# 2005 AIME II Problems

## 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 with radius . 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 *k*, let denote the increasing arithmetic sequence of integers whose first term is 1 and whose common difference is *k*. For example, is the squence . For how many values of *k* does contain the term 2005?

## Problem 3

How many positive integers have exactly three proper divisors, 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 distunguishable arrangements of the 8 coins.

## Problem 6

Let be the product of nonreal roots of . Find

## Problem 7

In quadrilateral , , , and . Given that , where *p* and *q* are positive integers, find .