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Ibero American 2012 - Problem 3

Source: Ibero American 2012

October 2, 2012
algebrapolynomialmodular arithmeticinductioncombinatorics proposedcombinatorics

Problem Statement

Let nn to be a positive integer. Given a set {a1,a2,…,an}\{ a_1, a_2, \ldots, a_n \} of integers, where ai∈{0,1,2,3,…,2n−1},a_i \in \{ 0, 1, 2, 3, \ldots, 2^n -1 \}, ∀i\forall i, we associate to each of its subsets the sum of its elements; particularly, the empty subset has sum of its elements equal to 00. If all of these sums have different remainders when divided by 2n2^n, we say that {a1,a2,…,an}\{ a_1, a_2, \ldots, a_n \} is nn-complete.
For each nn, find the number of nn-complete sets.
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