I was given the following problem by a colleague who is studying for the GRE. The question, I believe, comes from a study guide.
A rope is cut into five pieces of differing lengths. After the rope is cut, the length of the longest piece is twice the average length of the five cut pieces and is four times the length of the shortest piece. Show that the median length of the five cut pieces is less than the average length of the five cut pieces
I’ve racked my brain for a few hours and haven’t been able to show how the median piece length is less than the average piece length (which is obviously L/5 for a rope of length L). I was able to show that the longest piece has a length of 2L/5 while the shortest piece has a length of L/10. Any assistance would be greatly appreciated.
Suppose that the rope is 100 units long and that it is cut into lengths $$m_1<m_2<m_3<m_4<m_5$$ The average length is $20$, so $m_5=40$ and $m_1 =10$. Since $m_2>m_1=10$ and $m_4>m_3,$ we have $$100=m_1+m_2+m_3+m_4+m_5>60+2m_3\implies m_3<20.$$