MathDB

2

Part of 2024 India IMOTC

Problems(1)

Asymmetric inequality

Source: India IMOTC 2024 Day 1 Problem 2

5/31/2024
Let x1,x2…,x2024x_1, x_2 \dots, x_{2024} be non-negative real numbers such that x1≤x2⋯≤x2024x_1 \le x_2\cdots \le x_{2024}, and x13+x23+⋯+x20243=2024x_1^3 + x_2^3 + \dots + x_{2024}^3 = 2024. Prove that
∑1≤i<j≤2024(−1)i+jxi2xj≥−1012.\sum_{1 \le i < j \le 2024} (-1)^{i+j} x_i^2 x_j \ge -1012.
Proposed by Shantanu Nene
inequalities