Proving Inequality in Mathematics: Vacation Edition

[mathworker]

i found this problem interesting in stack exchange unfortunately i will participate in discussion for 4 days(vacation)
inequality - Prove $\sum_{i=1}^{n}\frac{a_{i}}{a_{i+1}}\ge\sum_{i=1}^{n}\frac{1-a_{i+1}}{1-a_{i}}$ if $a_{i}>0$ and $a_{1}+a_{2}+\cdots+a_{n}=1$ - Mathematics Stack Exchange
edit:hoping this to be an educational thread

 

[MarkFL]

Re: inequality

The actual problem statement here may be written as:

Given \(\displaystyle a_i>0\), \(\displaystyle \sum_{i=1}^n a_i=1\) and \(\displaystyle a_{n+1}=a_{1}\)

Prove:

\(\displaystyle \sum_{i=1}^{n}\dfrac{a_{i}}{a_{i+1}}\ge\sum_{i=1}^{n}\dfrac{1-a_{i+1}}{1-a_{i}}\)

Note: Normally, when a problem is posted in this sub-forum, the OP is expected to have a solution ready to post. However, the OP did not originally post the topic here and during a staff discussion, it was felt that this sub-forum would be best as it really does not fit into any neat category. So, consider this problem a challenge for our membership as a whole. (Cool)