Subcontests
(3)Ibero American 2012 - Problem 3
Let n to be a positive integer. Given a set {a1,a2,…,an} of integers, where ai∈{0,1,2,3,…,2n−1}, ∀i, we associate to each of its subsets the sum of its elements; particularly, the empty subset has sum of its elements equal to 0. If all of these sums have different remainders when divided by 2n, we say that {a1,a2,…,an} is n-complete.For each n, find the number of n-complete sets. Ibero American 2012 - Problem 4
Let a,b,c,d be integers such that the number a−b+c−d is odd and it divides the number a2−b2+c2−d2. Show that, for every positive integer n, a−b+c−d divides an−bn+cn−dn.