2

Part of 2005 AIME Problems

Problems(2)

Arithmetic Series and Common Differences

Source:

For each positive integer

k

S_k

denote the increasing arithmetic sequence of integers whose first term is

1

and whose common difference is

k

S_3

is the sequence

1,4,7,10,...

. For how many values of

k

S_k

contain the term

2005

arithmetic sequencenumber theoryprime factorization

Breakfast at a Hotel

Source:

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

\frac{m}{n}

m

n

are relatively prime integers, find

m+n

countingdistinguishabilityprobabilitynumber theoryrelatively prime