Exchange of candies amongst n pupils in a round table
Source: Vietnamese TST 2011 P6
April 27, 2011
invariantfunctioncombinatorics unsolvedcombinatorics
Problem Statement
Let be an integer greater than pupils are seated around a round table, each having a certain number of candies (it is possible that some pupils don't have a candy) such that the sum of all the candies they possess is a multiple of They exchange their candies as follows: For each student's candies at first, there is at least a student who has more candies than the student sitting to his/her right side, in which case, the student on the right side is given a candy by that student. After a round of exchanging, if there is at least a student who has candies greater than the right side student, then he/she will give a candy to the next student sitting to his/her right side. Prove that after the exchange of candies is completed (ie, when it reaches equilibrium), all students have the same number of candies.