ΒΆ 2023 AMC12A Problems and Solutions
Problem Set Workbook
Access the downloadable workbook for 2023 AMC12A problems here.
Discussion Forum
Engage in discussion about the 2023 AMC12A math contest by visiting Random Math AMC12A 2023 Forum
Individual Problems and Solutions
For problems and detailed solutions to each of the 2023 AMC12A problems, please refer below:
Problem 1: Cities and are miles apart. Alicia lives in and Beth lives in . Alicia bikes towards at miles per hour. Leaving at the same time, Beth bikes toward at miles per hour. How many miles from City will they be when they meet?
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Problem 2: The weight of of a large pizza together with cups of orange slices is the same as the weight of of a large pizza together with cup of orange slices. A cup of orange slices weighs of a pound. What is the weight, in pounds, of a large pizza?
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Problem 3: How many positive perfect squares less than are divisible by ?
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Problem 4: How many digits are in the base-ten representation of ?
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Problem 5: Janet rolls a standard -sided die times and keeps a running total of the numbers she rolls. What is the probability that at some point, her running total will equal ?
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Problem 6: Points and lie on the graph of . The midpoint of is . What is the positive difference between the -coordinates of and ?
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Problem 7: A digital display shows the current date as an -digit integer consisting of a -digit year, followed by a -digit month, followed by a -digit date within the month. For example, Arbor Day this year is displayed as . For how many dates in does each digit appear an even number of times in the -digit display for that date?
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Problem 8: Maureen is keeping track of the mean of her quiz scores this semester. If Maureen scores an on the next quiz, her mean will increase by . If she scores an on each of the next three quizzes, her mean will increase by . What is the mean of her quiz scores currently?
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Problem 9: A square of area is inscribed in a square of area , creating four congruent triangles, as shown below. What is the ratio of the shorter leg to the longer leg in the shaded right triangle?
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Problem 10: Positive real numbers and satisfy and . What is ?
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Problem 11: What is the degree measure of the acute angle formed by lines with slopes and ?
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Problem 12: What is the value of
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Problem 13: In a table tennis tournament every participant played every other participant exactly once. Although there were twice as many right-handed players as left-handed players, the number of games won by left-handed players was more than the number of games won by right-handed players (there were no ties and no ambidextrous players). What is the total number of games played?
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Problem 14: How many complex numbers satisfy the equation , where is the conjugate of the complex number ?
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Problem 15: Usain is walking for exercise by zigzagging across a -meter by -meter rectangular field, beginning at point and ending on the segment . He wants to increase the distance walked by zigzagging as shown in the figure below . What angle will produce a length that is meters? (The figure is not drawn to scale. Do not assume that the zigzag path has exactly four segments as shown; it could be more or fewer.)
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Problem 16: Consider the set of complex numbers satisfying . The maximum value of the imaginary part of can be written in the form , where and are relatively prime positive integers. What is ?
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Problem 17: Flora the frog starts at on the number line and makes a sequence of jumps to the right. In any one jump, independent of previous jumps, Flora leaps a positive integer distance with probability . What is the probability that Flora will eventually land at ?
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Problem 18: Circles and each have radius , and the distance between their centers is . Circle is the largest circle internally tangent to both and . Circle is internally tangent to both and and externally tangent to . What is the radius of ?
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Problem 19: What is the product of all solutions to the equation
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Problem 20: Rows , and of a triangular array of integers are shown below:
Each row after the first row is formed by placing a at each end of the row, and each interior entry is greater than the sum of the two numbers diagonally above it in the previous row. What is the units digit of the sum of the numbers in the rd row?
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Problem 21: If and are vertices of a polyhedron, define the distance to be the minimum number of edges of the polyhedron one must traverse in order to connect and . For example, if is an edge of the polyhedron, then , but if and are edges and is not an edge, then . Let , and be randomly chosen distinct vertices of a regular icosahedron (regular polyhedron made up of equilateral triangles). What is the probability that ?
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Problem 22: Let be the unique function defined on the positive integers such that
for all positive integers , where the sum is taken over all positive divisors of . What is ?
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Problem 23: How many ordered pairs of positive real numbers satisfy the equation
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Problem 24: Let be the number of sequences such that is a positive integer less than or equal to , each is a subset of , and is a subset of for each between and , inclusive. For example, is one such sequence, with . What is the remainder when is divided by ?
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Problem 25: There is a unique sequence of integers such that
whenever is defined. What is ?
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The problems on this page are the property of the MAA's American Mathematics Competitions