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Parabolas and a line

Source: Finnish Mathematics Competition 2001, Final Round, Problem 2

November 14, 2011

conicsparabolageometryperpendicular bisectoralgebra unsolvedalgebra

Problem Statement

Equations of non-intersecting curves are

y = ax^2 + bx + c

and

y = dx^2 + ex + f

where

ad < 0.

Prove that there is a line of the plane which does not meet either of the curves.