ΒΆ 2024 AMC10A Problems and Solutions
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Individual Problems and Solutions
For problems and detailed solutions to each of the 2024 AMC10A problems, please refer below:
Problem 1: What is the value of
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Problem 2: A model used to estimate the time it will take to hike to the top of a mountain on a trail is of the form , where and are constants, is the time in minutes, is the length of the trail in miles, and is the altitude gain in feet. The model estimates that it will take minutes to hike to the top if a trail is miles long and ascends feet, as well as if a trail is miles long and ascends feet. How many minutes does the model estimate it will take to hike to the top if the trail is miles long and ascends feet?
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Problem 3: Let be the least prime number that can be written as the sum of 5 distinct prime numbers. What is the sum of the digits of
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Problem 4: The number is written as the sum of not necessarily distinct two-digit numbers. What is the least number of two-digit numbers needed to write this sum?
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Problem 5: What is the least value of such that is a multiple of
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Problem 6: What is the minimum number of successive swaps of adjacent letters in the string that are needed to change the string to (For example, swaps are required to change to ; one such sequence of swaps is .)
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Problem 7: The product of three integers is . What is the least possible positive sum of the three integers?
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Problem 8: Amy, Bomani, Charlie, and Daria work in a chocolate factory. On Monday Amy, Bomani, and Charlie started working at and were able to pack , and packages, respectively, every minutes. At some later time, Daria joined the group, and Daria was able to pack packages every minutes. Together, they finished packing packages at exactly . At what time did Daria join the group?
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Problem 9: In how many ways can juniors and seniors form disjoint teams of people so that each team has juniors and $24 seniors?
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Problem 10: Consider the following operation. Given a positive integer , if is a multiple of , then you replace by . If is not a multiple of , then you replace by . Then continue this process. For example, beginning with , this procedure gives . Suppose you start with . What value results if you perform this operation exactly times?
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Problem 11: How many ordered pairs of integers satisfy
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Problem 12: Zelda played the Adventures of Math game on August and scored points. She continued to play daily over the next days. The bar chart below shows the daily change in her score compared to the day before. (For example, Zelda's score on August was points.) What was Zelda's average score in points over the days?
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Problem 13: Two transformations are said to commute if applying the first followed by the second gives the same result as applying the second followed by the first. Consider these four transformations of the coordinate plane:
- A translation units to the right,
- A -rotation counterclockwise about the origin,
- A reflection across the -axis, and
- A dilation centered at the origin with scale factor .
Of the pairs of distinct transformations from this list, how many commute?
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Problem 14: One side of an equilateral triangle of height lies on line . A circle of radius is tangent to and is externally tangent to the triangle. The area of the region exterior to the triangle and the circle and bounded by the triangle, the circle, and line can be written as , where , and are positive integers and is not divisible by the square of any prime. What is
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Problem 15: Let be the greatest integer such that both and are perfect squares. What is the units digit of
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Problem 16: All of the rectangles in the figure below, which is drawn to scale, are similar to the enclosing rectangle. Each number represents the area of its rectangle. What is length
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Problem 17: Two teams are in a best-two-out-of-three playoff: the teams will play at most games, and the winner of the playoff is the first team to win games. The first game is played on Team A's home field, and the remaining games are played on Team B's home field. Team A has a chance of winning at home, and its probability of winning when playing away from home is . Outcomes of the games are independent. The probability that Team A wins the playoff is . Then can be written in the form , where and are positive integers. What is
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Problem 18: There are exactly positive integers with such that the base- integer is divisible by (where is in base ten). What is the sum of the digits of
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Problem 19: The first three terms of a geometric sequence are the integers , , and , where . What is the sum of the digits of the least possible value of
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Problem 20: Let be a subset of such that the following two conditions hold:
- If and are distinct elements of , then .
- If and are distinct odd elements of , then .
What is the maximum possible number of elements in
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Problem 21: The numbers, in order, of each row and the numbers, in order, of each column of a array of integers form an arithmetic progression of length . The numbers in positions , and are , and , respectively. What number is in position
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Problem 22: Let be the kite formed by joining two right triangles with legs 1 and along a common hypotenuse. Eight copies of are used to form the polygon shown below. What is the area of triangle
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Problem 23: Integers , and satisfy , and . What is
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Problem 24: A bee is moving in three-dimensional space. A fair six-sided die with faces labeled , , and is rolled. Suppose the bee occupies the point . If the die shows , then the bee moves to the point , and if the die shows , then the bee moves to the point . Analogous moves are made with the other four outcomes. Suppose the bee starts at the point and the die is rolled four times. What is the probability that the bee traverses four distinct edges of some unit cube?
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Problem 25: The figure below shows a dotted grid cells wide and cells tall consisting of squares. Carl places -inch toothpicks along some of the sides of the squares to create a closed loop that does not intersect itself. The numbers in the cells indicate the number of sides of that square that are to be covered by toothpicks, and any number of toothpicks are allowed if no number is written. In how many ways can Carl place the toothpicks?
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The problems on this page are the property of the MAA's American Mathematics Competitions