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n parabolas cut Ox - All-Russian MO 2002 Regional (R4) 10.5

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September 17, 2024
conicsparabolaanalytic geometry

Problem Statement

Various points x1,...,xnx_1,..., x_n (n≥3n \ge 3) are randomly located on the OxOx axis. Construct all parabolas defined by the monic square trinomials and intersecting the Ox axis at these points (and not intersecting axis at other points). Lety=f1 y = f_1, ...... , y=fmy = f_m are functions that define these parabolas. Prove that the parabola y=f1+...+fmy = f_1 +...+ f_m intersects the OxOx axis at two points.
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