Harvard-MIT Mathematics Tournament

Solution:
This is the heptagon case of what is known as the "Happy Ending" or "Erdős-Szekeres" problem, which in general asks, For any integer $n \geq 3$, what is the smallest $N(n)$, such that any $N(n)$ points in the plane in general position determine a convex $n$-gon? It is known that such an $N(n)$ always exists and is finite (in fact a specific upper bound has been found). The best known lower bound is $N(n) \geq 2^{n-2}+1$; Erdős and Szekeres conjectured that this bound is tight. The $n \leq 5$ cases have been known for some time. According to the Wikipedia, the $n=6$ case is solved but unpublished, and for $n \geq 7$, the problem remains open.
For a discussion, see
W. Morris and V. Soltan. The Erdős-Szekeres Problem on Points in Convex Position-A Survey, Bulletin of the American Math Monthly. 37 (2000), 437-458.
This article is available at
http://www.ams.org/bull/2000-37-04/S0273-0979-00-00877-6/home.html.
If $N(7)=33$, the highest sure score on this problem would be $32-6=26$. It is not known whether there exist arbitrarily large sets of points that will fool the graders.