View source for 2020 AMC 10A Problems

{{AMC10 Problems|year=2020|ab=A}}

==Problem 1==

What value of <math>x</math> satisfies

<cmath>x- \frac{3}{4} = \frac{5}{12} - \frac{1}{3}?</cmath>
<math>\textbf{(A)}\ -\frac{2}{3}\qquad\textbf{(B)}\ \frac{7}{36}\qquad\textbf{(C)}\ \frac{7}{12}\qquad\textbf{(D)}\ \frac{2}{3}\qquad\textbf{(E)}\ \frac{5}{6}</math>

[[2020 AMC 10A Problems/Problem 1|Solution]]

==Problem 2==

The numbers <math>3, 5, 7, a,</math> and <math>b</math> have an average (arithmetic mean) of <math>15</math>. What is the average of <math>a</math> and <math>b</math>?

<math>\textbf{(A) } 0 \qquad\textbf{(B) } 15 \qquad\textbf{(C) } 30 \qquad\textbf{(D) } 45 \qquad\textbf{(E) } 60</math>

[[2020 AMC 10A Problems/Problem 2|Solution]]

==Problem 3==

Assuming <math>a\neq3</math>, <math>b\neq4</math>, and <math>c\neq5</math>, what is the value in simplest form of the following expression?
<cmath>\frac{a-3}{5-c} \cdot \frac{b-4}{3-a} \cdot \frac{c-5}{4-b}</cmath>

<math>\textbf{(A) } -1 \qquad \textbf{(B) } 1 \qquad \textbf{(C) } \frac{abc}{60} \qquad \textbf{(D) } \frac{1}{abc} - \frac{1}{60} \qquad \textbf{(E) } \frac{1}{60} - \frac{1}{abc}</math>

[[2020 AMC 10A Problems/Problem 3|Solution]]

==Problem 4==

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 10A Problems/Problem 4|Solution]]

==Problem 5==

What is the sum of all real numbers <math>x</math> for which <math>|x^2-12x+34|=2?</math>

<math>\textbf{(A) } 12 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 18 \qquad \textbf{(D) } 21 \qquad \textbf{(E) } 25</math>

[[2020 AMC 10A Problems/Problem 5|Solution]]

==Problem 6==

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 10A Problems/Problem 6|Solution]]

==Problem 7==

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 10A Problems/Problem 7|Solution]]

==Problem 8==

What is the value of

<cmath>1+2+3-4+5+6+7-8+\cdots+197+198+199-200?</cmath>

<math>\textbf{(A) } 9,800 \qquad \textbf{(B) } 9,900 \qquad \textbf{(C) } 10,000 \qquad \textbf{(D) } 10,100 \qquad \textbf{(E) } 10,200</math>

[[2020 AMC 10A Problems/Problem 8|Solution]]

==Problem 9==

A single bench section at a school event can hold either <math>7</math> adults or <math>11</math> children. When <math>N</math> bench sections are connected end to end, an equal number of adults and children seated together will occupy all the bench space. What is the least possible positive integer value of <math>N?</math>

<math>\textbf{(A) } 9 \qquad \textbf{(B) } 18 \qquad \textbf{(C) } 27 \qquad \textbf{(D) } 36 \qquad \textbf{(E) } 77</math>

[[2020 AMC 10A Problems/Problem 9|Solution]]

==Problem 10==

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 10A Problems/Problem 10|Solution]]

==Problem 11==

What is the median of the following list of <math>4040</math> numbers<math>?</math>

<cmath>1, 2, 3, ..., 2020, 1^2, 2^2, 3^2, ..., 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 10A Problems/Problem 11|Solution]]

==Problem 12==

math is stupid

==Problem 13==

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>\textbf{(A)}\ \frac12\qquad\textbf{(B)}\ \frac 58\qquad\textbf{(C)}\ \frac 23\qquad\textbf{(D)}\ \frac34\qquad\textbf{(E)}\ \frac 78</math>

[[2020 AMC 10A Problems/Problem 13|Solution]]

==Problem 14==

Real numbers <math>x</math> and <math>y</math> satisfy <math>x + y = 4</math> and <math>x \cdot y = -2</math>. What is the value of<cmath>x + \frac{x^3}{y^2} + \frac{y^3}{x^2} + y?</cmath><math>\textbf{(A)}\ 360\qquad\textbf{(B)}\ 400\qquad\textbf{(C)}\ 420\qquad\textbf{(D)}\ 440\qquad\textbf{(E)}\ 480</math>

[[2020 AMC 10A Problems/Problem 14|Solution]]

==Problem 15==

A positive integer divisor of <math>12!</math> is chosen at random. The probability that the divisor chosen is a perfect square can be expressed as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. What is <math>m+n</math>?

<math>\textbf{(A)}\ 3\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 23</math>

[[2020 AMC 10A 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>\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 10A Problems/Problem 16|Solution]]

==Problem 17==

Define<cmath>P(x) =(x-1^2)(x-2^2)\cdots(x-100^2).</cmath>How many integers <math>n</math> are there such that <math>P(n)\leq 0</math>?

