MathNet 2012 Question 76

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