Difference between revisions of "2005 AIME II Problems"
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== Problem 6 == | == Problem 6 == | ||
− | + | The cards in a stack of <math> 2n </math> cards are numbered consecutively from 1 through <math> 2n </math> from top to bottom. The top <math> n </math> cards are removed, kept in order, and form pile <math> A. </math> The remaining cards form pile <math> B. </math> The cards are then restacked by taking cards alternately from the tops of pile <math> B </math> and <math> A, </math> respectively. In this process, card number <math> (n+1) </math> becomes the bottom card of the new stack, card number 1 is on top of this card, and so on, until piles <math> A </math> and <math> B </math> 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. | |
[[2005 AIME II Problems/Problem 6|Solution]] | [[2005 AIME II Problems/Problem 6|Solution]] | ||
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== Problem 7 == | == Problem 7 == | ||
In quadrilateral <math>ABCD</math>, <math>BC=8</math>, <math>CD=12</math>, <math>AD=10</math> and <math>m\angle A=m\angle B=60\circ</math>. Given that <math>AB=p+\sqrt{q}</math>, where ''p'' and ''q'' are positive integers, find <math>p+q</math>. | In quadrilateral <math>ABCD</math>, <math>BC=8</math>, <math>CD=12</math>, <math>AD=10</math> and <math>m\angle A=m\angle B=60\circ</math>. Given that <math>AB=p+\sqrt{q}</math>, where ''p'' and ''q'' are positive integers, find <math>p+q</math>. |
Revision as of 23:22, 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
In quadrilateral ,
,
,
and
. Given that
, where p and q 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
.
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
.