Problem Set Workbook
Access the downloadable workbook for 2001 AIME I problems here.
Discussion Forum
Engage in discussion about the 2001 AIME I math contest by visiting Random Math AIME I 2001 Forum
Individual Problems and Solutions
For problems and detailed solutions to each of the 2001 AIME I problems, please refer below:
Problem 1: Find the sum of all positive two-digit integers that are divisible by each of their digits.
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Problem 2: A finite set of distinct real numbers has the following properties: the mean of is less than the mean of , and the mean of is more than the mean of .
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Problem 3: Find the sum of all the roots, real and non-real, of the equation , given that there are no multiple roots.
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Problem 4: In triangle , angles and measure degrees and degrees, respectively. The bisector of angle intersects at , and . The area of the triangle can be written in the form , where , and are positive integers, and is not divisible by the square of any prime. Find .
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Problem 5: An equilateral triangle is inscribed in the ellipse whose equationis . One vertex of the triangle is , one altitude is contained in the -axis, and the length of each side is , where and are relatively prime positive integers. Find .
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Problem 6: A fair die is rolled four times. The probability that each of the final three rolls is at least as large as the roll preceding it may be expressed in the form , where and are relatively prime positive integers. Find .
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Problem 7: Triange has , and . Points and are located on and , respectively, such that is parallel to and contains the center of the inscribed circle of triangle . Then , where and are relatively prime positive integers. Find .
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Problem 8: Call a positive integer a double if the digits of the base- representation of form a base- number that is twice . For example, is a double because its base- representation is . What is the largest double?
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Problem 9: In triangle and . Point is on is on , and is on . Let , and , where , and are positive and satisfy and . The ratio of the area of triangle to the area of triangle can be written in the form , where and are relatively prime positive integers. Find .
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Problem 10: Let be the set of points whose coordinates , and are integers that satisfy , and . Two distinct points are randomly chosen from . The probability that the midpoint of the segment they determine also belongs to is , where and are relatively prime positive integers. Find .
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Problem 11: In a rectangular array of points, with rows and columns, the points are numbered consecutively from left to right beginning with the top row. Thus the top row is numbered through , the second row is numbered through , and so forth. Five points, , and , are selected so that each is in row . Let be the number associated with . Now renumber the array consecutively from top to bottom, beginning with the first column. Let be the number associated with after renumbering. It is found that , , and . Find the smallest possible value of .
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Problem 12: A sphere is inscribed in the tetrahedron whose vertices are , , and . The radius of the sphere is , where and are relatively prime positive integers. Find .
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Problem 13: In a certain circle, the chord of a -degree arc is centimeters longer than the chord of a -degree arc, where . The length of the chord of a -degree arc is centimeters, where and are positive integers. Find .
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Problem 14: A mail carrier delivers mail to the nineteen houses on the east side of Elm Street. The carrier notices that no two adjacent houses ever get mail on the same day, but that there are never more than two houses in a row that get no mail on the same day. How many different patterns of mail delivery are possible?
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Problem 15: The numbers , and are randomly written on the faces of a regular octahedron so that each face contains a different number. The probability that no two consecutive numbers, where and are considered to be consecutive, are written on faces that share an edge is , 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