2020 AMC 12A Problems
2020 AMC 12A (Answer Key) Printable version: | AoPS Resources • PDF | ||
Instructions
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Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 See also
Problem 1
Carlos took of a whole pie. Maria took one third of the remainder. What portion of the whole pie was left?
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
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?
Problem 4
How many -digit positive integers (that is, integers between and , inclusive) having only even digits are divisible by
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?
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
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?
Problem 8
What is the median of the following list of numbers
Problem 9
How many solutions does the equation have on the interval
Problem 10
There is a unique positive integer such thatWhat is the sum of the digits of
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
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
Problem 13
There are integers , , and , each greater than 1, such thatfor all . What is ?
Problem 14
Regular octagon has area . Let be the area of quadrilateral . What is
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
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
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?
Problem 18
Quadrilateral satisfies , and . Diagonals and intersect at point , and . What is the area of quadrilateral ?
Problem 19
There exists a unique strictly increasing sequence of nonnegative integers such thatWhat is
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.)
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
Problem 22
Let and be the sequences of real numbers such that for all integers , where . What is
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?
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
Problem 25
The number , where and are relatively prime positive integers, has the property that the sum of all real numbers satisfyingis , where denotes the greatest integer less than or equal to and denotes the fractional part of . What is
See also
2020 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by 2019 AMC 12B Problems |
Followed by 2020 AMC 12B Problems |
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All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.