We call the sum of any $k$ of $n$ given numbers (with distinct indices) a $k$-sum. Given $n$, find all $k$ such that, whenever more than half of $k$-sums of numbers $a_{1},a_{2},...,a_{n}$ are positive, the sum $a_{1}+a_{2}+...+a_{n}$ is positive as well.
1. **Understanding the problem**: We need to find all values of \( k \) such that if more than half of the \( k \)-sums of \( n \) given numbers \( a_1, a_2, \ldots, a_n \) are positive, then the sum \( a_1 + a_2 + \cdots + a_n \) is also positive.

2. **Analyzing the \( k \)-sums**: A \( k \)-sum is the sum of any \( k \) numbers chosen from the \( n \) numbers. The total number of \( k \)-sums is given by the binomial coefficient \( \binom{n}{k} \).

3. **Condition for positivity**: We are given that more than half of these \( k \)-sums are positive. This means that the number of positive \( k \)-sums is greater than \( \frac{\binom{n}{k}}{2} \).

4. **Considering the case when \( k < \frac{n}{2} \)**: Suppose \( k < \frac{n}{2} \). We can construct a counterexample where \( a_1 \) is a large negative number and the rest \( a_2, a_3, \ldots, a_n \) are positive numbers such that their sum is positive. In this case, the sum \( a_1 + a_2 + \cdots + a_n \) can be negative, but many \( k \)-sums (which do not include \( a_1 \)) can be positive. Specifically, the number of \( k \)-sums that do not include \( a_1 \) is \( \binom{n-1}{k} \), and if \( k < \frac{n}{2} \), we have:
   \[
   \binom{n-1}{k} > \frac{\binom{n}{k}}{2}
   \]
   This inequality holds because \( \binom{n-1}{k} \) represents the number of \( k \)-sums from \( n-1 \) elements, which is more than half of the total \( k \)-sums from \( n \) elements when \( k \) is small.

5. **Considering the case when \( k \geq \frac{n}{2} \)**: If \( k \geq \frac{n}{2} \), then any \( k \)-sum will include at least half of the elements. If more than half of these \( k \)-sums are positive, it implies that the majority of the elements themselves must be positive. Therefore, the sum \( a_1 + a_2 + \cdots + a_n \) will also be positive.

6. **Conclusion**: The value of \( k \) must be such that \( k \geq \frac{n}{2} \) to ensure that if more than half of the \( k \)-sums are positive, the total sum \( a_1 + a_2 + \cdots + a_n \) is also positive.

\(\blacksquare\)

The final answer is \( \boxed{ k \geq \frac{n}{2} } \)