11. C4 (POL) Consider a matrix of size $n \times n$ whose entries are real numbers of absolute value not exceeding 1 , and the sum of all entries 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.
11. Consider the matrix $A=\left(a_{i j}\right)_{i, j=1}^{n}$ such that $a_{i j}$ is equal to 1 if $i, j \leq n / 2$, -1 if $i, j>n / 2$, and 0 otherwise. This matrix satisfies the conditions from the problem and all row sums and column sums are equal to $\pm n / 2$. Hence $C \geq n / 2$.
Let us show that $C=n / 2$. Assume to the contrary that there is a matrix $B=\left(b_{i j}\right)_{i, j=1}^{n}$ all of whose row sums and column sums are either greater than $n / 2$ or smaller than $-n / 2$. We may assume w.l.o.g. that at least $n / 2$ row sums are positive and, permuting rows if necessary, that the first $n / 2$ rows have positive sums. The sum of entries in the $n / 2 \times n$ submatrix $B^{\prime}$ consisting of first $n / 2$ rows is greater than $n^{2} / 4$, and since each column of $B^{\prime}$ has sum at most $n / 2$, it follows that more than $n / 2$ column sums of $B^{\prime}$, and therefore also of $B$, are positive. Again, suppose w.l.o.g. that the first $n / 2$ column sums are positive. Thus the sums $R^{+}$and $C^{+}$of entries in the first $n / 2$ rows and in the first $n / 2$ columns respectively are greater than $n^{2} / 4$. Now the sum of all entries of $B$ can be written as
$$
\sum a_{i j}=R^{+}+C^{+}+\sum_{\substack{\Delta=n / 2 \\ j>=/ 2}} a_{i j}-\sum_{\substack{1 \leq=/ 2 \\ j \leq m / 2}} a_{i j}>\frac{n^{2}}{2}-\frac{n^{2}}{4}-\frac{n^{2}}{4}=0,
$$
a contradiction. Hence $C=n / 2$, as claimed.