Sum of every row, column and diagonal is equal to 0. Is it possible that none of the numbers is eqaul to zero? A square with 2015 rows and 2015 columns is filled with integers. The sum of every row is equal to zero, the sum of every column is equal to zero, and sum of the two main diagonals is equal to zero. Is it possible that none of the numbers in square is equal to zero? How can I prove it?
 A: Michael beat me to writing a similar general idea, but here are the specifics. Consider this $5\times 5$ square:
$$\begin{bmatrix}
  17& 17& -8&-18& -8\\
   7&-23&  2& 12&  2\\
  -8&  2&  2&  2&  2\\
  -8&  2&  2&  2&  2\\
  -8&  2&  2&  2&  2\\
\end{bmatrix}$$
Each row, column, and main diagonal sums to zero. Fill up the $2015\times 2015$ square with this, $403$ times across and $403$ times down, and the entire square will also have those properties. This is true since each row, column, and main diagonal of the large square will be made of $403$ copies of rows, columns, and main diagonals of this $5\times 5$ square.
I found this square by solving the $12$ linear equations in $25$ variables that come from the condition. The $2$'s are the basis of the solution, while the others are determined from the $2$'s. The $2$'s could be changed to get other suitable squares.
A: Try to find a $5\times5$ matrix of this form:
$$\left[\begin{array}{ccccc}a&b&c&b&a\\b&d&e&d&b\\c&e&f&e&c\\b&d&e&d&b\\a&b&c&b&a\end{array}\right]$$
that obeys your rules.  Your have four equations in six variables.
Now fill your $2015\times2015$  square with these smaller squares.
