Given 2005 distinct numbers $a_1,\,a_2,\dots,a_{2005}$. By one question, we may take three different indices $1\le i<j<k\le 2005$ and find out the set of numbers $\{a_i,\,a_j,\,a_k\}$ (unordered, of course).  Find the minimal number of questions, which are necessary to find out all numbers $a_i$.
1. **Understanding the Problem:**
   We are given 2005 distinct numbers \(a_1, a_2, \dots, a_{2005}\). By asking a question, we can select three different indices \(1 \le i < j < k \le 2005\) and find out the set of numbers \(\{a_i, a_j, a_k\}\). We need to determine the minimal number of questions necessary to identify all the numbers \(a_i\).

2. **Establishing the Upper Bound:**
   To show that 1003 questions suffice, we can use the same method as **grobber** did. By asking 1003 questions, we can ensure that each number appears in at least one of the sets, allowing us to identify all 2005 numbers.

3. **Proving the Lower Bound:**
   We need to show that 1003 questions are necessary. Suppose, for contradiction, that we could determine all the numbers with only 1002 questions. Let the questions be \((x_1, y_1, z_1), (x_2, y_2, z_2), \ldots, (x_{1002}, y_{1002}, z_{1002})\), where \(x_i, y_i, z_i \in [1, 2005]\).

4. **Defining Sketchy and Rigorous Numbers:**
   - A number \(n\) is **sketchy** if it appears only once among the \(x_i\)'s, \(y_i\)'s, and \(z_i\)'s.
   - A number \(n\) is **rigorous** if it appears more than once.

5. **Multiset \(M\):**
   The multiset \(M\) contains all the \(x_i\)'s, \(y_i\)'s, and \(z_i\)'s, with a total size of \(3 \times 1002 = 3006\).

6. **Function \(f(n)\):**
   Define \(f(n)\) for \(1 \le n \le 2005\) as \(f(n) = \frac{1}{a}\), where \(a\) is the number of times \(n\) appears in \(M\).

7. **Sum of \(f\):**
   The sum \(\sum_{k=1}^{1002} (f(x_k) + f(y_k) + f(z_k))\) is exactly equal to 2005.

8. **Inequality for \(f\):**
   Observe that \(f(x_k) + f(y_k) + f(z_k) \le 2\) for all \(1 \le k \le 1002\). If \(f(x_k) + f(y_k) + f(z_k) > 2\), then at least two of \(f(x_k), f(y_k), f(z_k)\) must be 1, implying that the values of \(a_{x_k}\) and \(a_{y_k}\) are indistinguishable, which is impossible.

9. **Contradiction:**
   Since \(f(x_k) + f(y_k) + f(z_k) \le 2\), we have:
   \[
   2005 = \sum_{k=1}^{1002} (f(x_k) + f(y_k) + f(z_k)) \le 2 \times 1002 = 2004
   \]
   This is a contradiction, proving that our assumption of finishing in 1002 questions is incorrect.

10. **Conclusion:**
    Therefore, the minimal number of questions necessary to determine all the numbers is \(\boxed{1003}\).