Let a_j,b_j,c_j be integers for 1 <= j <= N. Assume, for each j, at least one of a_j, b_j, c_j is odd. Show that there exist integers r,s,t s.t. r a_j + s b_j +t c_j is odd for at least 4N/7 value of j.
Let 0 represent even numbers and 1 represent odd numbers since everything is mod 2.
We can put each ordered triple (a_j, b_j, c_j) in one of the 7 bins: (1,1,1) (1,1,0) (1,0,1) (1,0,0) (0,1,1) (0,1,0) (0,0,1)
Now I can prove that some set of 4 of those bins must contain 4N/7 ordered pairs. We need only prove that, given a set of 4 of those bins, we can find r,s,t that makes those 4 bins odd. Does anyone know how to do that? Is that a good approach? Will that work?