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
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
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
Find the number of positive integers that are divisors of at least one of
Determine the number of ordered pairs of integers such that and
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.
- Not the actual problem statement, but to clarify: "retains its original position" means as we count from the top of the stack, the first number is position 1, the second is position 2, and so the 131st number down is position 131. This question is saying that the 131st number down is equal to 131. (From someone with interpretation issues, I thought it would make this question clearer.)
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
For how many positive integers less than or equal to 1000 is true for all real ?
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
Let be a positive integer, and let be a sequence of reals such that and for Find
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
Let be a polynomial with integer coefficients that satisfies and Given that has two distinct integer solutions and find the product
In triangle and Point is on with Point is on such that Given that where and are relatively prime positive integers, find
Let and denote the circles and respectively. Let be the smallest positive 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