I am going to ask here a generalization of this other question:
Problem Fix $\varepsilon>0$ and let $p$ be a sufficiently large prime. Then, show that, for every $X\subseteq \{1,\ldots,p\}$ with $|X|\ge \varepsilon p$, there exist $a,b,c,x,y,z \in X$ such that $p$ divides $ax+by+cz$.
Motivation: I know a solution for this problem, but I feel that a simpler answer exists, as in the other case (which was just a pigenhole)
Intuition for the result: Usually a "large" subset of a finite field is not a proper subfield.