HMMT 2020 U30

Let $S=\{(x, y) \mid x>0, y>0, x+y<200$, and $x, y \in \mathbb{Z}\}$. Find the number of parabolas $\mathcal{P}$ with vertex $V$ that satisfy the following conditions:

• $\mathcal{P}$ goes through both $(100,100)$ and at least one point in $S$,
• $V$ has integer coordinates, and
• $\mathcal{P}$ is tangent to the line $x+y=0$ at $V$.