2004 AIME I Problems
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Instructions
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Contents
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 37?
Problem 2
Set consists of consecutive integers whose sum is 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 99. Find
Problem 3
A convex polyhedron has 26 vertices, 60 edges, and 36 faces, 24 of which are triangular, and 12 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?
Problem 4
A square has sides of length 2. Set is the set of all line segments that have length 2 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
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 rate (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 achieve is where and are relatively prime positive integers. What is ?
Problem 6
An integer is called snakelike if its decimal representation satisfies if is odd and if is even. How many snakelike integers between 1000 and 9999 have four distinct digits?
Problem 7
Let be the coefficient of in the expansion of the product Find
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 180 degrees at each vertex.
There are no regular 3-pointed, 4-pointed, or 6-pointed stars. All regular 5-pointed stars are similar, but there are two non-similar regular 7-pointed stars. How many non-similar regular 1000-pointed stars are there?
Problem 9
Let be a triangle with sides 3, 4, and 5, and be a 6-by-7 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
Problem 10
A circle of radius 1 is randomly placed in a 15-by-36 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
Problem 11
A solid in the shape of a right circular cone is 4 inches tall and its base has a 3-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
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
Problem 13
The polynomial has 34 complex roots of the form with and Given that where and are relatively prime positive integers, find
Problem 14
A unicorn is tethered by a 20-foot silver rope to the base of a magician's cylindrical tower whose radius is 8 feet. The rope is attached to the tower at ground level and to the unicorn at a height of 4 feet. The unicorn has pulled the rope taut, the end of the rope is 4 feet from the nearest point on the tower, and the length of the rope that is touching the tower is feet, where and are positive integers, and is prime. Find
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, and .) Let be the number of positive integers such that . Find the sum of the distinct prime factors of .
See Also
2004 AIME I (Problems • Answer Key • Resources) | ||
Preceded by 2003 AIME II Problems |
Followed by 2004 AIME II Problems | |
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All AIME Problems and Solutions |
- 2004 AIME I
- American Invitational Mathematics Examination
- AIME Problems and Solutions
- Mathematics competition resources
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.