IMO Shortlist 2013 problem C4

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Dodao/la: arhiva
21. rujna 2014.
Let n be a positive integer, and A be a subset of \{1, \ldots, n\}. An A-partition of n into k parts is a representation of n as a sum n = a_1 + \cdots a_k, where the parts a_1, \ldots, a_k belong to A and are not necessarily distinct. The number of different parts in such a partition is the number of (distinct) elements in the set \{a_1, a_2, \ldots, a_k\}.

We say that an A-partition of n into k parts is optimal if there is no A-partition of n into r parts with r < k. Prove that any optimal A-partition of n containts at most \sqrt[3]{6n} different parts.
Izvor: Germany