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Discerning Sets

Source: APMO 2014 Problem 4

March 28, 2014
pigeonhole principlefloor functionnumber theoryalgebracombinatorics

Problem Statement

Let nn and bb be positive integers. We say nn is bb-discerning if there exists a set consisting of nn different positive integers less than bb that has no two different subsets UU and VV such that the sum of all elements in UU equals the sum of all elements in VV.
(a) Prove that 88 is 100100-discerning. (b) Prove that 99 is not 100100-discerning.
Senior Problems Committee of the Australian Mathematical Olympiad Committee
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