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third order points in finite field

Source: miklos schweitzer 2006 q5

September 3, 2021
linear algebra

Problem Statement

let FqF_q be a finite field with char ≠ 2, and let V=Fq×FqV = F_q \times F_q be the 2-dimensional vector space over FqF_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.
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