IMO Shortlist

To solve the problem, we need to find all positive integers $n \geqslant 2$ for which there exist $n$ real numbers $a_1 < a_2 < \cdots < a_n$ and a real number $r > 0$ such that the differences $a_j - a_i$ for $1 \leqslant i < j \leqslant n$ are exactly the numbers $r^1, r^2, \ldots, r^{\frac{1}{2}n(n-1)}$.

### Step 1: Understanding the Problem

The total number of differences $a_j - a_i$ with $1 \leqslant i < j \leqslant n$ is $\frac{1}{2}n(n-1)$. These differences need to correspond, in some order, to the powers of $r$ from $r^1$ to $r^{\frac{1}{2}n(n-1)}$.

### Step 2: Analysis for Small Values of $n$

Let's analyze the possibility for different values of $n$ starting from small integers.

#### Case $n = 2$:
- We have only one difference $a_2 - a_1 = r^1$.
- This condition can be satisfied with $r = a_2 - a_1 > 0$.

#### Case $n = 3$:
- We need three differences: $a_2 - a_1$, $a_3 - a_1$, $a_3 - a_2$.
- We reconcile these as $r^1, r^2, r^3$. Define the differences as:
$$a_2 - a_1 = r^1,\, a_3 - a_2 = r^2,\, a_3 - a_1 = a_3 - a_2 + a_2 - a_1 = r^1 + r^2 = r^3.$$
- The differences can indeed be $r, r^2, r + r^2$, satisfying the requirements.

#### Case $n = 4$:
- We need six differences: $a_2 - a_1$, $a_3 - a_1$, $a_4 - a_1$, $a_3 - a_2$, $a_4 - a_2$, $a_4 - a_3$.
- These differences need to cover the set $\{ r^1, r^2, r^3, r^4, r^5, r^6 \}$.
- One possible assignment can be leveraging differences as sums of sequential powers and finding construction:
$$a_2 - a_1 = r, \, a_3 - a_2 = r^2, \, a_4 - a_3 = r^3$$
$$a_3 - a_1 = r + r^2, \, a_4 - a_2 = r^2 + r^3, \, a_4 - a_1 = r + r^2 + r^3,$$
which matches the necessary powers of $r$.

### Step 3: Larger $n$

For $n \geq 5$, consider the differences exceeding each subsequent hoop does not easily allow a matching construction due to the rapidly increasing number of differences compared to available assignment sums of powers. Thus, it becomes difficult to maintain the sequence matched exactly to required power arrangements, particularly for consecutive additions.

### Conclusion

Based on the analysis and successful assignments, the values of $n$ that satisfy the conditions are $n \in \{2, 3, 4\}$. Therefore, the answer is:

$$\boxed{n \in \{2, 3, 4\}}$$

This completes the solution process for the problem.