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$P(a)+P(b)+P(c)\ge 2021$

Source: Thailand Online MO 2021 P5

April 5, 2021
algebrapolynomialThailand online MO

Problem Statement

Prove that there exists a polynomial P(x)P(x) with real coefficients and degree greater than 1 such that both of the following conditions are true ⋅\cdot P(a)+P(b)+P(c)≥2021P(a)+P(b)+P(c)\ge 2021 holds for all nonnegative real numbers a,b,ca,b,c such that a+b+c=3a+b+c=3 ⋅\cdot There are infinitely many triples (a,b,c)(a,b,c) of nonnegative real numbers such that a+b+c=3a+b+c=3 and P(a)+P(b)+P(c)=2021P(a)+P(b)+P(c)= 2021
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