Problem 1367

1. Initial Setup and Definitions:
Let $n$ be an integer greater than $1$. We have $n$ pupils seated around a round table, each with a certain number of candies. Let $c_i$ denote the number of candies that the $i$-th pupil has, where $i = 1, 2, \ldots, n$. The sum of all candies, $\sum_{i=1}^n c_i$, is a multiple of $n$.

2. Exchange Process:
The exchange process is defined such that if a pupil $A$ has more candies than the pupil $B$ sitting to their right, $A$ gives one candy to $B$. This process continues until no pupil has more candies than the pupil to their right.

3. Semi-Invariant:
Define the semi-invariant $I$ as the sum of the squares of the number of candies each pupil has:
$$I = \sum_{i=1}^n c_i^2$$
We need to show that $I$ is a decreasing function during the exchange process.

4. **Effect of Exchange on $I$:**
Consider two adjacent pupils $A$ and $B$ with $a$ and $b$ candies respectively, where $a > b$. After $A$ gives one candy to $B$, the new number of candies are $a' = a - 1$ and $b' = b + 1$. The change in $I$ is:
$$I' = \sum_{i \neq A, B} c_i^2 + (a-1)^2 + (b+1)^2$$
$$I' = \sum_{i \neq A, B} c_i^2 + (a^2 - 2a + 1) + (b^2 + 2b + 1)$$
$$I' = \sum_{i=1}^n c_i^2 - 2a + 2b + 2$$
Since $a > b$, we have $-2a + 2b < 0$, thus:
$$I' < I$$
Therefore, $I$ decreases with each exchange.

5. Convergence to Equilibrium:
Since $I$ is a sum of squares of integers, it is a non-negative integer. As $I$ decreases with each exchange and is bounded below by zero, the process must eventually reach a point where $I$ cannot decrease further. At this point, no pupil has more candies than the pupil to their right, meaning the system is in equilibrium.

6. Equilibrium Condition:
Assume that at equilibrium, not all pupils have the same number of candies. This would imply that there exists at least one pupil with more candies than the pupil to their right, contradicting the equilibrium condition. Therefore, at equilibrium, all pupils must have the same number of candies.

7. Sum of Candies:
Since the total number of candies is a multiple of $n$, and at equilibrium, all pupils have the same number of candies, each pupil must have:
$$\frac{\sum_{i=1}^n c_i}{n}$$
candies, which is an integer.