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Marking positive integers with 0,1, or 2

Source: 8th European Mathematical Cup, Junior Category, Q1

December 26, 2019
number theorycombinatorics

Problem Statement

Every positive integer is marked with a number from the set {0,1,2}\{ 0,1,2\}, according to the following rule: if a positive integer k is marked with j, then the integer k+j is marked with 0.\text{if a positive integer }k\text{ is marked with }j,\text{ then the integer }k+j\text{ is marked with }0. Let SS denote the sum of marks of the first 20192019 positive integers. Determine the maximum possible value of SS.
Proposed by Ivan Novak
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