A set $P$ consists of $2005$ distinct prime numbers. Let $A$ be the set of all possible products of $1002$ elements of $P$ , and $B$ be the set of all products of $1003$ elements of $P$ . Find a one-to-one correspondance $f$ from $A$ to $B$ with the property that $a$ divides $f (a)$ for all $a \in A.$
To solve this problem, we need to construct a one-to-one correspondence \( f \) from the set \( A \) to the set \( B \) such that for every \( a \in A \), \( a \) divides \( f(a) \). Here is a detailed step-by-step solution:

1. **Define the Sets \( A \) and \( B \)**:
   - Let \( P \) be a set of 2005 distinct prime numbers.
   - Let \( A \) be the set of all possible products of 1002 elements of \( P \).
   - Let \( B \) be the set of all possible products of 1003 elements of \( P \).

2. **Understand the Problem**:
   - We need to find a function \( f: A \to B \) such that \( a \) divides \( f(a) \) for all \( a \in A \).
   - Since \( A \) consists of products of 1002 primes and \( B \) consists of products of 1003 primes, \( f(a) \) should be a product of \( a \) and one additional prime from \( P \).

3. **Construct the Function \( f \)**:
   - Consider a subset \( S \) of size 1002 from the set of 2005 primes.
   - Let \( S = \{p_1, p_2, \ldots, p_{1002}\} \).
   - The product of the elements in \( S \) is an element of \( A \).

4. **Find the Missing Prime**:
   - To map \( S \) to a subset of size 1003, we need to add one more prime from \( P \) that is not in \( S \).
   - Let \( T \) be the set of all primes in \( P \) that are not in \( S \).
   - \( T \) will have \( 2005 - 1002 = 1003 \) elements.

5. **Define the Function \( f \)**:
   - We need to choose one prime from \( T \) to multiply with the product of \( S \).
   - Let \( f(S) = S \cup \{q\} \) where \( q \) is a prime in \( T \).

6. **Ensure One-to-One Correspondence**:
   - We need to ensure that each subset of size 1002 maps to a unique subset of size 1003.
   - Since \( \binom{2005}{1002} = \binom{2005}{1003} \), there is a one-to-one correspondence between subsets of size 1002 and subsets of size 1003.

7. **Construct the Function Explicitly**:
   - Let \( f(S) = S \cup \{q\} \) where \( q \) is chosen such that \( q \equiv -(p_1 + p_2 + \cdots + p_{1002}) \mod 1003 \).
   - This ensures that \( q \) is uniquely determined for each subset \( S \).

8. **Verify the Divisibility**:
   - For any \( a \in A \), \( a \) is the product of 1002 primes.
   - \( f(a) \) is the product of \( a \) and one additional prime \( q \).
   - Therefore, \( a \) divides \( f(a) \).

The final answer is \( \boxed{ f(x_1, x_2, \cdots, x_{1002}) = -(x_1 + x_2 + \cdots + x_{1002}) \mod 1003 } \).