ΒΆ 2020 AMC 12A Problems and Solutions
Problem Set Workbook
Access the downloadable workbook for 2020 AMC 12A problems here.
Discussion Forum
Engage in discussion about the 2020 AMC 12A math contest by visiting Random Math AMC 12A 2020 Forum
Individual Problems and Solutions
For problems and detailed solutions to each of the 2020 AMC 12A problems, please refer below:
Problem 1: Carlos took of a whole pie. Maria took one third of the remainder. What portion of the whole pie was left?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 2: The acronym AMC is shown in the rectangular grid below with grid lines spaced unit apart. In units, what is the sum of the lengths of the line segments that form the acronym AMC?
.jpg)
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 3: A driver travels for hours at miles per hour, during which her car gets miles per gallon of gasoline. She is paid per mile, and her only expense is gasoline at per gallon. What is her net rate of pay, in dollars per hour, after this expense?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 4: How many -digit positive integers (that is, integers between and , inclusive) having only even digits are divisible by ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 5: The integers from to , inclusive, can be arranged to form a -by- 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?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 6: In the plane figure shown below, 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?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 7: Seven cubes, whose volumes are and 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?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 8: What is the median of the following list of numbers?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 9: How many solutions does the equation have on the interval ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 10: There is a unique positive integer such that
What is the sum of the digits of ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 11: A frog sitting at the point begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length , 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 , and . What is the probability that the sequence of jumps ends on a vertical side of the square?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 12: Line in the coordinate plane has the equation . This line is rotated counterclockwise about the point to obtain line . What is the -coordinate of the intercept of line ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 13: There are integers , and , each greater than , such that
for all . What is ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 14: Regular octagon has area . Let be the area of quadrilateral . What is ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 15: In the complex plane, let be the set of solutions to and let be the set of solutions to . What is the greatest distance between a point of and a point of ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 16: A point is chosen at random within the square in the coordinate plane whose vertices are , and . The probability that the point is within units of a lattice point is . (A point is a lattice point if and are both integers.) What is to the nearest tenth?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 17: The vertices of a quadrilateral lie on the graph of , and the -coordinates of these vertices are consecutive positive integers. The area of the quadrilateral is . What is the -coordinate of the leftmost vertex?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 18: Quadrilateral satisfies , and . Diagonals and intersect at point , and . What is the area of quadrilateral ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 19: There exists a unique strictly increasing sequence of nonnegative integers such that
What is ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 20: Let be the triangle in the coordinate plane with vertices , and . Consider the following five isometries (rigid transformations) of the plane: rotations of , and counterclockwise around the origin, reflection across the -axis, and reflection across the -axis. How many of the sequences of three of these transformations (not necessarily distinct) will return to its original position? (For example, a rotation, followed by a reflection across the -axis, followed by a reflection across the -axis will return to its original position, but a rotation, followed by a reflection across the -axis, followed by another reflection across the -axis will not return to its original position.)
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 21: How many positive integers are there such that is a multiple of , and the least common multiple of ! and equals times the greatest common divisor of ! and ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 22: Let and be the sequences of real numbers such that
for all integers , where . What is
Answer Choices:
A.
B.
C.
D.
E.
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 . Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 24: Suppose that is an equilateral triangle of side length , with the property that there is a unique point inside the triangle such that , and . What is ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 25: The number , where and are relatively prime positive integers, has the property that the sum of all real numbers satisfying
is , where denotes the greatest integer less than or equal to and denotes the fractional part of . What is ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
The problems on this page are the property of the MAA's American Mathematics Competitions