ΒΆ 2023 AMC10B Problems and Solutions
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Individual Problems and Solutions
For problems and detailed solutions to each of the 2023 AMC10B problems, please refer below:
Problem 1: Mrs. Jones is pouring orange juice into four identical glasses for her four sons. She fills the first three glasses completely but runs out of juice when the fourth glass is only full. What fraction of a glass must Mrs. Jones pour from each of the first three glasses into the fourth glass so that all four glasses will have the same amount of juice?
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Problem 2: Carlos went to a sports store to buy running shoes. Running shoes were on sale, with prices reduced by on every pair of shoes. Carlos also knew that he had to pay a sales tax on the discounted price. He had . What is the original (before discount) price of the most expensive shoes he could afford to buy?
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Problem 3: A right triangle is inscribed in circle , and a right triangle is inscribed in circle . What is the ratio of the area of circle to the area of circle
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Problem 4: Jackson's paintbrush makes a narrow strip with a width of millimeters. Jackson has enough paint to make a strip meters long. How many square centimeters of paper could Jackson cover with paint?
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Problem 5: Maddy and Lara see a list of numbers written on a blackboard. Maddy adds to each number in the list and finds that the sum of her new numbers is . Lara multiplies each number in the list by and finds that the sum of her new numbers is also . How many numbers are written on the blackboard?
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Problem 6: Let , and for . How many terms in the sequence are even?
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Problem 7: Square is rotated clockwise about its center to obtain square , as shown below. What is the degree measure of
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Problem 8: What is the units digit of
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Problem 9: The numbers and are a pair of consecutive positive perfect squares whose difference is . How many pairs of consecutive positive perfect squares have a difference of less than or equal to
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Problem 10: You are playing a game. A rectangle covers two adjacent squares (oriented either horizontally or vertically) of a grid of squares, but you are not told which two squares are covered. Your goal is to find at least one square that is covered by the rectangle. A "turn" consists of you guessing a square, after which you are told whether that square is covered by the hidden rectangle. What is the minimum number of turns you need to ensure that at least one of your guessed squares is covered by the rectangle?
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Problem 11: Suzanne went to the bank and withdrew . The teller gave her this amount using bills, bills, and bills, with at least one of each denomination. How many different collections of bills could Suzanne have received?
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Problem 12: When the roots of the polynomial
are removed from the real number line, what remains is the union of disjoint open intervals. On how many of these intervals is positive?
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Problem 13: What is the area of the region in the coordinate plane defined by
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Problem 14: How many ordered pairs of integers satisfy the equation
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Problem 15: What is the least positive integer such that is a perfect square?
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Problem 16: Define an upno to be a positive integer of or more digits where the digits are strictly increasing moving left to right. Similarly, define a downno to be a positive integer of or more digits where the digits are strictly decreasing moving left to right. For instance, the number is an upno and is a downno. Let equal the total number of upnos and equal the total number of downnos. What is
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Problem 17: A rectangular box has distinct edge lengths , and . The sum of the lengths of all edges of is , the sum of the areas of all faces of is , and the volume of is . What is the length of the longest interior diagonal connecting two vertices of
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Problem 18: Suppose that , and are positive integers such that
Which of the following statements are necessarily true?
. If or or both, then .
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. if and only if and .
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Problem 19: Sonya the frog chooses a point uniformly at random lying within the square in the coordinate plane and hops to that point. She then chooses a distance uniformly at random in the interval and a direction uniformly at random from . All her choices are independent. She now hops the chosen distance in the chosen direction. What is the probability that she lands outside the square?
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Problem 20: Four congruent semicircles are drawn on the surface of a sphere with radius , as shown, creating a closed curve that divides its surface into two congruent regions. The length of the curve is . What is
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Problem 21: Each of balls is randomly placed into one of bins. Which of the following is closest to the probability that each of the bins will contain an odd number of balls?
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Problem 22: How many distinct values of satisfy , where denotes the greatest integer less than or equal to
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Problem 23: An arithmetic sequence of positive integers has terms, initial term , and common difference . Carl wrote down all the terms in this sequence correctly except for one term, which was off by . The sum of the terms he wrote down was . What is
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Problem 24: What is the length of the boundary of the region in the plane consisting of points of the form where , and
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Problem 25: A regular pentagon with area is printed on paper and cut out. All five vertices are folded to the center of the pentagon, creating a smaller pentagon. What is the area of the new pentagon?
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The problems on this page are the property of the MAA's American Mathematics Competitions