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Find all triples such that a quadratic bound implies a constant function

Source: 2019 Philippine IMO TST2 Problem 3

May 4, 2022

quadraticsalgebrainequalitiesFunctional inequalityfunctionabsolute valueconstant

Problem Statement

Determine all ordered triples

(a, b, c)

of real numbers such that whenever a function

f : \mathbb{R} \to \mathbb{R}

satisfies

|f(x) - f(y)| \le a(x - y)^2 + b(x - y) + c

for all real numbers

x

and

y

, then

f

must be a constant function.