Let $n$ be an even positive integer. We fill in a number on each cell of a rectangle table of $n$ columns and multiple rows as following: i. Each row is assigned to some positive integer $a$ and its cells are filled by $0$ or $a$ (in any order); ii. The sum of all numbers in each row is $n$. Note that we cannot add any more row to the table such that the conditions (i) and (ii) still hold. Prove that if the number of $0$’s on the table is odd then the maximum odd number on the table is a perfect square.