Let $n > 3$ be a positive integer. Find all integers $k$ such that $1 \le k \le n$ and for
which the following property holds:
If $x_1, . . . , x_n$ are $n$ real numbers such that $x_i + x_{i + 1} + ... + x_{i + k - 1} = 0$ for all integers $i > 1$ (indexes are taken modulo $n$), then $x_1 = . . . = x_n = 0$.

Proposed by [i]Vincent Jugé and Théo Lenoir, France[/i]
1. **Understanding the Problem:**
   We need to find all integers \( k \) such that \( 1 \le k \le n \) and for which the following property holds: If \( x_1, x_2, \ldots, x_n \) are \( n \) real numbers such that \( x_i + x_{i+1} + \cdots + x_{i+k-1} = 0 \) for all integers \( i \) (with indices taken modulo \( n \)), then \( x_1 = x_2 = \cdots = x_n = 0 \).

2. **Case 1: \( k \) is not relatively prime to \( n \):**
   - Suppose \(\gcd(k, n) = d > 1\).
   - We can construct a counterexample where the sequence \( x_1, x_2, \ldots, x_n \) is not all zeros.
   - Consider a sequence where we have blocks of length \( d \) of the form \( 1, 0, 0, \ldots, 0, -1 \). This sequence satisfies the condition \( x_i + x_{i+1} + \cdots + x_{i+k-1} = 0 \) but is not identically zero.
   - Therefore, if \( k \) is not relatively prime to \( n \), the property does not hold.

3. **Case 2: \( k \) is relatively prime to \( n \):**
   - Suppose \(\gcd(k, n) = 1\).
   - By Bézout's identity, there exist integers \( x \) and \( y \) such that \( kx \equiv 1 \pmod{n} \).
   - This implies that for any \( i \), \( x_i = x_{i+k} = x_{i+2k} = \cdots \).
   - Since \( k \) is relatively prime to \( n \), the sequence \( x_i, x_{i+k}, x_{i+2k}, \ldots \) will eventually cover all indices modulo \( n \).
   - Therefore, \( x_i = x_{i+1} \) for all \( i \), implying that all \( x_i \) are equal.
   - Given that the sum of any \( k \) consecutive terms is zero, and all terms are equal, each term must be zero.
   - Hence, \( x_1 = x_2 = \cdots = x_n = 0 \).

Conclusion:
- The property holds if and only if \( k \) is relatively prime to \( n \).

The final answer is \( \boxed{ k } \) such that \(\gcd(k, n) = 1\).