Problem Set Workbook
Access the downloadable workbook for 2003 AIME I problems here.
Discussion Forum
Engage in discussion about the 2003 AIME I math contest by visiting Random Math AIME I 2003 Forum
Individual Problems and Solutions
For problems and detailed solutions to each of the 2003 AIME I problems, please refer below:
Problem 1: Given that , where and are positive integers and is as large as possible, find .
Solution:
Problem 2: One hundred concentric circles with radii are drawn in a plane. The interior of the circle of radius is colored red, and each region bounded by consecutive circles is colored either red or green, with no two adjacent regions the same color. The ratio of the total area of the green regions to the area of the circle of radius can be expressed as , where and are relatively prime positive integers. Find .
Solution:
Problem 3: Let the set . Susan makes a list as follows: for each two-element subset of , she writes on her list the greater of the set's two elements. Find the sum of the numbers on the list.
Solution:
Problem 4: Given that and that , find .
Solution:
Problem 5: Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures by by units. Given that the volume of this set is , where , and are positive integers, and and are relatively prime, find .
Solution:
Problem 6: The sum of the areas of all triangles whose vertices are also vertices of a by by cube is , where , and are integers. Find .
Solution:
Problem 7: Point is on with and . Point is not on so that , and and are integers. Let be the sum of all possible perimeters of .
Find .
Solution:
Problem 8: In an increasing sequence of four positive integers, the first three terms form an arithmetic progression, the last three terms form a geometric progression, and the first and fourth terms differ by . Find the sum of the four terms.
Solution:
Problem 9: An integer between and , inclusive, is called balanced if the sum of its two leftmost digits equals the sum of its two rightmost digits. How many balanced integers are there?
Solution:
Problem 10: Triangle is isosceles with and . Point is in the interior of the triangle so that and . Find the number of degrees in .
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Problem 11: An angle is chosen at random from the interval . Let be the probability that the numbers , and are not the lengths of the sides of a triangle. Given that , where is the number of degrees in and and are positive integers with , find .
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Problem 12: In convex quadrilateral , and . The perimeter of is . Find . (The notation means the greatest integer that is less than or equal to .)
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Problem 13: Let be the number of positive integers that are less than or equal to and whose base- representation has more 's than 's. Find the remainder when is divided by .
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Problem 14: The decimal representation of , where and are relatively prime positive integers and , contains the digits , and consecutively, and in that order. Find the smallest value of for which this is possible.
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Problem 15: In , and . Let be the midpoint of , and let be the point on such that bisects angle . Let be the point on such that . Suppose that meets at . The ratio can be written in the form , where and are relatively prime positive integers. Find .
Solution:
The problems on this page are the property of the MAA's American Mathematics Competitions