Here are a few questions that I thought of while teaching high-school mathletes, which I thought were perhaps appropriate as training problems but then found them too hard. My geometry is also quite dusty.
Question 1: Let $T$ be an equilateral triangle in the plane, with unit area, with vertices $A_1, A_2, A_3$ say. Let $T_i$ be a triangle of unit area whose centroid is equal to $A_i$. What is the maximum possible value of the area of the region $T_1 \cap T_2 \cap T_3$, and what configuration of the $T_i$'s achieves this maximum?
Question 2: Same question as Question 1, except $T$ is now an arbitrary (non-degenerate) triangle of unit area.
(Cross posted on MathOverflow)