2020 AMC 10A Exam Problems
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Used with permission of the Mathematical Association of America.
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1.
What value of satisfies
Answer: E
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2.
The numbers and have an average (arithmetic mean) of What is the average of and
Answer: C
Solution(s):
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3.
Assuming and what is the value in simplest form of the following expression?
Answer: A
Solution(s):
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4.
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: E
Solution(s):
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5.
What is the sum of all real numbers for which
Answer: C
Solution(s):
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6.
How many -digit positive integers (that is, integers between and inclusive) having only even digits are divisible by
Answer: B
Solution(s):
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7.
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: C
Solution(s):
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8.
What is the value of
Answer: B
Solution(s):
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9.
A single bench section at a school event can hold either adults or children. When 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
Answer: B
Solution(s):
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10.
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: B
Solution(s):
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11.
What is the median of the following list of numbers
Answer: C
Solution(s):
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12.
Triangle is isosceles with Medians and are perpendicular to each other, and What is the area of
Answer: C
Solution(s):
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13.
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: B
Solution(s):
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14.
Real numbers and satisfy and What is the value of
Answer: D
Solution(s):
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15.
A positive integer divisor of is chosen at random. The probability that the divisor chosen is a perfect square can be expressed as where and are relatively prime positive integers. What is
Answer: E
Solution(s):
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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: B
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17.
Define How many integers are there such that
Answer: E
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18.
Let be an ordered quadruple of not necessarily distinct integers, each one of them in the set For how many such quadruples is it true that is odd? (For example, is one such quadruple, because is odd.)
Answer: C
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19.
As shown in the figure below, a regular dodecahedron (the polyhedron consisting of congruent regular pentagonal faces) floats in 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?
Answer: E
Solution(s):
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20.
Quadrilateral satisfies and Diagonals and intersect at point and What is the area of quadrilateral
Answer: D
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21.
There exists a unique strictly increasing sequence of nonnegative integers such thatWhat is
Answer: C
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22.
For how many positive integers isnot divisible by (Recall that is the greatest integer less than or equal to )
Answer: A
Solution(s):
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23.
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: A
Solution(s):
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24.
Let be the least positive integer greater than for which
and
What is the sum of the digits of
Answer: C
Solution(s):
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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 Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice?
Answer: A
Solution(s):
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