ΒΆ 1996 AHSME Problems and Solutions
Individual Problems and Solutions
For problems and detailed solutions to each of the 1996 AHSME problems, please refer below:
Problem 1: The addition below is incorrect. What is the largest digit that can be changed to make the addition correct?
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Problem 2: Each day Walter gets for doing his chores or for doing them exceptionally well. After days of doing his chores daily, Walter has received a total of . On how many days did Walter do them exceptionally well?
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Problem 3:
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Problem 4: Six numbers from a list of nine integers are , and . The largest possible value of the median of all nine numbers in this list is
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Problem 5: Given that , which of the following is the largest?
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Problem 6: If then
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Problem 7: A father takes his twins and a younger child out to dinner on the twins' birthday. The restaurant charges for the father and for each year of a child's age, where age is defined as the age at the most recent birthday. If the bill is , which of the following could be the age of the youngest child?
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Problem 8: If and , then
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Problem 9: Triangle and square are in perpendicular planes. Given that , and , what is
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Problem 10: How many line segments have both their endpoints located at the vertices of a given cube?
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Problem 11: Given a circle of radius , there are many line segments of length that are tangent to the circle at their midpoints. Find the area of the region consisting of all such line segments.
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Problem 12: A function from the integers to the integers is defined as follows:
Suppose is odd and . What is the sum of the digits of
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Problem 13: Sunny runs at a steady rate, and Moonbeam runs times as fast, where is a number greater than . If Moonbeam gives Sunny a head start of meters, how many meters must Moonbeam run to overtake Sunny?
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Problem 14: Let denote the sum of the even digits of . For example, . Find .
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Problem 15: Two opposite sides of a rectangle are each divided into congruent segments, and the endpoints of one segment are joined to the center to form triangle . The other sides are each divided into congruent segments, and the endpoints of one of these segments are joined to the center to form triangle . [See figure for .] What is the ratio of the area of triangle to the area of triangle
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Problem 16: A fair standard six-sided die is tossed three times. Given that the sum of the first two tosses equals the third, what is the probability that at least one "" is tossed?
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Problem 17: In rectangle , angle is trisected by and , where is on is on , and . Which of the following is closest to the area of the rectangle
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Problem 18: A circle of radius has center at . A circle of radius has center at . A line is tangent to the two circles at points in the first quadrant. Which of the following is closest to the -intercept of the line?
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Problem 19: The midpoints of the sides of a regular hexagon are joined to form a smaller hexagon. What fraction of the area of is enclosed by the smaller hexagon?
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Problem 20: In the -plane, what is the length of the shortest path from to that does not go inside the circle
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Problem 21: Triangles and are isosceles with , and intersects at . If , then is
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Problem 22: Four distinct points, , and , are to be selected from points evenly spaced around a circle. All quadruples are equally likely to be chosen. What is the probability that the chord intersects the chord
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Problem 23: The sum of the lengths of the twelve edges of a rectangular box is , and the distance from one corner of the box to the farthest corner is . The total surface area of the box is
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Problem 24: The sequence
consists of 's separated by blocks of 's with 's in the block. The sum of the first terms of this sequence is
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Problem 25: Given that , what is the largest possible value that can have?
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Problem 26: An urn contains marbles of four colors: red, white, blue, and green. When four marbles are drawn without replacement, the following events are equally likely:
the selection of four red marbles;
the selection of one white and three red marbles;
the selection of one white, one blue, and two red marbles; and
the selection of one marble of each color.
What is the smallest number of marbles satisfying the given condition?
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Problem 27: Consider two solid spherical balls, one centered at with radius , and the other centered at with radius . How many points with only integer coordinates (lattice points) are there in the intersection of the balls?
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Problem 28: On a rectangular parallelepiped, vertices , and are adjacent to vertex . The perpendicular distance from to the plane containing , and is closest to
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Problem 29: If is a positive integer such that has positive divisors and has positive divisors, then how many positive divisors does have?
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Problem 30: A hexagon inscribed in a circle has three consecutive sides each of length and three consecutive sides each of length . The chord of the circle that divides the hexagon into two trapezoids, one with three sides each of length and the other with three sides each of length , has length equal to , where and are relatively prime positive integers. Find .
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The problems on this page are the property of the MAA's American Mathematics Competitions