SMO 2012 Open Round 2 Question 3

For each $i=1,2,\ldots,N$, let $a_i,b_i,c_i$ be integers such that at least one is odd. Show that there exist integers $x,y,z$ such that $xa_i+yb_i+zc_i$ is odd for at least $4N/7$ different values of $i$.