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 |