To solve the problem, we need to analyze the structure of a $10 \times 10$ grid filled with numbers $+1$ and $-1$.

1. Definitions:

   • Let $k$ be the number of rows with a positive sum.
   • Consequently, there will be $10 - k$ rows with a negative sum.
   • Similarly, $k$ will also be the number of columns with a negative sum, leading to $10 - k$ columns with a positive sum.

2. Total Sum Calculation:

   • The total sum of all the entries in the grid can be expressed as: $S = (+1) \cdot a + (-1) \cdot b$ where $a$ is the count of $+1$s and $b$ is the count of $-1$s. Since the grid contains 100 entries, we have: $a + b = 100$

3. Row and Column Sums:

   • For each row with a positive sum, the number of $+1$s must exceed the number of $-1$s. Therefore, if a row has $n_i$ $+1$s, then: $n_i > 5 \quad \text{(for positive sum)}$
   • Conversely, for rows with negative sums, the opposite must hold, so: $n_i < 5 \quad \text{(for negative sum)}$

4. Balancing Rows and Columns:

   • The total number of $+1$s in $k$ rows and $10 - k$ rows can be denoted as: $\text{Total from } k \text{ rows} > 5k$ $\text{Total from } (10 - k) \text{ rows} < 5(10 - k)$
   • Thus, we can set up the inequality for the total number of $+1$s: $a > 5k \quad \text{and} \quad a < 5(10 - k)$ Combining these: $5k < a < 50 - 5k$

5. Combining Inequalities:

   • Therefore: $5k < 50 - 5k \implies 10k < 50 \implies k < 5$
   • Since $k$ must also be a non-negative integer, we find: $k \leq 4$

6. Finding Maximum $k$:

   • Testing $k = 4$:

     • If $k = 4$, we have $4$ rows with positive sums and $6$ rows with negative sums.
     • The count of $+1$s will need to satisfy: $a > 5 \cdot 4 = 20 \quad \text{and} \quad a < 50 - 5 \cdot 4 = 30$ Thus, $a$ can take values $21, 22, \ldots, 29$.

   • The configuration is possible, ensuring that in the remaining columns (with $10 - k = 6$), there are enough $-1$s to maintain the conditions.

In conclusion, the maximum $k$ such that there are exactly $k$ rows with a positive sum and exactly $k$ columns with a negative sum is:

$\boxed{4}$

Further Exploration

Would you like to explore:

1. Examples of specific configurations of the grid?
2. A different approach to the problem?
3. How varying $k$ affects the sums?
4. Similar problems with different grid sizes?
5. Detailed proofs of the inequalities used?

Tip

When tackling problems involving combinatorial configurations, always consider the implications of constraints on sums and distributions within the grid.