USAMO 2020 — Days 1 and 2 Question 5

A finite set $S$ of plane points is overdetermined if $|S|\ge2$ and a nonzero real polynomial $P$ of degree at most $|S|-2$ satisfies $P(x)=y$ at every $(x,y)\in S$. For each $n\ge2$, find the largest number of overdetermined subsets that an n-point set can have while the whole set is not overdetermined.

Proposed by Carl Schildkraut. Source: Evan Chen’s third-party solution notes.

Source video link: https://youtu.be/r7j0oRtpErA?t=5420