Every possible length
Discrete Mathematics
Level
4
Let \(S\) be a subset of \([0,1]\) consisting of a union of \(10\) disjoint closed intervals \(I_1, I_2, \ldots, I_{10}.\)
Suppose \(S\) has the property that for every \(d \in [0,1],\) there are two points \(x,y \in S\) such that \(|x-y|=d.\)
Letting \(s = \sum\limits_{n=1}^{10} \text{length}(I_n),\) what is the minimum possible value of \(s?\)
Your answer should be a rational number \(\frac{p}{q},\) where \(p\) and \(q\) are coprime positive integers.
Find \(p+q.\)
Bonus: Describe the sets \(S\) for which the minimum is attained.
by
Patrick Corn