Let $ a_1,a_2, \cdots ,a_{2015} $ be $2015$-tuples of positive integers (not necessary distinct) and let $ k $ be a positive integers. Denote $\displaystyle f(i)=a_i+\frac{a_1a_2 \cdots a_{2015}}{a_i} $. a) Prove that if $ k=2015^{2015} $, there exist $ a_1, a_2, \cdots , a_{2015} $ such that $ f(i)= k $ for all $1\le i\le 2015 $. b) Find the maximum $k_0$ so that for $k\le k_0$, there are no $k$ such that there are at least $ 2 $ different $2015$-tuples which fulfill the above condition.