbf{(E) } 306$

Solution

Problem 20

Let $T$ be the triangle in the coordinate plane with vertices $\left(0,0\right)$, $\left(4,0\right)$, and $\left(0,3\right)$. Consider the following five isometries (rigid transformations) of the plane: rotations of $90^{\circ}$, $180^{\circ}$, and $270^{\circ}$ counterclockwise around the origin, reflection across the $x$-axis, and reflection across the $y$-axis. How many of the $125$ sequences of three of these transformations (not necessarily distinct) will return $T$ to its original position? (For example, a $180^{\circ}$ rotation, followed by a reflection across the $x$-axis, followed by a reflection across the $y$-axis will return $T$ to its original position, but a $90^{\circ}$ rotation, followed by a reflection across the $x$-axis, followed by another reflection across the $x$-axis will not return $T$ to its original position.)

$\textbf{(A) } 12\qquad\textbf{(B) } 15\qquad\textbf{(C) }17 \qquad\textbf{(D) }20 \qquad\textbf{(E) }25$

Solution

Problem 21

How many positive integers $n$ are there such that $n$ is a multiple of $5$, and the least common multiple of $5!$ and $n$ equals $5$ times the greatest common divisor of $10!$ and $n?$

$\textbf{(A) } 12 \qquad \textbf{(B) } 24 \qquad \textbf{(C) } 36 \qquad \textbf{(D) } 48 \qquad \textbf{(E) } 72$

Solution

Problem 22

Let $(a_n)$ and $(b_n)$ be the sequences of real numbers such that \[ (2 + i)^n = a_n + b_ni \]for all integers $n\geq 0$, where $i = \sqrt{-1}$. What is\[\sum_{n=0}^\infty\frac{a_nb_n}{7^n}\,?\] $\textbf{(A) }\frac 38\qquad\textbf{(B) }\frac7{16}\qquad\textbf{(C) }\frac12\qquad\textbf{(D) }\frac9{16}\qquad\textbf{(E) }\frac47$

Solution

Problem 23

Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly $7$. Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice?

$\textbf{(A) } \frac{7}{36} \qquad\textbf{(B) } \frac{5}{24} \qquad\textbf{(C) } \frac{2}{9} \qquad\textbf{(D) } \frac{17}{72} \qquad\textbf{(E) } \frac{1}{4}$

Solution

Problem 24

Suppose that $\triangle ABC$ is an equilateral triangle of side length $s$, with the property that there is a unique point $P$ inside the triangle such that $AP = 1$, $BP = \sqrt{3}$, and $CP = 2$. What is $s?$

$\textbf{(A) } 1 + \sqrt{2} \qquad \textbf{(B) } \sqrt{7} \qquad \textbf{(C) } \frac{8}{3} \qquad \textbf{(D) } \sqrt{5 + \sqrt{5}} \qquad \textbf{(E) } 2\sqrt{2}$

Solution

Problem 25

The number $a = \tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers, has the property that the sum of all real numbers $x$ satisfying\[\lfloor x \rfloor \cdot \{x\} = a \cdot x^2\]is $420$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$ and $\{x\} = x - \lfloor x \rfloor$ denotes the fractional part of $x$. What is $p + q?$

$\textbf{(A) } 245 \qquad \textbf{(B) } 593 \qquad \textbf{(C) } 929 \qquad \textbf{(D) } 1331 \qquad \textbf{(E) } 1332$

Solution

See also

2020 AMC 12A (Problems • Answer Key • Resources)
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2019 AMC 12B Problems 		Followed by
2020 AMC 12B Problems
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All AMC 12 Problems and Solutions

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