2001 AIME I Problems
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Instructions
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Contents
Problem 1
Find the sum of all positive two-digit integers that are divisible by each of their digits.
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 . Find the mean of .
Problem 3
Find the sum of the roots, real and non-real, of the equation , given that there are no multiple roots.
Problem 4
In triangle , angles and measure degrees and degrees, respectively. The bisector of angle intersects at , and . The area of triangle can be written in the form , where , , and are positive integers, and is not divisible by the square of any prime. Find .
Problem 5
An equilateral triangle is inscribed in the ellipse whose equation is . One vertex of the triangle is , one altitude is contained in the y-axis, and the length of each side is , where and are relatively prime positive integers. Find .
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 .
Problem 7
Triangle 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 .
Problem 8
Call a positive integer a if the digits of the base-7 representation of form a base-10 number that is twice . For example, is a 7-10 double because its base-7 representation is . What is the largest 7-10 double?
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 .
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
Problem 11
In a rectangular array of points, with 5 rows and columns, the points are numbered consecutively from left to right beginning with the top row. Thus the top row is numbered 1 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 the renumbering. It is found that and Find the smallest possible value of
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
Problem 13
In a certain circle, the chord of a -degree arc is 22 centimeters long, and the chord of a -degree arc is 20 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
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?
Problem 15
The numbers 1, 2, 3, 4, 5, 6, 7, and 8 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 8 and 1 are considered to be consecutive, are written on faces that share an edge is where and are relatively prime positive integers. Find
See also
2001 AIME I (Problems • Answer Key • Resources) | ||
Preceded by 2000 AIME II Problems |
Followed by 2001 AIME II Problems | |
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All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.