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partition of {1,2,...,3n} into 3 subsets such as a_i +b_i = c_i

Source: Romania IMO TST 1990 p6

February 19, 2020
partitionSubsetscombinatorics

Problem Statement

Prove that there are infinitely many n’s for which there exists a partition of {1,2,...,3n}\{1,2,...,3n\} into subsets {a1,...,an},{b1,...,bn},{c1,...,cn}\{a_1,...,a_n\}, \{b_1,...,b_n\}, \{c_1,...,c_n\} such that ai+bi=cia_i +b_i = c_i for all ii, and prove that there are infinitely many nn’s for which there is no such partition.
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