Question

Suppose S is a union of finitely many disjoint subintervals of [0, 1] such that no two points in S have distance 1/10. Show that the total length of the intervals comprising S is at most 1/2.

do the first one asap

Answer 1

Cut the given segment into 5 Segments of length 13. Let AB be one of these segments and M its midpoint. Translate Each point of AM by the vector Miß . No colored point can be placed in MB w/o overlap. and their total length therefore does not exceed 1/10 of the to each desired Applying this reasoning 5 segments gives the result.