Školjka

%V0 Let $n$ be a positive integer, and $A$ be a subset of $\{1, \ldots, n\}$. An $A$-[i]partition[/i] of $n$ [i]into[/i] $k$ [i]parts[/i] 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 [i]number of different parts[/i] 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 [i]optimal[/i] 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.