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parabolas and tangency, points conparabolic

Source: VJIMC 2014 1.4

May 20, 2021
geometryconic sectionsconics

Problem Statement

Let P1,P2,P3,P4P_1,P_2,P_3,P_4 be the graphs of four quadratic polynomials drawn in the coordinate plane. Suppose that P1P_1 is tangent to P2P_2 at the point q2,P2q_2,P_2 is tangent to P3P_3 at the point q3,P3q_3,P_3 is tangent to P4P_4 at the point q4q_4, and P4P_4 is tangent to P1P_1 at the point q1q_1. Assume that all the points q1,q2,q3,q4q_1,q_2,q_3,q_4 have distinct xx-coordinates. Prove that q1,q2,q3,q4q_1,q_2,q_3,q_4 lie on a graph of an at most quadratic polynomial.
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