Question 11
Real numbers are written in the cells of a rectangular table with $n$ rows and $m$ columns. Assume that for every row or column, the sum of the numbers written in that row or column is an integer. Prove that it is possible to replace each real number $x$ by the integer $\lfloor x\rfloor$ or $\lceil x\rceil$ so that the sums of each column and of each row remain unchanged.
Remark: Recall that if $x$ is a real number, $\lfloor x\rfloor$ is the unique integer such that $\lfloor x\rfloor \leqslant x\lt\lfloor x\rfloor+1$, and $\lceil x\rceil$ is the unique integer such that $\lceil x\rceil-1\lt x \leqslant\lceil x\rceil$.
Remark: Recall that if $x$ is a real number, $\lfloor x\rfloor$ is the unique integer such that $\lfloor x\rfloor \leqslant x\lt\lfloor x\rfloor+1$, and $\lceil x\rceil$ is the unique integer such that $\lceil x\rceil-1\lt x \leqslant\lceil x\rceil$.