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simple polynomial -- prove an inequality

Source: Indonesia National Math Olympiad 2021 Problem 5 (INAMO 2021/5)

November 9, 2021

algebrapolynomialinequalities

Problem Statement

Let

P(x) = x^2 + rx + s

be a polynomial with real coefficients. Suppose

P(x)

has two distinct real roots, both of which are less than

-1

and the difference between the two is less than

2

. Prove that

P(P(x)) > 0

for all real

x