Question 5
Fix an integer $n \ge 3$. Let $S$ be a set of $n$ points in the plane, no three of which are collinear. Given pairwise distinct points $A, B, C$ in $S$, the triangle $ABC$ is suitable for $AB$ if area $ABC \le$ area $ABX$ for all $X$ in $S$ different from $A$ and $B$. (Note that for a segment $AB$ there could be several suitable triangles.) A triangle is adequate if its vertices are all in $S$ and it is suitable for at least two of its sides. Prove that there are at least $\frac{1}{2}(n-1)$ adequate triangles.
Bulgaria, Alexadar Ivanov
Bulgaria, Alexadar Ivanov