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Self-bounded Polynomial Inequalities

Source: Indonesian National Mathematical Olympiad 2024, Problem 7

August 29, 2024
algebrapolynomialinequalitiesIndonesiaIndonesia MO

Problem Statement

Suppose P(x)=xn+an−1xn−1+⋯+a1x+a0P(x) = x^n + a_{n-1} x^{n-1} + \cdots + a_1x + a_0 where a0,a1,…,an−1a_0, a_1, \ldots, a_{n-1} are reals for n≥1n\geq 1 (monic nnth-degree polynomial with real coefficients). If the inequality 3(P(x)+P(y))≥P(x+y) 3(P(x)+P(y)) \geq P(x+y) holds for all reals x,yx,y, determine the minimum possible value of P(2024)P(2024).
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