HMIC 2025 Problem 1

Let $A B C D$ be a convex quadrilateral. Define parabolas $\mathcal{P}_{A}$ $\mathcal{P}_{B}$ $\mathcal{P}_{C}$ , and $\mathcal{P}_{D}$ to have directrices $B D$ $C A$ $D B$ , and $A C$ , and foci $A$ $B$ $C$ , and $D$ , respectively. Prove that no two of these parabolas intersect more than once.

(A parabola with directrix $\ell$ and focus $P$ consists of all points $X$ for which $P X$ equals the distance from $P$ to $\ell$ .)