Problem Set Workbook
Access the downloadable workbook for 2020 AMC 10A problems here.
Discussion Forum
Engage in discussion about the 2020 AMC 10A math contest by visiting Random Math AMC 10A 2020 Forum
Individual Problems and Solutions
For problems and detailed solutions to each of the 2020 AMC 10A problems, please refer below:
Problem 1: What value of satisfies
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Problem 2: The numbers , and have an average (arithmetic mean) of . What is the average of and ?
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Problem 3: Assuming , and , what is the value in simplest form of the following expression?
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Problem 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?
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Problem 5: What is the sum of all real numbers for which
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Problem 6: How many -digit positive integers (that is, integers between and , inclusive) having only even digits are divisible by ?
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Problem 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?
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Problem 8: What is the value of
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Problem 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 ?
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Problem 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?
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Problem 11: What is the median of the following list of numbers?
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Problem 12: Triangle is isosceles with . Medians and are perpendicular to each other, and . What is the area of ?
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Problem 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?
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Problem 14: Real numbers and satisfy and . What is the value of
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Problem 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 ?
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Problem 16: A point is chosen at random within the square in the coordinate plane whose vertices are , and . The probability that the point lies within units of a lattice point is . (A point is a lattice point if and are both integers.) What is to the nearest tenth?
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Problem 17: Define
How many integers are there such that ?
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Problem 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.)
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Problem 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?
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Problem 20: Quadrilateral satisfies , and . Diagonals and intersect at point , and . What is the area of quadrilateral ?
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Problem 21: There exists a unique strictly increasing sequence of nonnegative integers such that
What is ?
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Problem 22: For how many positive integers is
not divisible by ? (Recall that is the greatest integer less than or equal to .)
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Problem 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.)
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Problem 24: Let be the least positive integer greater than for which
What is the sum of the digits of ?
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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 . Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice?
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The problems on this page are the property of the MAA's American Mathematics Competitions