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2013 PUMaC Geometry A5

Source:

November 22, 2013
geometry3D geometrysphereconics

Problem Statement

Suppose you have a sphere tangent to the xyxy-plane with its center having positive zz-coordinate. If it is projected from a point P=(0,b,a)P=(0,b,a) to the xyxy-plane, it gives the conic section y=x2y=x^2. If we write a=pqa=\tfrac pq where p,qp,q are integers, find p+qp+q.