MathDB

3

Part of 2012 IberoAmerican

Problems(2)

Ibero American 2012 - Problem 3

Source: Ibero American 2012

10/2/2012
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.
algebrapolynomialmodular arithmeticinductioncombinatorics proposedcombinatorics
Ibero American 2012 - Problem 6

Source:

10/3/2012
Show that, for every positive integer nn, there exist nn consecutive positive integers such that none is divisible by the sum of its digits.
(Alternative Formulation: Call a number good if it's not divisible by the sum of its digits. Show that for every positive integer nn there are nn consecutive good numbers.)
logarithmsfloor functionnumber theory proposednumber theory