Each of segments intersects with at least half of the others, prove that there is a segment that intersects with all segments

Question

There are a finite number of segments in $R^1$, and each of them intersects with at least half of the others, prove that there is a segment that intersects with all of segments

My ideas: 1)There is a similar problem where any two segments intersect, which is solved using the Helly theorem, but I don't know how to use Helly in this situation. 2) Use some olympiad ideas

Answer 1

Idea: Let $I_l$ be the interval with the smallest right endpoint, and $I_r$ be the interval with the largest left endpoint. By pigeonhole, there exists an interval $I$ that intersects with both $I_l$ and $I_r$. Now show $I$ intersects with all the segments.