2009 IMO Problems/Problem 1
Let be a positive integer and let be distinct integers in the set such that divides for . Prove that doesn't divide .
Author: Ross Atkins, Australia
Let such that and . Suppose divides . Note implies and hence . Similarly one has for all 's, in particular, and force . Now gives , similarly one has for all 's, that is 's satisfy and , but there should be at most one such integer satisfies them within the range of for and . A contradiction!!!