Problem
ALG-B2-M04-P023 Indices as coefficients
#23
★★★★★ Level 5 of 5
Let \(x_1\le x_2\le\cdots\le x_n\) and \(x_1+x_2+\cdots+x_n=0\). Prove \[\sum_{i=1}^n i\,x_i\ge0.\]
Hint. The sequences \(1,2,\ldots,n\) and \(x_1,\ldots,x_n\) are similarly ordered.
By Chebyshev, \[\frac1n\sum_{i=1}^n i\,x_i\ge\left(\frac1n\sum_{i=1}^n i\right)\left(\frac1n\sum_{i=1}^n x_i\right)=0.\] Therefore \(\sum i x_i\ge0\).
The difficulty is not computation but recognizing the coefficients as a second ordered sequence.