Given a finite sequence $x_{1,1}, x_{2,1}, \dots , x_{n,1}$ of integers $(n\ge 2)$, not all equal, define the sequences $x_{1,k}, \dots , x_{n,k}$ by
\[ x_{i,k+1}=\frac{1}{2}(x_{i,k}+x_{i+1,k})\quad\text{where }x_{n+1,k}=x_{1,k}.\]
Show that if $n$ is odd, then not all $x_{j,k}$ are integers. Is this also true for even $n$?
1. **Assume for contradiction**: Suppose all \( x_{j,k} \) are integers for all \( j \) and \( k \).

2. **Claim**: There does not exist a \( k \) such that \( x_{1,k} = x_{2,k} = \dots = x_{n,k} \). 
   - If such a \( k \) exists, consider the smallest such \( k \). 
   - By the definition of the sequence, we have:
     \[
     x_{1,k} = \frac{x_{1,k-1} + x_{2,k-1}}{2}, \quad x_{2,k} = \frac{x_{2,k-1} + x_{3,k-1}}{2}, \quad \dots, \quad x_{n,k} = \frac{x_{n,k-1} + x_{1,k-1}}{2}
     \]
   - Since \( x_{1,k} = x_{2,k} = \dots = x_{n,k} \), it follows that:
     \[
     x_{1,k-1} = x_{2,k-1} = \dots = x_{n,k-1}
     \]
   - This contradicts the minimality of \( k \), as \( k \neq 1 \).

3. **Define \( m \)**: Let \( m \) be the smallest integer such that \( x_{j,1} \leq m \) for all \( j \). Since not all \( x_{j,1} \) are equal, there exists some \( k \) such that \( x_{j,k} \leq m-1 \).

4. **Inequality propagation**:
   - Consider:
     \[
     x_{j-1,k+1} = \frac{x_{j-1,k} + x_{j,k}}{2} \leq \frac{2m-1}{2}
     \]
     \[
     x_{j,k+1} = \frac{x_{j,k} + x_{j+1,k}}{2} \leq \frac{2m-1}{2}
     \]
   - Since \( x_{j-1,k+1} \) and \( x_{j,k+1} \) are integers, they must be at most \( m-1 \).

5. **Iterative argument**:
   - Repeating this argument, for all \( k \) large enough, \( x_{j,k} \) are at most \( m-1 \).
   - Continuing this process, for \( k \) sufficiently large, all \( x_{j,k} \) are at most \( m-2 \), and so on.
   - Eventually, for \( k \) sufficiently large, all \( x_{j,k} \) will be arbitrarily small, contradicting the fact that their sum \( x_{1,1} + x_{2,1} + \dots + x_{n,1} \) is fixed.

6. **Conclusion for odd \( n \)**:
   - Thus, there exist indices \( j \) and \( k \) for which \( x_{j,k} \) is not an integer when \( n \) is odd.

7. **Case for even \( n \)**:
   - Consider the sequence \( 1, -1, 1, -1, \dots, 1, -1 \) for even \( n \).
   - For this sequence, all \( x_{j,k} \) remain integers, showing that the statement is not true for even \( n \).

The final answer is True for odd \( \boxed{ n } \) and False for even \( n \).