Question 82
Let $S=\{(x, y) \mid x\gt 0, y\gt 0, x+y\lt 200$, 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 $V$.
- $\mathcal{P}$ 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 $V$.