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Determine the greatest positive integer

k

that satisfies the following property: The set of positive integers can be partitioned into

k

subsets

A_1, A_2, \ldots, A_k

such that for all integers

n \geq 15

and all

i \in \{1, 2, \ldots, k\}

there exist two distinct elements of

A_i

whose sum is

n.

Proposed by Igor Voronovich, Belarus

Problem Statement