Fix a seat, and then label these delegates from $0$ to $N-1$ clockwise based on their seats. After the break, label these delegates' seat as $a_0$, $a_1$, $\cdots$, $a_{n-1}$.

Now, let's consider $b_i = (a_i - i) \pmod{n}$, where $i=0, 1, \cdots, N-1$.

If there exist $m$ and $k$, such that $b_k = a_m$, then $a_k - a_m = k - m$. This means the intervals between the $k^{th}$ and the $m^{th}$ delegates keeps the same.

Otherwise, if all $b_i$'s are different, they must take distinct values from $\{0$, $1$, $2$, $\cdots$, $N-1\}$. However this is impossible because, on one hand, $\displaystyle\sum_{i=0}^{N-1} b_i = \frac{N(N-1)}{2} \equiv \frac{N}{2} \pmod{N}$. On the other hand, the result should be $0$ by definition because $\displaystyle\sum_{i=0}^{N-1}b_i = \sum_{i=0}^{N-1}(a_i - i) = 0$.