US-EGMO-TST 2026 Problem 3

Let $S$ be a subset of $\mathbb{R}^2$ such that any triangle of area 1 contains at least 1 point of $S$ in its strict interior. Prove that for any positive integer $n$ and real number $\varepsilon > 0$, there exists a triangle with area at most $\varepsilon$ containing at least $n$ points of $S$ in its strict interior.