<math>\textbf{(A) } 4900 \qquad \textbf{(B) } 4950\qquad \textbf{(C) } 5000\qquad \textbf{(D) } 5050 \qquad \textbf{(E) } 5100</math>

[[2020 AMC 10A Problems/Problem 17|Solution]]

==Problem 18==

Let <math>(a,b,c,d)</math> be an ordered quadruple of not necessarily distinct integers, each one of them in the set <math>{0,1,2,3}.</math> For how many such quadruples is it true that <math>a\cdot d-b\cdot c</math> is odd? (For example, <math>(0,3,1,1)</math> is one such quadruple, because <math>0\cdot 1-3\cdot 1 = -3</math> is odd.)

<math>\textbf{(A) } 48 \qquad \textbf{(B) } 64 \qquad \textbf{(C) } 96 \qquad \textbf{(D) } 128 \qquad \textbf{(E) } 192</math>

[[2020 AMC 10A Problems/Problem 18|Solution]]

==Problem 19==

As shown in the figure below, a regular dodecahedron (the polyhedron consisting of <math>12</math> congruent regular pentagonal faces) floats in empty space with two horizontal faces. Note that there is a ring of five slanted faces adjacent to the top face, and a ring of five slanted faces adjacent to the bottom face. How many ways are there to move from the top face to the bottom face via a sequence of adjacent faces so that each face is visited at most once and moves are not permitted from the bottom ring to the top ring?

<asy>
import graph;
unitsize(5cm);
pair A = (0.082, 0.378);
pair B = (0.091, 0.649);
pair C = (0.249, 0.899);
pair D = (0.479, 0.939);
pair E = (0.758, 0.893);
pair F = (0.862, 0.658);
pair G = (0.924, 0.403);
pair H = (0.747, 0.194);
pair I = (0.526, 0.075);
pair J = (0.251, 0.170);
pair K = (0.568, 0.234);
pair L = (0.262, 0.449);
pair M = (0.373, 0.813);
pair N = (0.731, 0.813);
pair O = (0.851, 0.461);
path[] f;
f[0] = A--B--C--M--L--cycle;
f[1] = C--D--E--N--M--cycle;
f[2] = E--F--G--O--N--cycle;
f[3] = G--H--I--K--O--cycle;
f[4] = I--J--A--L--K--cycle;
f[5] = K--L--M--N--O--cycle;
draw(f[0]);
axialshade(f[1], white, M, gray(0.5), (C+2*D)/3);
draw(f[1]);
filldraw(f[2], gray);
filldraw(f[3], gray);
axialshade(f[4], white, L, gray(0.7), J);
draw(f[4]);
draw(f[5]);
</asy>

<math>\textbf{(A) } 125 \qquad \textbf{(B) } 250 \qquad \textbf{(C) } 405 \qquad \textbf{(D) } 640 \qquad \textbf{(E) } 810</math>

[[2020 AMC 10A Problems/Problem 19|Solution]]

==Problem 20==

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 10A Problems/Problem 20|Solution]]

==Problem 21==

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 10A Problems/Problem 21|Solution]]

==Problem 22==

For how many positive integers <math>n \le 1000</math> is<cmath>\left\lfloor \dfrac{998}{n} \right\rfloor+\left\lfloor \dfrac{999}{n} \right\rfloor+\left\lfloor \dfrac{1000}{n}\right \rfloor</cmath>not divisible by <math>3</math>? (Recall that <math>\lfloor x \rfloor</math> is the greatest integer less than or equal to <math>x</math>.)

<math>\textbf{(A) } 22 \qquad\textbf{(B) } 23 \qquad\textbf{(C) } 24 \qquad\textbf{(D) } 25 \qquad\textbf{(E) } 26</math>

[[2020 AMC 10A Problems/Problem 22|Solution]]

==Problem 23==

Let <math>T</math> be the triangle in the coordinate plane with vertices <math>(0,0), (4,0),</math> and <math>(0,3).</math> Consider the following five isometries (rigid transformations) of the plane: rotations of <math>90^{\circ}, 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 10A Problems/Problem 23|Solution]]

==Problem 24==

Let <math>n</math> be the least positive integer greater than <math>1000</math> for which<cmath>\gcd(63, n+120) =21\quad \text{and} \quad \gcd(n+63, 120)=60.</cmath>What is the sum of the digits of <math>n</math>?

<math>\textbf{(A) } 12 \qquad\textbf{(B) } 15 \qquad\textbf{(C) } 18 \qquad\textbf{(D) } 21\qquad\textbf{(E) } 24</math>

[[2020 AMC 10A Problems/Problem 24|Solution]]

==Problem 25==

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 10A Problems/Problem 25|Solution]]

==See also==
{{AMC10 box|year=2020|ab=A|before=[[2019 AMC 10B Problems]]|after=[[2020 AMC 10B Problems]]}}
{{MAA Notice}}