2003 AIME I Problems
2003 AIME I (Answer Key) | AoPS Contest Collections • PDF | ||
Instructions
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Contents
Problem 1
Given that
![$\frac{((3!)!)!}{3!} = k \cdot n!,$](http://latex.artofproblemsolving.com/d/2/9/d29e6d13e0d8b213ee70519a8250268a6a19a4c1.png)
where and
are positive integers and
is as large as possible, find
Problem 2
One hundred concentric circles with radii are drawn in a plane. The interior of the circle of radius 1 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 100 can be expressed as
where
and
are relatively prime positive integers. Find
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.
Problem 4
Given that and that
find
Problem 5
Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures 3 by 4 by 5 units. Given that the volume of this set is where
and
are positive integers, and
and
are relatively prime, find
Problem 6
The sum of the areas of all triangles whose vertices are also vertices of a 1 by 1 by 1 cube is where
and
are integers. Find
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
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 30. Find the sum of the four terms.
Problem 9
An integer between 1000 and 9999, 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?
Problem 10
Triangle is isosceles with
and
Point
is in the interior of the triangle so that
and
Find the number of degrees in
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
Problem 12
In convex quadrilateral and
The perimeter of
is 640. Find
(The notation
means the greatest integer that is less than or equal to
)
Problem 13
Let be the number of positive integers that are less than or equal to 2003 and whose base-2 representation has more 1's than 0's. Find the remainder when
is divided by 1000.
Problem 14
The decimal representation of where
and
are relatively prime positive integers and
contains the digits 2, 5, and 1 consecutively, and in that order. Find the smallest value of
for which this is possible.
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