ΒΆ 2004 AMC 10B Problems and Solutions
Problem Set Workbook
Access the downloadable workbook for 2004 AMC 10B problems here.
Discussion Forum
Engage in discussion about the 2004 AMC 10B math contest by visiting Random Math AMC 10B 2004 Forum
Individual Problems and Solutions
For problems and detailed solutions to each of the 2004 AMC 10B problems, please refer below:
Problem 1: Each row of the Misty Moon Amphitheater has seats. Rows through are reserved for a youth club. How many seats are reserved for this club?
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Problem 2: How many two-digit positive integers have at least one as a digit?
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Problem 3: At each basketball practice last week, Jenny made twice as many free throws as she made at the previous practice. At her fifth practice she made free throws. How many free throws did she make at the first practice?
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Problem 4: A standard six-sided die is rolled, and is the product of the five numbers that are visible. What is the largest number that is certain to divide ?
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Problem 5: In the expression , the values of , , , and are , , , and , although not necessarily in that order. What is the maximum possible value of the result?
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Problem 6: Which of the following numbers is a perfect square?
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Problem 7: On a trip from the United States to Canada, Isabella took U.S. dollars. At the border she exchanged them all, receiving Canadian dollars for every U.S. dollars. After spending Canadian dollars, she had Canadian dollars left. What is the sum of the digits of ?
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Problem 8: Minneapolis-St. Paul International Airport is miles southwest of downtown St. Paul and miles southeast of downtown Minneapolis. Which of the following is closest to the number of miles between downtown St. Paul and downtown Minneapolis?
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Problem 9: A square has sides of length , and a circle centered at one of its vertices has radius . What is the area of the union of the regions enclosed by the square and the circle?
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Problem 10: A grocer makes a display of cans in which the top row has one can and each lower row has two more cans than the row above it. If the display contains cans, how many rows does it contain?
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Problem 11: Two eight-sided dice each have faces numbered through . When the dice are rolled, each face has an equal probability of appearing on the top. What is the probability that the product of the two top numbers is greater than their sum?
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Problem 12: An annulus is the region between two concentric circles. The concentric circles in the ο¬gure have radii and , with . Let be a radius of the larger circle, let be tangent to the smaller circle at , and let be the radius of the larger circle that contains . Let , , and . What is the area of the annulus?
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Problem 13: In the United States, coins have the following thicknesses: penny, ; nickel, ; dime, ; quarter, . If a stack of these coins is exactly high, how many coins are in the stack?
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Problem 14: A bag initially contains red marbles and blue marbles only, with more blue than red. Red marbles are added to the bag until only of the marbles in the bag are blue. Then yellow marbles are added to the bag until only of the marbles in the bag are blue. Finally, the number of blue marbles in the bag is doubled. What fraction of the marbles now in the bag are blue?
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Problem 15: Patty has coins consisting of nickels and dimes. If her nickels were dimes and her dimes were nickels, she would have cents more. How much are her coins worth?
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Problem 16: Three circles of radius are externally tangent to each other and internally tangent to a larger circle. What is the radius of the large circle?
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Problem 17: The two digits in Jack's age are the same as the digits in Bill's age, but in reverse order. In five years Jack will be twice as old as Bill will be then. What is the difference in their current ages?
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Problem 18: In right triangle , we have , and . Points , and are located on , and , respectively, so that , , and . What is the ratio of the area of to that of ?
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Problem 19: In the sequence , each term after the third is found by subtracting the previous term from the sum of the two terms that precede that term. For example, the fourth term is . What is the term in this sequence?
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Problem 20: In points and lie on and , respectively. If and intersect at so that and , what is ?
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Problem 21: Let and be two arithmetic progressions. The set is the union of the first terms of each sequence. How many distinct numbers are in
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Problem 22: A triangle with sides of , and has both an inscribed and a circumscribed circle. What is the distance between the centers of those circles?
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Problem 23: Each face of a cube is painted either red or blue, each with probability . The color of each face is determined independently. What is the probability that the painted cube can be placed on a horizontal surface so that the four vertical faces are all the same color?
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Problem 24: In we have , , and . Point is on the circumscribed circle of the triangle so that bisects . What is the value of ?
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Problem 25: A circle of radius is internally tangent to two circles of radius at points and , where is a diameter of the smaller circle. What is the area of the region, shaded in the picture, that is outside the smaller circle and inside each of the two larger circles?
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The problems on this page are the property of the MAA's American Mathematics Competitions