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a - b= one of the numbers m, n, m - n, m + n when coloring integers in 4 colors

Source: Oliforum Contest III 2012 p6 https://artofproblemsolving.com/community/c2487525_oliforum_contes

September 28, 2021
Coloringcombinatorics

Problem Statement

Suppose that every integer is colored using one of 44 colors. Let m,nm, n be distinct odd integers such that m+n≠0m + n \ne 0. Prove that there exist integers aa, b b of the same color such that a−b a - b equals one of the numbers mm, nn, m−nm - n, m+nm + n.
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