MathDB

Asymmetric inequality

Source: India IMOTC 2024 Day 1 Problem 2

May 31, 2024

inequalities

Problem Statement

Let

x_1, x_2 \dots, x_{2024}

be non-negative real numbers such that

x_1 \le x_2\cdots \le x_{2024}

, and

x_1^3 + x_2^3 + \dots + x_{2024}^3 = 2024

. Prove that

\sum_{1 \le i < j \le 2024} (-1)^{i+j} x_i^2 x_j \ge -1012.

Proposed by Shantanu Nene