Every possible length

Probability Level 4

Let SS be a subset of [0,1][0,1] consisting of a union of 1010 disjoint closed intervals I1,I2,,I10.I_1, I_2, \ldots, I_{10}.

Suppose SS has the property that for every d[0,1],d \in [0,1], there are two points x,ySx,y \in S such that xy=d.|x-y|=d.

Letting s=n=110length(In),s = \sum\limits_{n=1}^{10} \text{length}(I_n), what is the minimum possible value of s?s?

Your answer should be a rational number pq,\frac{p}{q}, where pp and qq are coprime positive integers.

Find p+q.p+q.


Bonus: Describe the sets SS for which the minimum is attained.

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