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Let n≥1n\ge 1n≥1 be an integer and let a1,…,ana_1,\ldots,a_na1,…,an be real numbers. For every nonempty subset S⊆{1,…,n}S\subseteq\{1,\ldots,n\}S⊆{1,…,n}, record the average
1∣S∣∑i∈Sai.\frac{1}{|S|}\sum_{i\in S}a_i.∣S∣1∑i∈Sai.
Sort all 2n−12^n-12n−1 entries, keeping every repetition.
Prove that this list uniquely determines the original sequence up to permutation.