Suppose that $x_1 < x_2 < \dots < x_n$ is a sequence of positive integers such that $x_k$ divides $x_{k+2}$ for each $k = 1, 2, \dots, n-2$.  Given that $x_n = 1000$, what is the largest possible value of $n$?

[i]Proposed by Evan Chen[/i]
1. **Factorize 1000**: 
   \[
   1000 = 2^3 \times 5^3
   \]
   This factorization will help us understand the possible divisors of 1000.

2. **Sequence Construction**: 
   We need to construct a sequence \( x_1 < x_2 < \dots < x_n \) such that \( x_k \) divides \( x_{k+2} \) for each \( k = 1, 2, \dots, n-2 \) and \( x_n = 1000 \).

3. **Divisibility Condition**: 
   Since \( x_k \) must divide \( x_{k+2} \), we can consider the prime factorization of each term. Each term must be a divisor of 1000 and must be able to divide a term two places ahead in the sequence.

4. **Constructing the Sequence**: 
   We start with \( x_n = 1000 \) and work backwards, ensuring each term divides the term two places ahead. We can either divide by 2 or 5 at each step:
   \[
   1000, 200, 40, 20, 10, 5, 1
   \]
   This sequence has length 7, but we need to ensure it is strictly increasing and meets the conditions.

5. **Ensuring Monotonicity**: 
   We need to ensure the sequence is strictly increasing and each term divides the term two places ahead. We can construct a sequence that meets these conditions:
   \[
   x_1 = 1, x_2 = 2, x_3 = 5, x_4 = 8, x_5 = 10, x_6 = 16, x_7 = 20, x_8 = 32, x_9 = 40, x_{10} = 64, x_{11} = 200, x_{12} = 320, x_{13} = 1000
   \]
   Each term divides the term two places ahead, and the sequence is strictly increasing.

6. **Verification**: 
   - \( x_1 = 1 \) divides \( x_3 = 5 \)
   - \( x_2 = 2 \) divides \( x_4 = 8 \)
   - \( x_3 = 5 \) divides \( x_5 = 10 \)
   - \( x_4 = 8 \) divides \( x_6 = 16 \)
   - \( x_5 = 10 \) divides \( x_7 = 20 \)
   - \( x_6 = 16 \) divides \( x_8 = 32 \)
   - \( x_7 = 20 \) divides \( x_9 = 40 \)
   - \( x_8 = 32 \) divides \( x_{10} = 64 \)
   - \( x_9 = 40 \) divides \( x_{11} = 200 \)
   - \( x_{10} = 64 \) divides \( x_{12} = 320 \)
   - \( x_{11} = 200 \) divides \( x_{13} = 1000 \)

   All conditions are satisfied.

Conclusion:
\[
\boxed{13}
\]