Problem

Source: Nov 28, 2023

Tags: inequalities, algebra, CMO, China



Find the largest real number $c$ such that $$\sum_{i=1}^{n}\sum_{j=1}^{n}(n-|i-j|)x_ix_j \geq c\sum_{j=1}^{n}x^2_i$$for any positive integer $n $ and any real numbers $x_1,x_2,\dots,x_n.$