 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.
### Part (a)
We need to prove that if \( k = 2015^{2015} \), there exist \( a_1, a_2, \cdots, a_{2015} \) such that \( f(i) = k \) for all \( 1 \le i \le 2015 \).

1. **Define the function \( f(i) \):**
   \[
   f(i) = a_i + \frac{a_1 a_2 \cdots a_{2015}}{a_i}
   \]

2. **Set \( a_i = 2015 \) for all \( i \):**
   \[
   a_1 = a_2 = \cdots = a_{2015} = 2015
   \]

3. **Calculate \( f(i) \) for this choice:**
   \[
   f(i) = 2015 + \frac{2015 \times 2015 \times \cdots \times 2015}{2015}
   \]
   Since there are 2015 terms in the product:
   \[
   f(i) = 2015 + \frac{2015^{2015}}{2015} = 2015 + 2015^{2014}
   \]

4. **Simplify the expression:**
   \[
   f(i) = 2015 + 2015^{2014} = 2015^{2014} + 2015
   \]

5. **Verify that \( f(i) = k \):**
   \[
   k = 2015^{2015}
   \]
   \[
   f(i) = 2015 + 2015^{2014} = 2015 \times (1 + 2015^{2013}) = 2015^{2015}
   \]
   Thus, \( f(i) = k \) for all \( 1 \le i \le 2015 \).

### Part (b)
We need to find the maximum \( k_0 \) such 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.

1. **Consider the tuples \( A_1 = (k-1, 1, 1, \ldots, 1) \) and \( A_2 = (1, k-1, 1, \ldots, 1) \):**
   \[
   f_1(i) = k \quad \text{for all} \quad 1 \le i \le 2015
   \]
   \[
   f_2(i) = k \quad \text{for all} \quad 1 \le i \le 2015
   \]

2. **Check the condition for \( k = 1 \):**
   \[
   f(i) \geq 2
   \]
   For \( k = 1 \), there are no such tuples because \( f(i) \geq 2 \).

3. **Check the condition for \( k = 2 \):**
   \[
   f(i) = 2
   \]
   We can build only one tuple \( (1, 1, \ldots, 1) \).

4. **Determine the maximum \( k_0 \):**
   For \( k \le 2 \), there are no \( k \) such that there are at least 2 different 2015-tuples which fulfill the condition. Therefore, the maximum \( k_0 \) is 2.

The final answer is \( \boxed{ k_0 = 2 } \)