8. people from $n$ different countries are sitting at a round table, so that for every two people from the same country, their immediate neighbors to their right are from different countries. What is the largest possible number of people that can sit at the table?

## 1st solution
The people from a fixed country have neighbors from pairs of different countries on the right, so there can be at most $n$ people per country at the table. This results in the upper limit $n^{2}$.

We now show inductively that this limit can be reached. In the following, we represent people from the $k$th country with a $k$. If there really are $n^{2}$ people sitting at the table, then each pair $(k, l), 1 \leq k, l \leq n$ must occur exactly once as a neighboring pair. The cyclic arrangement 1122 works for $n=2$. Assume that we have already found an arrangement for $n-1$. Now replace the neighboring pair 11 with $1 n n 11$ and replace the pair $k k$ with $k n k k$ for $2 \leq k \leq n-1$. The new arrangement now contains each pair $(k, l), 1 \leq k, l \leq n$ exactly once again. The induction step is complete.

## 2nd solution

We give an alternative construction of a seating arrangement for $n^{2}$ people. For $n=2$ we again use the arrangement 1122. We now assume that we have already constructed such an arrangement for $n-1$, so that in addition the first person
comes from country 1 and the last person from country $n-1$. Supplement this arrangement with

$$
1 n 2 n 3 n \ldots n(n-1) n n
$$

to obtain one for $n$.