Question 22
(UKR) A rectangular array of numbers is given. In each row and each column, the sum of all numbers is an integer. Prove that each nonintegral number $x$ in the array can be changed into either $\lceil x\rceil$ or $\lfloor x\rfloor$ so that the row sums and column sums remain unchanged. (Note that $\lceil x\rceil$ is the least integer greater than or equal to $x$, while $\lfloor x\rfloor$ is the greatest integer less than or equal to $x$.)