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Prime divisors in a set

Source: Iranian National Olympiad (3rd Round) 2004

January 9, 2009
number theoryleast common multiplenumber theory proposed

Problem Statement

Prove that for any n n, there is a subset {a1,…,an} \{a_1,\dots,a_n\} of N \mathbb N such that for each subset S S of {1,…,n} \{1,\dots,n\}, ∑i∈Sai \sum_{i\in S}a_i has the same set of prime divisors.
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