Let $A$ be positive real number and $k$ a positive integer.
How to find the analytical solution of this system? Find the $a_i$ \begin{align} \begin{cases} \displaystyle\sum_{i=1}^n\ln\left(1+a_i\right)\geq k\ln\left(1+A/k\right),\\ \displaystyle\sum_{i=1}^na_i\leq A,\\ \displaystyle a_i>0\\ \end{cases} \end{align}