Determine all composite positive integers $n$ with the following property: If $1 = d_1 < d_2 < \cdots < d_k = n$ are all the positive divisors of $n$, then

$$(d_2 - d_1) : (d_3 - d_2) : \cdots : (d_k - d_{k-1}) = 1:2: \cdots :(k-1)$$

(Walther Janous)
To determine all composite positive integers \( n \) with the given property, we need to analyze the sequence of divisors and their differences. Let's break down the problem step by step.

1. **Identify the divisors and their differences:**
   Let \( 1 = d_1 < d_2 < \cdots < d_k = n \) be all the positive divisors of \( n \). The given property states that:
   \[
   (d_2 - d_1) : (d_3 - d_2) : \cdots : (d_k - d_{k-1}) = 1 : 2 : \cdots : (k-1)
   \]
   This means:
   \[
   d_2 - d_1 = 1, \quad d_3 - d_2 = 2, \quad \ldots, \quad d_k - d_{k-1} = k-1
   \]

2. **Express the divisors in terms of \( k \):**
   Since \( d_1 = 1 \) and \( d_2 - d_1 = 1 \), we have \( d_2 = 2 \). Similarly, \( d_3 - d_2 = 2 \) implies \( d_3 = 4 \), and so on. We can generalize this pattern:
   \[
   d_i = d_{i-1} + (i-1)
   \]
   For example:
   \[
   d_3 = d_2 + 2 = 2 + 2 = 4
   \]
   \[
   d_4 = d_3 + 3 = 4 + 3 = 7
   \]
   However, this pattern does not hold for all \( i \). Instead, we need to find a consistent pattern that fits the given ratio.

3. **Analyze the pattern for small values of \( k \):**
   Let's consider small values of \( k \) to find a consistent pattern:
   - For \( k = 3 \):
     \[
     d_1 = 1, \quad d_2 = 2, \quad d_3 = 4
     \]
     The differences are:
     \[
     d_2 - d_1 = 1, \quad d_3 - d_2 = 2
     \]
     This matches the ratio \( 1 : 2 \).

4. **Verify if \( n = 4 \) is a solution:**
   For \( n = 4 \), the divisors are \( 1, 2, 4 \). The differences are:
   \[
   2 - 1 = 1, \quad 4 - 2 = 2
   \]
   This matches the ratio \( 1 : 2 \).

5. **Check for larger values of \( k \):**
   If \( k > 3 \), the pattern \( d_i = i \) for each integer \( i \) does not hold because the differences \( d_3 - d_2 \) would not be equal to 1. Therefore, \( k \) cannot be greater than 3.

6. **Conclusion:**
   The only composite positive integer \( n \) that satisfies the given property is \( n = 4 \).

The final answer is \( \boxed{ n = 4 } \).