Problem Set Workbook
Access the downloadable workbook for 2004 AIME I problems here.
Discussion Forum
Engage in discussion about the 2004 AIME I math contest by visiting Random Math AIME I 2004 Forum
Individual Problems and Solutions
For problems and detailed solutions to each of the 2004 AIME I problems, please refer below:
Problem 1: The digits of a positive integer are four consecutive integers in decreasing order when read from left to right. What is the sum of the possible remainders when is divided by ?
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Problem 2: Set consists of consecutive integers whose sum is m$, and set consists of consecutive integers whose sum is . The absolute value of the difference between the greatest element of and the greatest element of is . Find .
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Problem 3: A convex polyhedron has vertices, edges, and faces, of which are triangular, and of which are quadrilaterals. A space diagonal is a line segment connecting two non-adjacent vertices that do not belong to the same face. How many space diagonals does have?
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Problem 4: A square has sides of length . Set is the set of all line segments that have length and whose endpoints are on adjacent sides of the square. The midpoints of the line segments in set enclose a region whose area to the nearest hundredth is . Find .
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Problem 5: Alpha and Beta both took part in a two-day problem-solving competition. At the end of the second day, each had attempted questions worth a total of points. Alpha scored points out of points attempted on the first day, and scored points out of points attempted on the second day. Beta, who did not attempt points on the first day, had a positive integer score on each of the two days, and Beta's daily success ratio (points scored divided by points attempted) on each day was less than Alpha's on that day. Alpha's two-day success ratio was . The largest possible two-day success ratio that Beta could have achieved is , where and are relatively prime positive integers. What is ?
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Problem 6: An integer is called snakelike if its decimal representation satisfies if is odd and if is even. How many snakelike integers between and have four distinct digits?
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Problem 7: Let be the coefficient of in the expansion of the product
Find .
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Problem 8: Define a regular -pointed star to be the union of line segments such that
the points are coplanar and no three of them are collinear,
each of the line segments intersects at least one of the other line segments at a point other than an endpoint,
all of the angles at are congruent,
all of the line segments are congruent, and
the path turns counterclockwise at an angle of less than at each vertex.
There are no regular -pointed, -pointed, or -pointed stars. All regular -pointed stars are similar, but there are two non-similar regular -pointed stars. How many non-similar regular -pointed stars are there?
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Problem 9: Let be a triangle with sides , and , and be a -by- rectangle. A segment is drawn to divide triangle into a triangle and a trapezoid , and another segment is drawn to divide rectangle into a triangle and a trapezoid such that is similar to and is similar to . The minimum value of the area of can be written in the form , where and are relatively prime positive integers. Find .
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Problem 10: A circle of radius is randomly placed in a -by- rectangle so that the circle lies completely within the rectangle. Given that the probability that the circle will not touch diagonal is , where and are relatively prime positive integers, find .
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Problem 11: A solid in the shape of a right circular cone is inches tall and its base has a -inch radius. The entire surface of the cone, including its base, is painted. A plane parallel to the base of the cone divides the cone into two solids, a smaller cone-shaped solid and a frustum-shaped solid , in such a way that the ratio between the areas of the painted surfaces of and and the ratio between the volumes of and are both equal to . Given that , where and are relatively prime positive integers, find .
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Problem 12: Let be the set of ordered pairs such that , and and are both even. Given that the area of the graph of is , where and are relatively prime positive integers, find . The notation denotes the greatest integer that is less than or equal to .
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Problem 13: The polynomial
has complex zeros of the form , , with and . Given that , where and are relatively prime positive integers, find .
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Problem 14: A unicorn is tethered by a -foot silver rope to the base of a magician's cylindrical tower whose radius is feet. The rope is attached to the tower at ground level and to the unicorn at a height of feet. The unicorn has pulled the rope taut, the end of the rope is feet from its nearest point on the tower, and the length of rope that is touching the tower is feet, where , and are positive integers, and is prime. Find .
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Problem 15: For all positive integers , let
and define a sequence as follows: and for all positive integers . Let be the smallest such that . (For example, 3 and .) Let be the number of positive integers such that . Find the sum of the distinct prime factors of .
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The problems on this page are the property of the MAA's American Mathematics Competitions