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Each positive integer is colored either red, blue and green. Show that there always exists distinct positive integers $a,b,c$ such that $a,b,c,a+b,b+c,a+c,a+b+c$ are all of the same color.

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closed as off-topic by user26857, zz20s, Xander Henderson, Feng Shao, nmasanta Oct 10 at 0:32

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