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partition of 1-15, or 1-30 into 3-element sets with x+y=3z

Source: Netherlands - Dutch NMO 1981 p3

January 28, 2023

number theorycombinatorics

Problem Statement

We want to split the set of natural numbers from

1

3n

, where

n

is a natural number, into

n

mutually disjoint sets

\{x,y,z\}

of three elements such that always holds:

x + y = 3z

. Is this possible for : a)

n = 5

? b)

n=10

? In both cases, provide either such a split or proof that such a split is not possible.