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|a|,|b| >= 2017 , P(a^2+b^2) >= P(2ab) , P trinomial

Source: Dutch IMO TST1 2019 p1

January 10, 2020
trinomialquadratic trinomialalgebrapolynomial

Problem Statement

Let P(x)P(x) be a quadratic polynomial with two distinct real roots. For all real numbers aa and bb satisfying ∣a∣,∣b∣≥2017|a|,|b| \ge 2017, we have P(a2+b2)≥P(2ab)P(a^2+b^2) \ge P(2ab). Show that at least one of the roots of PP is negative.
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