2005年AIME I 真题：
Problem 1
Six congruent circles form a ring with each circle externally tangent to two circles adjacent to it. All circles are internally tangent to a circle 
with radius 30. Let 
be the area of the region inside circle and outside of the six circles in the ring. Find 
Problem 2
For each positive integer 
let 
denote the increasing arithmetic sequence of integers whose first term is 1 and whose common difference is 
For example, 
is the sequence 
For how many values of 
does contain the term 2005?
Problem 3
How many positive integers have exactly three proper divisors (positive integral divisors excluding itself), each of which is less than 50?
Problem 4
The director of a marching band wishes to place the members into a formation that includes all of them and has no unfilled positions. If they are arranged in a square formation, there are 5 members left over. The director realizes that if he arranges the group in a formation with 7 more rows than columns, there are no members left over. Find the maximum number of members this band can have.
Problem 5
Robert has 4 indistinguishable gold coins and 4 indistinguishable silver coins. Each coin has an engraving of one face on one side, but not on the other. He wants to stack the eight coins on a table into a single stack so that no two adjacent coins are face to face. Find the number of possible distinguishable arrangements of the 8 coins.
Problem 6
Let 
be the product of the nonreal roots of 
Find 
Problem 7
In quadrilateral 
and 
Given that 
where 
and 
are positive integers, find 
Problem 8
The equation 
has three real roots. Given that their sum is 
where 
and 
are relatively prime positive integers, find 
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