SCHWEITZER 2006 Problem 5

let $F_q$ be a finite field with char ≠ 2, and let $V = F_q \times F_q$ be the 2-dimensional vector space over $F_q$ . Let L ⊂ V be a subset containing lines in all directions. The order of a point in V is the number of lines in L that pass through the point. Prove that L contains at least q lines having a third-order point.