I have found out this very nice inequality, which I have not been able to solve and I think it is an interesting challenge. Simulation results seem to verify it.
Consider $n$ positive (strictly larger than zero) real numbers $x_1,x_2,...,x_n$ such that $x_1+x_2+...+x_n=1$ and $n$ other arbitrary positive (strictly larger than zero) real numbers, denoted as $y_1,y_2,...y_n$. Show that: $$\left(\frac{1}{\sum_{i=1}^n{\frac{x_i}{y_i}}}+1\right) \cdot \sum_{i=1}^n{\frac{x_i}{y_i+1}} \leq 1.$$
Let me know if you can help me.