Difference between revisions of "2005 AIME II Problems"
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== Problem 7 == | == Problem 7 == | ||
| − | + | Let <math> x=\frac{4}{(\sqrt{5}+1)(\sqrt[4]{5}+1)(\sqrt[8]{5}+1)(\sqrt[16]{5}+1)}. </math> Find <math> (x+1)^{48}. </math> | |
[[2005 AIME II Problems/Problem 7|Solution]] | [[2005 AIME II Problems/Problem 7|Solution]] | ||
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== Problem 8 == | == Problem 8 == | ||
The equation '''<math>2^{333x-2}+2^{111x+2}=2^{222x+1}+1</math>''' has three real roots. Given that their sum is <math>\frac{m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime positive integers, find <math>m+n</math>. | The equation '''<math>2^{333x-2}+2^{111x+2}=2^{222x+1}+1</math>''' has three real roots. Given that their sum is <math>\frac{m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime positive integers, find <math>m+n</math>. | ||
Revision as of 23:23, 8 July 2006
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
The equation 
has three real roots. Given that their sum is 
where and are relatively prime positive integers, find 
.
Problem 9
Twenty seven unit cubes are painted orange on a set of four faces so that two non-painted faces share an edge. The 
cubes are randomly arranged to form a 
cube. Given the probability of the entire surface area of the larger cube is orange is 
where 
and 
are distinct primes and 
and 
are positive integers, find 
.
Problem 10
Triangle 
lies in the Cartesian Plane and has an area of 70. The coordinates of and 
are 
and 
respectively, and the coordinates of are 
The line containing the median to side 
has slope 
. Find the largest possible value of 
.
Problem 11
A semicircle with diameter 
is contained in a square whose sides have length 8. Given the maximum value of is 
, find .
