Each cell of a 1997x1997 square grid contains either +1 or -1, with no cell being vacant.
The product of all the numbers in the ith row, and
the product of all the numbers in the ith column are respectively denoted by Ri and Ci.
Prove that Σi=1(Ri + Ci) is always nonzero.
(In reply to re: a simple proof
You are right,Daniel.
Still, the same proof is applicable to the product case.
I have erased one figure and now I paste the new text, just solving two problems for the price of one....And here we go:
Imagine all squares having +1 as a content. The overall sum-the one called sigma- will equal 2*1997 which is 2 mod 4. <br> Changing square a(i,j) to -1 will reduce the sum in row i by 2 , and in column j by 2 , leaving the overall sum still equal to 2 mod 4.<br><br>
if we continue and make further changes (1 into -1) striving to reduce the overall sum and get 0,- one of the following will happen:<br>
a) both the row's and column's sums will decrease by 2.<br>
b) both the row's and column's sums will increase by 2.<br>
c) one sum will increase and the other decrease by 2.<br>
In all those cases the overall sum will remain 2 mod 4 and 0 mod 4 cannot be reached.<br><br>
Since any matrix out of (1997*1997)! existing can be constructed by the process I described - no matrix with sigma=0 exists. q.e.d.