Problem 2: Given some m x n table, and some numbers in its fields. You are allowed to change the sign in one row or one column simultaneously. Prove that You can obtain a table, with non-negative sums over each row and over each column.
Update (11th June 2011):
Solution to both problems posted by NG [Nikhil Garg (CSE IIT Delhi final year undergraduate student)] in comments! Thanks.
Confession: I could never solve Problem 1 even after spending hours. :P :(