2005 AIME II Problems
Contents
Problem 1
A game uses a deck of 
different cards, where is an integer and 
The number of possible sets of 6 cards that can be drawn from the deck is 6 times the number of possible sets of 3 cards that can be drawn. Find 
Problem 2
A hotel packed breakfast for each of three guests. Each breakfast should have consisted of three types of rolls, one each of nut, cheese, and fruit rolls. The preparer wrapped each of the nine rolls and once wrapped, the rolls were indistinguishable from one another. She then randomly put three rolls in a bag for each of the guests. Given that the probability each guest got one roll of each type is 
where 
and are relatively prime integers, find 
Problem 3
An infinite geometric series has sum 2005. A new series, obtained by squaring each term of the original series, has 10 times the sum of the original series. The common ratio of the original series is 
where and are relatively prime integers. Find
Problem 4
Find the number of positive integers that are divisors of at least one of 
Problem 5
Determine the number of ordered pairs 
of integers such that 
and 
Problem 6
The cards in a stack of 
cards are numbered consecutively from 1 through from top to bottom. The top cards are removed, kept in order, and form pile 
The remaining cards form pile 
The cards are then restacked by taking cards alternately from the tops of pile 
and 
respectively. In this process, card number 
becomes the bottom card of the new stack, card number 1 is on top of this card, and so on, until piles 
and are exhausted. If, after the restacking process, at least one card from each pile occupies the same position that it occupied in the original stack, the stack is named magical. Find the number of cards in the magical stack in which card number 131 retains its original position.
Problem 7
Let ![$x=\frac{4}{(\sqrt{5}+1)(\sqrt[4]{5}+1)(\sqrt[8]{5}+1)(\sqrt[16]{5}+1)}.$](http://latex.artofproblemsolving.com/6/f/7/6f71d4755273a86295f916ae662a281e75af55c6.png)
Find 
Problem 8
Circles 
and 
are externally tangent, and they are both internally tangent to circle 
The radii of and are 4 and 10, respectively, and the centers of the three circles are all collinear. A chord of 
is also a common external tangent of and 
Given that the length of the chord is 
where 
and 
are positive integers, and are relatively prime, and is not divisible by the square of any prime, find 
Problem 9
For how many positive integers less than or equal to 1000 is 
true for all real 
?
Problem 10
Given that 
is a regular octahedron, that 
is the cube whose vertices are the centers of the faces of 
and that the ratio of the volume of to that of is where and are relatively prime integers, find
Problem 11
Let be a positive integer, and let 
be a sequence of integers such that 
and 
for 
Find 
Problem 12
Square 
has center 
and 
are on 
with 
and 
between and 
and 
Given that 
where 
and 
are positive integers and is not divisible by the square of any prime, find 
Problem 13
Let 
be a polynomial with integer coefficients that satisfies 
and 
Given that 
has two distinct integer solutions 
and 
find the product 
Problem 14
In triangle 
and 
Point 
is on 
with 
Point is on such that 
Given that 
where and 
are relatively prime positive integers, find 
Problem 15
Let 
and 
denote the circles 
and 
respectively. Let be the smallest possible value of 
for which the line 
contains the center of a circle that is externally tangent to and internally tangent to 
Given that 
where and are relatively prime integers, find 
