MathDB

4

Part of 2013 APMO

Problems(1)

Sets of integers with i+a in A or i-b in B

Source: APMO 2013, Problem 4

5/3/2013
Let aa and bb be positive integers, and let AA and BB be finite sets of integers satisfying (i) AA and BB are disjoint; (ii) if an integer ii belongs to either to AA or to BB, then either i+ai+a belongs to AA or i−bi-b belongs to BB. Prove that a∣A∣=b∣B∣a\left\lvert A \right\rvert = b \left\lvert B \right\rvert. (Here ∣X∣\left\lvert X \right\rvert denotes the number of elements in the set XX.)
algebrapolynomialcombinatoricsAPMO