Difference between revisions of "Mock AIME 1 20062007 Problems/Problem 5"
(No difference)

Revision as of 14:48, 3 April 2012
Modified Problem
For a prime number , define the function as follows: If there exists , , such that
set . Otherwise, set . Compute the sum .
Original Problem
Let be a prime and satisfy for all integers . is the greatest integer less than or equal to . If for fixed , there exists an integer such that:
then . If there is no such , then . If , find the sum: .
Solution
The definition of is equivalent to the following: "If has a multiplicative inverse mod , is the member of the set such that . Otherwise, ."
Note that this really gives a welldefined function because that set includes exactly one member from each congruence class modulo , and each invertible element has inverses in only one such class.
From this point onwards, it's clear: as cycles through , also cycles through the same values in some order. We cover those values 11 times. Thus the answer is .