Let $N \geq 3$ be an integer, and let $a_0, \dots, a_{N-1}$ be pairwise distinct reals so that $a_i \geq a_{2i}$ for all $i$ (indices are taken $\bmod~ N$). Find all possible $N$ for which this is possible.

[i]Proposed by Sutanay Bhattacharya[/i]
1. **Define the problem in terms of a digraph:**
   We are given an integer \( N \geq 3 \) and pairwise distinct reals \( a_0, a_1, \dots, a_{N-1} \) such that \( a_i \geq a_{2i \mod N} \) for all \( i \). We need to find all possible values of \( N \) for which this condition is possible.

2. **Construct the digraph \( G \):**
   Construct a digraph \( G \) on the set of vertices \( \mathbb{Z}/N\mathbb{Z} \) with directed edges from \( i \) to \( 2i \mod N \). This means there is a directed edge from vertex \( i \) to vertex \( 2i \mod N \).

3. **Topological sorting and cycles:**
   The condition \( a_i \geq a_{2i \mod N} \) implies that the graph \( G \) must admit a valid topological sort. A topological sort is possible if and only if the graph has no directed cycles of length greater than 1.

4. **Analyze the structure of \( G \):**
   Express \( N \) as \( N = 2^{\ell} m \) where \( \ell \geq 0 \) and \( m \) is an odd positive integer. Consider the order of 2 modulo \( m \), denoted as \( t = \text{ord}_m(2) \). This is the smallest positive integer \( t \) such that \( 2^t \equiv 1 \mod m \).

5. **Identify cycles in \( G \):**
   If \( t > 1 \), then there exists a cycle of length \( t \) in \( G \). Specifically, the vertices \( 2^{\ell}, 2^{\ell+1}, \dots, 2^{\ell+t-1} \) form a cycle because \( 2^{\ell+t} \equiv 2^{\ell} \mod N \). This implies that \( t = 1 \) for \( G \) to have no cycles of length greater than 1.

6. **Conclude \( m = 1 \):**
   Since \( t = 1 \), it follows that \( 2 \equiv 1 \mod m \), which implies \( m = 1 \). Therefore, \( N = 2^{\ell} \).

7. **Verify the converse:**
   When \( N = 2^{\ell} \), the graph \( G \) is a complete (upward directed) binary tree on \( N-1 \) vertices with its root directed at the self-loop \( 0 \mapsto 0 \). This graph has no cycles of length greater than 1, confirming that the condition \( a_i \geq a_{2i \mod N} \) can be satisfied.

Thus, the only possible values of \( N \) are powers of 2.

The final answer is \( \boxed{ N } \) must be a power of 2.