ΒΆ 2020 AMC 10B Problems and Solutions
Problem Set Workbook
Access the downloadable workbook for 2020 AMC 10B problems here.
Discussion Forum
Engage in discussion about the 2020 AMC 10B math contest by visiting Random Math AMC 10B 2020 Forum
Individual Problems and Solutions
For problems and detailed solutions to each of the 2020 AMC 10B problems, please refer below:
Problem 1: What is the value of
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Problem 2: Carl has cubes each having side length , and Kate has cubes each having side length . What is the total volume of these cubes?
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Problem 3: The ratio of to is , the ratio of to is , and the ratio of to is . What is the ratio of to ?
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Problem 4: The acute angles of a right triangle are and , where and both and are prime numbers. What is the least possible value of ?
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Problem 5: How many distinguishable arrangements are there of brown tile, purple tile, green tiles, and yellow tiles in a row from left to right? (Tiles of the same color are indistinguishable.)
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Problem 6: Driving along a highway, Megan noticed that her odometer showed (miles). This number is a palindrome - it reads the same forward and backward. Then hours later, the odometer displayed the next higher palindrome. What was her average speed, in miles per hour, during this -hour period?
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Problem 7: How many positive even multiples of less than are perfect squares?
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Problem 8: Points and lie in a plane with . How many locations for point in this plane are there such that the triangle with vertices , , and is a right triangle with area square units?
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Problem 9: How many ordered pairs of integers satisfy the equation
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Problem 10: A three-quarter sector of a circle of radius inches together with its interior can be rolled up to form the lateral surface of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches?
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Problem 11: Ms. Carr asks her students to read any of the books on a reading list. Harold randomly selects books from this list, and Betty does the same. What is the probability that there are exactly books that they both select?
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Problem 12: The decimal representation of
consists of a string of zeros after the decimal point, followed by a and then several more digits. How many zeros are in that initial string of zeros after the decimal point?
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Problem 13: Andy the Ant lives on a coordinate plane and is currently at facing east (that is, in the positive -direction). Andy moves unit and then turns left. From there, Andy moves units (north) and then turns left. He then moves units (west) and again turns left. Andy continues this process, increasing his distance each time by unit and always turning left. What is the location of the point at which Andy makes the th left turn?
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Problem 14: As shown in the figure below, six semicircles lie in the interior of a regular hexagon with side length so that the diameters of the semicircles coincide with the sides of the hexagon. What is the area of the shaded region-inside the hexagon but outside all of the semicircles?
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Problem 15: Steve wrote the digits , and in order repeatedly from left to right, forming a list of digits, beginning . He then erased every third digit from his list (that is, the rd, th, th, ... digits from the left), then erased every fourth digit from the resulting list (that is, the digits from the left in what remained), and then erased every fifth digit from what remained at that point. What is the sum of the three digits that were then in positions and ?
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Problem 16: Bela and Jenn play the following game on the closed interval of the real number line, where is a fixed integer greater than . They take turns playing, with Bela going first. At his first turn, Bela chooses any real number in the interval . Thereafter, the player whose turn it is chooses a real number that is more than one unit away from all numbers previously chosen by either player. A player unable to choose such a number loses. Using optimal strategy, which player will win the game?
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A. Bela will always win.
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C. Bela will win if and only if is odd.
D. Jenn will win if and only if is odd.
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Problem 17: There are people standing equally spaced around a circle. Each person knows exactly of the other people: the people standing next to her or him, as well as the person directly across the circle. How many ways are there for the people to split up into pairs so that the members of each pair know each other?
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Problem 18: An urn contains one red ball and one blue ball. A box of extra red and blue balls lies nearby. George performs the following operation four times: he draws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to the urn. After the four iterations the urn contains six balls. What is the probability that the urn contains three balls of each color?
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Problem 19: In a certain card game, a player is dealt a hand of cards from a deck of distinct cards. The number of distinct (unordered) hands that can be dealt to the player can be written as . What is the digit ?
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Problem 20: Let be a right rectangular prism (box) with edge lengths , and , together with its interior. For real , let be the set of points in -dimensional space that lie within a distance of some point in . The volume of can be expressed as , where , , and are positive real numbers. What is ?
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Problem 21: In square , points and lie on and , respectively, so that . Points and lie on and , respectively, and points and lie on so that and . See the figure below. Triangle , quadrilateral , quadrilateral , and pentagon each has area . What is ?
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Problem 22: What is the remainder when is divided by ?
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Problem 23: Square in the coordinate plane has vertices at the points , and . Consider the following four transformations:
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Each of these transformations maps the square onto itself, but the positions of the labeled vertices will change. For example, applying and then would send the vertex at to and would send the vertex at to itself. How many sequences of transformations chosen from will send all of the labeled vertices back to their original positions? (For example, is one sequence of transformations that will send the vertices back to their original positions.)
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Problem 24: How many positive integers satisfy
(Recall that is the greatest integer not exceeding .)
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Problem 25: Let denote the number of ways of writing the positive integer as a product
where , the are integers strictly greater than , and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are counted as distinct). For example, the number can be written as , and , so . What is ?
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The problems on this page are the property of the MAA's American Mathematics Competitions