2003年AIME II 真题：
Problem 1
The product 
of three positive integers is 6 times their sum, and one of the integers is the sum of the other two. Find the sum of all possible values of .
Problem 2
Let be the greatest integer multiple of 8, whose digits are all different. What is the remainder when is divided by 1000?
Problem 3
Define a 
as a sequence of letters that consists only of the letters 
, 
, and 
- some of these letters may not appear in the sequence - and in which is never immediately followed by , is never immediately followed by , and is never immediately followed by . How many seven-letter good words are there?
Problem 4
In a regular tetrahedron, the centers of the four faces are the vertices of a smaller tetrahedron. The ratio of the volume of the smaller tetrahedron to that of the larger is 
, where 
and 
are relatively prime positive integers. Find 
.
Problem 5
A cylindrical log has diameter 
inches. A wedge is cut from the log by making two planar cuts that go entirely through the log. The first is perpendicular to the axis of the cylinder, and the plane of the second cut forms a 
angle with the plane of the first cut. The intersection of these two planes has exactly one point in common with the log. The number of cubic inches in the wedge can be expressed as 
, where n is a positive integer. Find .
Problem 6
In triangle 
and point 
is the intersection of the medians. Points 
and 
are the images of 
and 
respectively, after a 
rotation about 
What is the area of the union of the two regions enclosed by the triangles 
and 
Problem 7
Find the area of rhombus 
given that the radii of the circles circumscribed around triangles 
and 
are 
and 
, respectively.
Problem 8
Find the eighth term of the sequence 
whose terms are formed by multiplying the corresponding terms of two arithmetic sequences.
Problem 9
Consider the polynomials 
and 
Given that 
and 
are the roots of 
find 
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