Let $m$ be a positive integer. Determine the smallest positive integer $n$ for which there exist real numbers $x_1, x_2,...,x_n \in (-1, 1)$ such that $|x_1| + |x_2| +...+ |x_n| = m + |x_1 + x_2 + ... + x_n|$.
1. **Understanding the problem**: We need to find the smallest positive integer \( n \) such that there exist real numbers \( x_1, x_2, \ldots, x_n \in (-1, 1) \) satisfying the equation:
   \[
   |x_1| + |x_2| + \cdots + |x_n| = m + |x_1 + x_2 + \cdots + x_n|
   \]
   where \( m \) is a given positive integer.

2. **Initial observation**: Let's try small values of \( m \) to find a pattern. For \( m = 1 \):
   - If \( n = 2 \), we can choose \( x_1 = 1/2 \) and \( x_2 = 1/2 \). Then:
     \[
     |x_1| + |x_2| = \frac{1}{2} + \frac{1}{2} = 1
     \]
     \[
     x_1 + x_2 = \frac{1}{2} + \frac{1}{2} = 1
     \]
     \[
     |x_1 + x_2| = 1
     \]
     \[
     |x_1| + |x_2| = 1 + |x_1 + x_2| = 1 + 1 = 2
     \]
     This does not satisfy the equation for \( m = 1 \).

3. **General case**: Let \( S^- \) be the sum of the non-positive numbers and \( S^+ \) be the sum of the non-negative numbers. Without loss of generality (WLOG), assume \( S^- + S^+ \geq 0 \). The given equation becomes:
   \[
   |S^-| + |S^+| = m + |S^- + S^+|
   \]
   Since \( S^- \leq 0 \) and \( S^+ \geq 0 \), we have \( |S^-| = -S^- \) and \( |S^+| = S^+ \). Thus, the equation simplifies to:
   \[
   -S^- + S^+ = m + S^- + S^+
   \]
   Rearranging terms, we get:
   \[
   -2S^- = m
   \]
   Therefore, \( S^- = -\frac{m}{2} \).

4. **Bounds on \( k \)**: Let \( k \) be the number of non-positive numbers. Then \( |S^-| = k \) and \( S^+ \leq n - k \). From the equation \( -2S^- = m \), we get:
   \[
   m \leq 2k
   \]
   Similarly, since \( S^+ \leq n - k \) and \( -S^- \leq n - k \), we have:
   \[
   \frac{m}{2} \leq k \quad \text{and} \quad \frac{m}{2} \leq n - k
   \]
   Adding these inequalities, we get:
   \[
   m \leq 2k \quad \text{and} \quad m \leq 2(n - k)
   \]
   Therefore:
   \[
   \frac{m}{2} \leq k \leq n - \frac{m}{2}
   \]

5. **Conclusion**: For the smallest \( n \), we need \( k \) to be an integer. The smallest \( n \) that satisfies these bounds is \( n = m + 1 \).

The final answer is \( \boxed{ n = m + 1 } \).