Problem

Source: Indonesian National Mathematical Olympiad 2024, Problem 7

Tags: algebra, polynomial, inequalities, Indonesia, Indonesia MO



Suppose $P(x) = x^n + a_{n-1} x^{n-1} + \cdots + a_1x + a_0$ where $a_0, a_1, \ldots, a_{n-1}$ are reals for $n\geq 1$ (monic $n$th-degree polynomial with real coefficients). If the inequality \[ 3(P(x)+P(y)) \geq P(x+y) \]holds for all reals $x,y$, determine the minimum possible value of $P(2024)$.