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Algebra with tangents to quadratics

Source: 2024 All-Russian Olympiad Regional Round

February 23, 2024

quadraticsderivativetangentconicsparabolaalgebra

Problem Statement

Graph

G_1

of a quadratic trinomial

y = px^2 + qx + r

with real coefficients intersects the graph

G_2

of a quadratic trinomial

y = x^2

in points

A

,

B

. The intersection of tangents to

G_2

in points

A

,

B

is point

C

. If

C \in G_1

, find all possible values of

p

.

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