Consider a matrix of size $n\times n$ whose entries are real numbers of absolute value not exceeding $1$. The sum of all entries of the matrix is $0$. Let $n$ be an even positive integer. Determine the least number $C$ such that every such matrix necessarily has a row or a column with the sum of its entries not exceeding $C$ in absolute value.

[i]Proposed by Marcin Kuczma, Poland[/i]
1. **Define the problem and notation:**
   Let \( A \) be an \( n \times n \) matrix with real entries such that \( |a_{ij}| \leq 1 \) for all \( i, j \) and the sum of all entries in \( A \) is zero. We need to determine the least number \( C \) such that there exists at least one row or column in \( A \) with the sum of its entries not exceeding \( C \) in absolute value. Given \( n \) is an even positive integer, let \( n = 2k \).

2. **Construct an example to find an upper bound for \( C \):**
   Consider a matrix \( A \) where the top left \( k \times k \) submatrix is filled with \( 1 \)'s and the bottom right \( k \times k \) submatrix is filled with \( -1 \)'s. The remaining entries are zero. This matrix satisfies the conditions:
   \[
   A = \begin{pmatrix}
   1 & 1 & \cdots & 1 & 0 & 0 & \cdots & 0 \\
   1 & 1 & \cdots & 1 & 0 & 0 & \cdots & 0 \\
   \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\
   1 & 1 & \cdots & 1 & 0 & 0 & \cdots & 0 \\
   0 & 0 & \cdots & 0 & -1 & -1 & \cdots & -1 \\
   0 & 0 & \cdots & 0 & -1 & -1 & \cdots & -1 \\
   \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\
   0 & 0 & \cdots & 0 & -1 & -1 & \cdots & -1 \\
   \end{pmatrix}
   \]
   The sum of the entries in any row or column of this matrix is either \( k \) or \( -k \). Therefore, \( C \leq k \).

3. **Prove that \( C = k \) is the least possible value:**
   Suppose for contradiction that \( C < k \). Then, every row and column sum must be greater than \( -k \) and less than \( k \) in absolute value. 

4. **Apply the pigeonhole principle:**
   Since the sum of all entries in the matrix is zero, at least half of the rows must have a non-negative sum and at least half of the rows must have a non-positive sum. Without loss of generality, assume the first \( k \) rows have non-negative sums. 

5. **Analyze the sums of the rows and columns:**
   The sum of the entries in the first \( k \) rows is at least zero. Since each row sum is at most \( k \), the total sum of the first \( k \) rows is at most \( k^2 \). Similarly, the sum of the entries in the first \( k \) columns is at most \( k^2 \).

6. **Consider the sum of the entire matrix:**
   The sum of the entries in the top left \( k \times k \) submatrix is at least zero, and the sum of the entries in the bottom right \( k \times k \) submatrix is at most zero. By the principle of inclusion-exclusion, the sum of the entire matrix is at most \( k^2 - k^2 = 0 \), which contradicts the assumption that the sum of all entries is zero.

7. **Conclude the proof:**
   Therefore, the least number \( C \) such that every \( n \times n \) matrix with the given properties has a row or column with the sum of its entries not exceeding \( C \) in absolute value is \( k \).

\(\blacksquare\)

The final answer is \( \boxed{ k } \).