Impossible Decagon

Geometry Level 5

Let \(D\) be the set of all of the multisets containing 10 positive integers such that these integers cannot be the side lengths of a decagon. If \(A\) is a set in \(D\) and \(S(A)\) is the sum of the elements of \(A\), find the smallest value that \(S(A)\) can have.

A multiset is an unordered collection of elements that may repeat; compare this with a set, which may not repeat elements, or with a sequence, which is ordered.

Bonus: Can you generalize this for the \(n\)-gon?

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