2005 AIME II Problems

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Problem 1

Six circles form a ring with with each circle externally tangent to two circles adjacent to it. All circles are internally tangent to a circle $C$ with radius $30$. Let $K$ be the area of the region inside circle $C$ and outside of the six circles in the ring. Find $\lfloor K \rfloor$.

Solution

Problem 2

For each positive integer k, let $S_k$ denote the increasing arithmetic sequence of integers whose first term is 1 and whose common difference is k. For example, $S_3$ is the squence $1,4,7,10 ...$. For how many values of k does $S_k$ contain the term 2005?

Solution

Problem 3

How many positive integers have exactly three proper divisors, each of which is less than 50?

Solution

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.

Solution

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 distunguishable arrangements of the 8 coins.

Solution

Problem 6

Let $P$ be the product of nonreal roots of $x^4-4x^3+6x^2-4x=2005$. Find $\lfloor P \rfloor$

Solution

Problem 7

In quadrilateral $ABCD$, $BC=8$, $CD=12$, $AD=10$ and $m\angle A=m\angle B=60\circ$. Given that $AB=p+\sqrt{q}$, where p and q are positive integers, find $p+q$.

Solution

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