ΒΆ 2024 AMC10B Problems and Solutions
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Individual Problems and Solutions
For problems and detailed solutions to each of the 2024 AMC10B problems, please refer below:
Problem 1: In a long line of people arranged left to right, the th person from the left is also the th person from the right. How many people are in the line?
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Problem 2: What is
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Problem 3: For how many integer values of is
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Problem 4: Balls numbered are deposited in bins, labeled , and , using the following procedure: ball is deposited in bin , balls and in bin , the next balls in bin , the next in bin , and so on, cycling back to bin after balls are deposited in bin (for example, balls numbered are deposited in bin at step of this process); in which bin is ball deposited?
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Problem 5: In the following expression, Melanie changed some of the plus signs to minus signs:
When the new expression was evaluated, it was negative. What is the least number of plus signs that Melanie could have changed to minus signs?
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Problem 6: A rectangle has integer length sides and an area of . What is the least possible perimeter of the rectangle?
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Problem 7: What is the remainder when is divided by
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Problem 8: Let be the product of all the positive integer divisors of . What is the units digit of
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Problem 9: Real numbers , and have arithmetic mean . The arithmetic mean of , and is . What is the arithmetic mean of , and
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Problem 10: Quadrilateral is a parallelogram, and is the midpoint of the side . Let be the intersection of lines and . What is the ratio of the area of quadrilateral to the area of
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Problem 11: In the figure below is a rectangle with and . Point lies on , point lies on , and is a right angle. The areas of triangles and are equal. What is the area of
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Problem 12: A group of students from different countries meet at a mathematics competition. Each student speaks the same number of languages, and, for every pair of students and , student speaks some language that student does not speak, and student speaks some language that student does not speak. What is the least possible total number of languages spoken by all the students?
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Problem 13: Positive integers and satisfy the equation . What is the minimum possible value of
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Problem 14: A dartboard is the region in the coordinate plane consisting of points such that . A target is the region where . A dart is thrown and lands at a random point in . The probability that the dart lands in can be expressed as , where and are relatively prime positive integers. What is
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Problem 15: A list of real numbers consists of , and , as well as , and with . The range of the list is , and the mean and the median are both positive integers. How many ordered triples are possible?
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Problem 16: Jerry likes to play with numbers. One day, he wrote all the integers from to on the whiteboard. Then he repeatedly chose four numbers on the whiteboard, erased them, and replaced them by either their sum or their product. (For example, Jerry's first step might have been to erase and and then write either their sum, or their product, on the whiteboard.) After repeatedly performing this operation, Jerry noticed that all the remaining numbers on the whiteboard were odd. What is the maximum possible number of integers on the whiteboard at that time?
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Problem 17: In a race among snails, there is at most one tie, but that tie can involve any number of snails. For example, the result of the race might be that Dazzler is first; Abby, Cyrus, and Elroy are tied for second; and Bruna is fifth. How many different results of the race are possible?
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Problem 18: How many different remainders can result when the th power of an integer is divided by
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Problem 19: In the following table, each question mark is to be replaced by "Possible" or "Not Possible" to indicate whether a nonvertical line with the given slope can contain the given number of lattice points (points both of whose coordinates are integers). How many of the entries will be "Possible"?
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Problem 20: Three different pairs of shoes are placed in a row so that no left shoe is next to a right shoe from a different pair. In how many ways can these six shoes be lined up?
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Problem 21: Two straight pipes (circular cylinders), with radii and , lie parallel and in contact on a flat floor. The figure below shows a head-on view. What is the sum of the possible radii of a third parallel pipe lying on the same floor and in contact with both?
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Problem 22: A group of people will be partitioned into indistinguishable -person committees. Each committee will have one chairperson and one secretary. The number of different ways to make these assignments can be written as , where and are positive integers and is not divisible by . What is
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Problem 23: The Fibonacci numbers are defined by , and for . What is
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Problem 24: Let
How many of the values , and are integers?
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Problem 25: Each of bricks (right rectangular prisms) has dimensions , where , and are pairwise relatively prime positive integers. These bricks are arranged to form a block, as shown on the left below. A th brick with the same dimensions is introduced, and these bricks are reconfigured into a block, shown on the right. The new block is unit taller, unit wider, and unit deeper than the old one. What is
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The problems on this page are the property of the MAA's American Mathematics Competitions