Solve Putnam 2000 b1: Odd Integers for 4N/7 Values of j

[ehrenfest]

Homework Statement

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?

Homework Equations

The Attempt at a Solution

 

[ehrenfest]

Does my attempted solution make sense to people?

 

[ehrenfest]

If you go here: http://www.unl.edu/amc/a-activities/a7-problems/putnam/-pdf/2000s.pdf
and look at the solution, does anyone get else get confused near then end?

In particular, shouldn't it be "exactly four of the seven" instead of "at least four of the seven" in the third sentence? And in the fourth sentence, shouldn't that be exactly instead of at least?