Difference between revisions of "2005 AIME II Problems"
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== Problem 2 == | == 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 <math> \frac mn, </math> where <math> m </math> and <math> n </math> are relatively prime integers, find <math> m+n. </math> | |
[[2005 AIME II Problems/Problem 2|Solution]] | [[2005 AIME II Problems/Problem 2|Solution]] | ||
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== Problem 3 == | == Problem 3 == | ||
How many positive integers have exactly three proper divisors, each of which is less than 50? | How many positive integers have exactly three proper divisors, each of which is less than 50? |
Revision as of 22:17, 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
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 .
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 .