Let $$S$$ be a set of five different positive integers, the largest of which is $$n$$. It is impossible to construct a quadrilateral with non zero area, whose side lengths are all distinct elements of $$S$$. The smallest possible value of $$n$$ is

A quadrilateral with side lengths $$a,b,c,d$$ requires the largest side to be less than the sum of the other three. The set $$\{1,2,3,6,11\}$$ has largest element $$11$$, while $$1+2+3<11$$ and the other choices of four elements also fail to form a non degenerate quadrilateral. No smaller largest element permits five distinct positive integers with this property, so the smallest possible value is $$11$$.