$2000$ people are standing on a line. Each one of them is either a [i]liar[/i], who will always lie, or a [i]truth-teller[/i], who will always tell the truth. Each one of them says: "there are more liars to my left than truth-tellers to my right". Determine, if possible, how many people from each class are on the line.
1. **Labeling and Initial Assumption:**
   - Label the people from left to right as \( P_1, P_2, \ldots, P_{2000} \).
   - We aim to show that the first \( 1000 \) people are liars and the remaining \( 1000 \) people are truth-tellers.

2. **Lemma and Proof:**
   - **Lemma:** If \( P_1, P_2, \ldots, P_k \) (where \( 1 \leq k < 1000 \)) are all liars, then \( P_{k+1} \) is also a liar.
   - **Proof:**
     - Assume for contradiction that \( P_{k+1} \) is a truth-teller.
     - \( P_{k+1} \) would see \( k \) liars to his left and at most \( 1999 - (k+1) \) people to his right, which includes at most \( 1999 - (k+1) - k = 1998 - 2k \) truth-tellers.
     - Since \( P_{k+1} \) is a truth-teller, he must see more liars to his left than truth-tellers to his right, i.e., \( k > 1998 - 2k \).
     - Simplifying, \( 3k > 1998 \) or \( k > 666 \).
     - This implies \( k \geq 667 \), which contradicts \( k < 1000 \).

3. **Base Case:**
   - \( P_1 \) is a liar because there are no people to his left, so he cannot truthfully claim there are more liars to his left than truth-tellers to his right.

4. **Inductive Step:**
   - By the lemma, if \( P_1, P_2, \ldots, P_k \) are liars, then \( P_{k+1} \) is also a liar for \( k < 1000 \).
   - Therefore, the first \( 1000 \) people are liars.

5. **Remaining People:**
   - Consider \( P_n \) where \( 1001 \leq n \leq 2000 \).
   - \( P_n \) sees at least \( 1000 \) liars to his left.
   - Since there are fewer than \( 1000 \) people to his right, he sees fewer than \( 1000 \) truth-tellers to his right.
   - Thus, \( P_n \) is a truth-teller because his statement is true.

6. **Conclusion:**
   - The first \( 1000 \) people are liars.
   - The remaining \( 1000 \) people are truth-tellers.

The final answer is \( \boxed{ 1000 } \) liars and \( 1000 \) truth-tellers.