2003 Indonesia MO Problems/Problem 4
Problem
Given a matrix, where each element is valued or . Let be the product of all elements at the row, and be the product of all elements at the column. Prove that:
Solution
In order for the sum to equal 0, rows or columns must have a product of and the other rows or columns must have a product of
On a given entry of a matrix, the product of its row and column is either or If an entry with a is switched to a then the product of its row and the product of its column change signs. That means the number of rows and columns where the product equals will either be always an even number or always an odd number since the number of rows and columns where the product equals will change by either 2, 0, or -2.
In a grid where all entries are , there is no that is equal to Thus, the number of rows and columns where the product equal will always be an even number. Since the number of rows and columns where the product equals can not equal we find that
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
See Also
2003 Indonesia MO (Problems) | ||
Preceded by Problem 3 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 | Followed by Problem 5 |
All Indonesia MO Problems and Solutions |