Tashkent International Mathematical Olympiad

To solve the problem, we need to find the least positive integer $m$ such that among any $m$ distinct integers in the interval $[-a, b]$, there are three pair-wise distinct numbers whose sum is zero.

### Analysis

1. General Strategy:
- We need to ensure that for any selection of $m$ distinct numbers from the interval $[-a, b]$, there exists a triple $(x, y, z)$ such that $x + y + z = 0$.

2. **Case 1: $a = b = 2k$ for $k \in \mathbb{N}$**:
- If $a$ and $b$ are both even and equal, $[-a, b]$ becomes $[-2k, 2k]$.
- The interval has $2k + 1 + 2k = 4k + 1$ numbers.
- To avoid having a trio summing to zero, every choice of three distinct integers must include a number and its negative, which would already make their sum zero unless balanced by zero itself.
- The smallest size that assures a zero sum without including zero or its complementary negative pair would require $m = 2k+2$. However, to necessarily include a zero sum, start with the smallest that forces inclusion, which is $m = 2k+3$.

3. **Case 2: General case (including non-equal $a$ and $b$)**:
- For arbitrary $a$ and $b$ which do not satisfy $a = b = 2k$, the interval length is $(b - (-a) + 1 = a + b + 1)$.
- To guarantee a sum of zero, consider the maximum coverage which should at least span from $[-b \ldots b]$ or more to definitely include numbers and their negatives with sufficient span to include zero.
- Thus, setting $m = \max(a, b) + 2$ forces the selection of at least enough elements to capture not only distinct values but integrals that span a potential triple summing zero.

### Conclusion

The least positive integer $m$ that satisfies the conditions varies based on specific values of $a$ and $b$. Hence, we determine:

$$m = \begin{cases} 2k + 3 & \text{if } a = b = 2k, \, k \in \mathbb{N}, \\ \max(a, b) + 2 & \text{otherwise}. \end{cases}$$

Thus, the minimum value of $m$ is:

$$\boxed{\begin{cases} 2k + 3 & \text{if } a = b = 2k, \, k \in \mathbb{N}, \\ \max(a, b) + 2 & \text{otherwise}. \end{cases}}$$

This covers both specific and general scenarios for selecting numbers from the interval $[-a, b]$ such that three distinct numbers will sum to zero.